利用深度神经网络解决偏微分问题论文

论文有如下，被分为两类。摘自Solving Electrical Impedance Tomography with Deep Learning论文citations。

1. [32] Y. Khoo, J. Lu, and L. Ying. Solving parametric PDE problems with artificial neural networks. arXiv preprint arXiv:1707.03351, 2017.
2. [8] J. Berg and K. Nystr¨om. A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing, 317:28–41, 2018.
3. [26] J. Han, A. Jentzen, and W. E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.
4. [22] Y. Fan, L. Lin, L. Ying, and L. Zepeda-Nu´n˜ez. A multiscale neural network based on hierarchical matrices. arXiv preprint arXiv:1807.01883, 2018.
5. [21] Y. Fan, J. Feliu-Fab`a, L. Lin, L. Ying, and L. Zepeda-Nu´n˜ez. A multiscale neural network based on hierarchical nested bases. Research in the Mathematical Sciences, 6(2):21, 2019.
6. [5] M. Araya-Polo, J. Jennings, A. Adler, and T. Dahlke. Deep-learning tomography. The Leading Edge, 37(1):58–66, 2018.
7. [42] M. Raissi and G. E. Karniadakis. Hidden physics models: Machine learning of nonlinear partial differ- ential equations. Journal of Computational Physics, 357:125 – 141, 2018.
8. [20] Y. Fan, C. O. Bohorquez, and L. Ying. BCR-Net: a neural network based on the nonstandard wavelet form. Journal of Computational Physics, 384:1–15, 2019.
9. [34] Y. Khoo and L. Ying. SwitchNet: a neural network model for forward and inverse scattering problems.
   arXiv preprint arXiv:1810.09675, 2018.

用DNNs表示高维PDEs的解

1. [43] K. Rudd and S. Ferrari. A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks. Neurocomputing, 155:277–285, 2015.
2. [14] G. Carleo and M. Troyer. Solving the quantum many-body problem with artificial neural networks.
   Science, 355(6325):602–606, 2017.
3. [26] J. Han, A. Jentzen, and W. E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.
4. [33] Y. Khoo, J. Lu, and L. Ying. Solving for high-dimensional committor functions using artificial neural networks. Research in the Mathematical Sciences, 6(1):1, 2019.
5. [19] W. E and B. Yu. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1–12, 2018.

解决带参数的PDEs，用DNNs表示从高维参数到解的映射

[21][22][20][34][32]

1. [40] Z. Long, Y. Lu, X. Ma, and B. Dong. PDE-net: Learning PDEs from data. In J. Dy and A. Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 3208–3216, Stockholmsmssan, Stockholm Sweden, 10–15 Jul 2018. PMLR.
2. [27] J. Han, L. Zhang, R. Car, et al. Deep potential: A general representation of a many-body potential energy surface. arXiv preprint arXiv:1707.01478, 2017.
3. [38] Y. Li, J. Lu, and A. Mao. Variational training of neural network approximations of solution maps for physical models. arXiv preprint arXiv:1905.02789, 2019.
4. [6] L. Bar and N. Sochen. Unsupervised deep learning algorithm for PDE-based forward and inverse problems. arXiv preprint arXiv:1904.05417, 2019.