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DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networks
Published 6 May 2023 in cs.CE and cs.LG | (2305.04107v2)
Abstract: We propose a direct mesh-free method for performing topology optimization by integrating a density field approximation neural network with a displacement field approximation neural network. We show that this direct integration approach can give comparable results to conventional topology optimization techniques, with an added advantage of enabling seamless integration with post-processing software, and a potential of topology optimization with objectives where meshing and Finite Element Analysis (FEA) may be expensive or not suitable. Our approach (DMF-TONN) takes in as inputs the boundary conditions and domain coordinates and finds the optimum density field for minimizing the loss function of compliance and volume fraction constraint violation. The mesh-free nature is enabled by a physics-informed displacement field approximation neural network to solve the linear elasticity partial differential equation and replace the FEA conventionally used for calculating the compliance. We show that using a suitable Fourier Features neural network architecture and hyperparameters, the density field approximation neural network can learn the weights to represent the optimal density field for the given domain and boundary conditions, by directly backpropagating the loss gradient through the displacement field approximation neural network, and unlike prior work there is no requirement of a sensitivity filter, optimality criterion method, or a separate training of density network in each topology optimization iteration.
- Zhou, M., Rozvany, G.: The coc algorithm, part ii: Topological, geometrical and generalized shape optimization. Computer methods in applied mechanics and engineering 89(1-3), 309–336 (1991) Chandrasekhar and Suresh [2021] Chandrasekhar, A., Suresh, K.: Tounn: Topology optimization using neural networks. Structural and Multidisciplinary Optimization 63 (2021) https://doi.org/10.1007/s00158-020-02748-4 Samaniego et al. [2020] Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T.: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362, 112790 (2020) Zehnder et al. [2021] Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Tounn: Topology optimization using neural networks. Structural and Multidisciplinary Optimization 63 (2021) https://doi.org/10.1007/s00158-020-02748-4 Samaniego et al. [2020] Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T.: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362, 112790 (2020) Zehnder et al. [2021] Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T.: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362, 112790 (2020) Zehnder et al. [2021] Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. 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[2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. 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Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. 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SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Chandrasekhar, A., Suresh, K.: Tounn: Topology optimization using neural networks. Structural and Multidisciplinary Optimization 63 (2021) https://doi.org/10.1007/s00158-020-02748-4 Samaniego et al. [2020] Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T.: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362, 112790 (2020) Zehnder et al. [2021] Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T.: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362, 112790 (2020) Zehnder et al. [2021] Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zehnder, J., Li, Y., Coros, S., Thomaszewski, B.: Ntopo: Mesh-free topology optimization using implicit neural representations. Advances in Neural Information Processing Systems 34, 10368–10381 (2021) Bendsøe and Kikuchi [1988] Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. 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Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. 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[2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. 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Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering 71(2), 197–224 (1988) Allaire et al. [2002] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. 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Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. 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[2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. 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SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. 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[2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002) Wang et al. [2003] Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. 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[2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192(1-2), 227–246 (2003) Woldseth et al. [2022] Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. 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Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. 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SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. 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[2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. 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SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. 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[2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. 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[2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Woldseth, R.V., Aage, N., Bærentzen, J.A., Sigmund, O.: On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization 65(10), 294 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) Nguyen-Thanh et al. [2020] Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids 80, 103874 (2020) Sitzmann et al. [2020] Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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[2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems 33, 7462–7473 (2020) Tancik et al. [2020] Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. 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[2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems 33, 7537–7547 (2020) Banga et al. [2018] Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Banga, S., Gehani, H., Bhilare, S., Patel, S., Kara, L.: 3d topology optimization using convolutional neural networks. arXiv preprint arXiv:1808.07440 (2018) Yu et al. [2019] Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Yu, Y., Hur, T., Jung, J., Jang, I.G.: Deep learning for determining a near-optimal topological design without any iteration. Structural and Multidisciplinary Optimization 59(3), 787–799 (2019) Nakamura and Suzuki [2020] Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nakamura, K., Suzuki, Y.: Deep learning-based topological optimization for representing a user-specified design area. arXiv preprint arXiv:2004.05461 (2020) Nie et al. [2021] Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Nie, Z., Lin, T., Jiang, H., Kara, L.B.: Topologygan: Topology optimization using generative adversarial networks based on physical fields over the initial domain. Journal of Mechanical Design 143(3) (2021) Behzadi and Ilieş [2021] Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Behzadi, M.M., Ilieş, H.T.: Real-time topology optimization in 3d via deep transfer learning. Computer-Aided Design 135, 103014 (2021) Mazé and Ahmed [2022] Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Mazé, F., Ahmed, F.: Diffusion models beat gans on topology optimization (2022) White et al. [2019] White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- White, D.A., Arrighi, W.J., Kudo, J., Watts, S.E.: Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering 346, 1118–1135 (2019) Chandrasekhar and Suresh [2021a] Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Chandrasekhar, A., Suresh, K.: Length scale control in topology optimization using fourier enhanced neural networks. CoRR abs/2109.01861 (2021) 2109.01861 Chandrasekhar and Suresh [2021b] Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. 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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chandrasekhar, A., Suresh, K.: Multi-material topology optimization using neural networks. CAD Computer Aided Design 136 (2021) https://doi.org/10.1016/j.cad.2021.103017 Deng and To [2020] Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Deng, H., To, A.C.: Topology optimization based on deep representation learning (drl) for compliance and stress-constrained design. Computational Mechanics 66(2), 449–469 (2020) Zhang et al. [2023] Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Zhang, Z., Yao, W., Li, Y., Zhou, W., Chen, X.: Topology optimization via implicit neural representations. Computer Methods in Applied Mechanics and Engineering 411, 116052 (2023) Hoyer et al. [2019] Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Hoyer, S., Sohl-Dickstein, J., Greydanus, S.: Neural reparameterization improves structural optimization. arXiv preprint arXiv:1909.04240 (2019) Chen et al. [2023a] Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Whitefoot, K.S., Kara, L.B.: Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks. Journal of Mechanical Design (2023) Chen et al. [2023b] Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. 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Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Chen, H., Joglekar, A., Kara, L.B.: Topology optimization using neural networks with conditioning field initialization for improved efficiency. In: ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering and Conference (2023). American Society of Mechanical Engineers He et al. [2022] He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., Jasiuk, I.: Deep energy method in topology optimization applications. Acta Mechanica, 1–15 (2022) Jeong et al. [2023] Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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[2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Jeong, H., Bai, J., Batuwatta-Gamage, C., Rathnayaka, C., Zhou, Y., Gu, Y.: A physics-informed neural network-based topology optimization (pinnto) framework for structural optimization. Engineering Structures 278, 115484 (2023) Lu et al. [2021] Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing 43(6), 1105–1132 (2021) Mai et al. [2023] Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Mai, H.T., Mai, D.D., Kang, J., Lee, J., Lee, J.: Physics-informed neural energy-force network: a unified solver-free numerical simulation for structural optimization. Engineering with Computers, 1–24 (2023) Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014) Abadi et al. [2016] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016) Andreassen et al. [2011] Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–16 (2011) Liu and Tovar [2014] Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014) Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
- Liu, K., Tovar, A.: An efficient 3d topology optimization code written in matlab. Structural and Multidisciplinary Optimization 50, 1175–1196 (2014)
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