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Label Propagation Training Schemes for Physics-Informed Neural Networks and Gaussian Processes
Published 8 Apr 2024 in cs.LG | (2404.05817v1)
Abstract: This paper proposes a semi-supervised methodology for training physics-informed machine learning methods. This includes self-training of physics-informed neural networks and physics-informed Gaussian processes in isolation, and the integration of the two via co-training. We demonstrate via extensive numerical experiments how these methods can ameliorate the issue of propagating information forward in time, which is a common failure mode of physics-informed machine learning.
- (1998). Combining labeled and unlabeled data with co-training. In Proceedings of the eleventh annual conference on computational learning theory (pp. 92–100). \NAT@swatrue Brefeld et al. (2006) Brefeld, U., Gärtner, T., Scheffer, T., and Wrobel, S. (2006). Efficient co-regularised least squares regression. In Proceedings of the 23rd international conference on machine learning (pp. 137–144). \NAT@swatrue Cai et al. (2021) Cai, S., Wang, Z., Lu, L., Zaki, T. A., and Karniadakis, G. E. (2021). DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296. \NAT@swatrue Chen et al. (2021) Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Brefeld, U., Gärtner, T., Scheffer, T., and Wrobel, S. (2006). Efficient co-regularised least squares regression. In Proceedings of the 23rd international conference on machine learning (pp. 137–144). \NAT@swatrue Cai et al. (2021) Cai, S., Wang, Z., Lu, L., Zaki, T. A., and Karniadakis, G. E. (2021). DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296. \NAT@swatrue Chen et al. (2021) Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Cai, S., Wang, Z., Lu, L., Zaki, T. A., and Karniadakis, G. E. (2021). DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296. \NAT@swatrue Chen et al. (2021) Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2006). Efficient co-regularised least squares regression. In Proceedings of the 23rd international conference on machine learning (pp. 137–144). \NAT@swatrue Cai et al. (2021) Cai, S., Wang, Z., Lu, L., Zaki, T. A., and Karniadakis, G. E. (2021). DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296. \NAT@swatrue Chen et al. (2021) Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Cai, S., Wang, Z., Lu, L., Zaki, T. A., and Karniadakis, G. E. (2021). DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296. \NAT@swatrue Chen et al. (2021) Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2021). DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296. \NAT@swatrue Chen et al. (2021) Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Chen, Y., Hosseini, B., Owhadi, H., and Stuart, A. M. (2021). Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447, 110668. \NAT@swatrue Coutinho et al. (2023) Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Coutinho, E. J. R., Dall’Aqua, M., McClenny, L., Zhong, M., Braga-Neto, U., and Gildin, E. (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2023). Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265. Retrieved from https://www.sciencedirect.com/science/article/pii/S0021999123003601 DOI: https://doi.org/10.1016/j.jcp.2023.112265 \NAT@swatrue Haitsiukevich and Ilin (2022) Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Haitsiukevich, K., and Ilin, A. (2022). Improved training of physics-informed neural networks with model ensembles. arXiv preprint arXiv:2204.05108. \NAT@swatrue Jin et al. (2021) Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
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Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951. \NAT@swatrue Krishnapriyan et al. (2021) Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
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A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34, 26548–26560. \NAT@swatrue Kunselman et al. (2020) Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. 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(2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
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(2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. 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Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Kunselman, C., Attari, V., McClenny, L., Braga-Neto, U., and Arroyave, R. (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2020). Semi-supervised learning approaches to class assignment in ambiguous microstructures. Acta Materialia, 188, 49–62. \NAT@swatrue Liu and Wang (2019) Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2019). Multi-fidelity physics-constrained neural network and its application in materials modeling. Journal of Mechanical Design, 141(12). \NAT@swatrue Liu and Wang (2021) Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Liu, D., and Wang, Y. (2021). A dual-dimer method for training physics-constrained neural networks with minimax architecture. Neural Networks, 136, 112–125. \NAT@swatrue McClenny and Braga-Neto (2020) McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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(2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. 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Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. McClenny, L., and Braga-Neto, U. (2020). Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544. \NAT@swatrue Rad et al. (2020) Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). 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Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rad, M. T., Viardin, A., Schmitz, G., and Apel, M. (2020). Theory-training deep neural networks for an alloy solidification benchmark problem. Computational Materials Science, 180, 109687. \NAT@swatrue Raissi and Karniadakis (2018) Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. 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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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). 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In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). 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Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. 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Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. 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(2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
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On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357, 125–141. \NAT@swatrue Raissi et al. (2019) Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. 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(2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). 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. \NAT@swatrue Rosenberg et al. (2005) Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Rosenberg, C., Hebert, M., and Schneiderman, H. (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). 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(2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. 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Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. 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Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. 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Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2005). Semi-supervised self-training of object detection models. \NAT@swatrue Särkkä (2013) Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- Särkkä, S. (2013). Bayesian filtering and smoothing. Cambridge University Press. \NAT@swatrue Shin et al. (2020) Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Shin, Y., Darbon, J., and Karniadakis, G. E. (2020). On the convergence and generalization of physics informed neural networks. arXiv preprint arXiv:2004.01806. \NAT@swatrue Wang et al. (2022) Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. 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(2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Sankaran, S., and Perdikaris, P. (2022). Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. \NAT@swatrue Wang, Teng, and Perdikaris (2020) Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. 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Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). 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Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Teng, Y., and Perdikaris, P. (2020). Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536. \NAT@swatrue Wang, Yu, and Perdikaris (2020) Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. 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Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). 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In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wang, S., Yu, X., and Perdikaris, P. (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2020). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527. \NAT@swatrue Wight and Zhao (2020) Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Wight, C. L., and Zhao, J. (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2020). Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542. \NAT@swatrue Yang et al. (2019) Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yang, X., Barajas-Solano, D., Tartakovsky, G., and Tartakovsky, A. M. (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2019). Physics-informed cokriging: A Gaussian-process-regression-based multifidelity method for data-model convergence. Journal of Computational Physics, 395, 410–431. \NAT@swatrue Yarowsky (1995) Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- Yarowsky, D. (1995). Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics (pp. 189–196). \NAT@swatrue Zhou et al. (2005) Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhou, Z.-H., Li, M., et al. (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2005). Semi-supervised regression with co-training. In Ijcai (Vol. 5, pp. 908–913). \NAT@swatrue Q. Zhu et al. (2021) Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, Q., Liu, Z., and Yan, J. (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- (2021). Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Computational Mechanics, 67(2), 619–635. \NAT@swatrue X. J. Zhu (2005) Zhu, X. J. (2005). Semi-supervised learning literature survey. Zhu, X. J. (2005). Semi-supervised learning literature survey.
- Zhu, X. J. (2005). Semi-supervised learning literature survey.
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