2000 character limit reached
Neural Fractional Differential Equations (2403.02737v2)
Published 5 Mar 2024 in cs.LG, cs.CE, cs.NA, and math.NA
Abstract: Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise representation of processes characterised by non-local and memory-dependent behaviours. This property is useful in systems where variables do not respond to changes instantaneously, but instead exhibit a strong memory of past interactions. Having this in mind, and drawing inspiration from Neural Ordinary Differential Equations (Neural ODEs), we propose the Neural FDE, a novel deep neural network architecture that adjusts a FDE to the dynamics of data. This work provides a comprehensive overview of the numerical method employed in Neural FDEs and the Neural FDE architecture. The numerical outcomes suggest that, despite being more computationally demanding, the Neural FDE may outperform the Neural ODE in modelling systems with memory or dependencies on past states, and it can effectively be applied to learn more intricate dynamical systems.
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(2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Herrmann, R.: Fractional Calculus: An Introduction for Physicists, 2. ed edn. World Scientific, New Jersey, NJ (2014). https://doi.org/10.1142/9789814551083 Coelho et al. [2024] Coelho, C., Costa, M.F.P., Ferrás, L.L.: Tracing footprints: Neural networks meet non-integer order differential equations for modelling systems with memory. In: Tiny Papers @ ICLR (2024). https://openreview.net/forum?id=8518dcW4hc He et al. [2016] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016) Haber et al. [2018] Haber, E., Ruthotto, L., Holtham, E., Jun, S.-H.: Learning across scales—multiscale methods for convolution neural networks. Proceedings of the AAAI Conference on Artificial Intelligence 32(1) (2018) https://doi.org/10.1609/aaai.v32i1.11680 Haber and Ruthotto [2017] Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Coelho, C., Costa, M.F.P., Ferrás, L.L.: Tracing footprints: Neural networks meet non-integer order differential equations for modelling systems with memory. In: Tiny Papers @ ICLR (2024). https://openreview.net/forum?id=8518dcW4hc He et al. [2016] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016) Haber et al. [2018] Haber, E., Ruthotto, L., Holtham, E., Jun, S.-H.: Learning across scales—multiscale methods for convolution neural networks. Proceedings of the AAAI Conference on Artificial Intelligence 32(1) (2018) https://doi.org/10.1609/aaai.v32i1.11680 Haber and Ruthotto [2017] Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016) Haber et al. [2018] Haber, E., Ruthotto, L., Holtham, E., Jun, S.-H.: Learning across scales—multiscale methods for convolution neural networks. Proceedings of the AAAI Conference on Artificial Intelligence 32(1) (2018) https://doi.org/10.1609/aaai.v32i1.11680 Haber and Ruthotto [2017] Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Haber, E., Ruthotto, L., Holtham, E., Jun, S.-H.: Learning across scales—multiscale methods for convolution neural networks. Proceedings of the AAAI Conference on Artificial Intelligence 32(1) (2018) https://doi.org/10.1609/aaai.v32i1.11680 Haber and Ruthotto [2017] Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? 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Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. 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In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. 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Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Haber, E., Ruthotto, L., Holtham, E., Jun, S.-H.: Learning across scales—multiscale methods for convolution neural networks. Proceedings of the AAAI Conference on Artificial Intelligence 32(1) (2018) https://doi.org/10.1609/aaai.v32i1.11680 Haber and Ruthotto [2017] Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. 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Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Haber, E., Ruthotto, L.: Stable architectures for deep neural networks. Inverse Problems 34(1), 014004 (2017) https://doi.org/10.1088/1361-6420/aa9a90 E [2017] E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 E, W.: A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5(1), 1–11 (2017) https://doi.org/10.1007/s40304-017-0103-z Lu et al. [2018] Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lu, Y., Zhong, A., Li, Q., Dong, B.: Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 3276–3285. PMLR, ??? (2018). https://proceedings.mlr.press/v80/lu18d.html Ruthotto and Haber [2019] Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. 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(2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Ruthotto, L., Haber, E.: Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62(3), 352–364 (2019) https://doi.org/10.1007/s10851-019-00903-1 Chen [2018] Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Chen, R.T.Q.: torchdiffeq (2018). https://github.com/rtqichen/torchdiffeq Diethelm [2010] Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. 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Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. 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Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
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Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, ??? (2010). https://doi.org/10.1007/978-3-642-14574-2 . http://dx.doi.org/10.1007/978-3-642-14574-2 Liu et al. [2018] Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. International Journal of Computer Mathematics 95(6-7), 1151–1169 (2018) https://doi.org/10.1080/00207160.2017.1381691 Lundstrom et al. [2008] Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
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- Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11(11), 1335–1342 (2008) https://doi.org/10.1038/nn.2212 Diethelm and Ford [2002] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229–248 (2002) https://doi.org/10.1006/jmaa.2000.7194 Barros et al. [2021] Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Barros, L.C.d., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.d., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics 40(3) (2021) https://doi.org/10.1007/s40314-021-01456-z Diethelm et al. [2002] Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Diethelm, K., Ford, N.J., Freed, A.D.: A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics 29(1), 3–22 (2002) Kaslik and Sivasundaram [2012] Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030 Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030
- Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks 32, 245–256 (2012) https://doi.org/10.1016/j.neunet.2012.02.030