DGM: a deep learning algorithm for solving partial differential equations. High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to 200 dimensions. The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE and Burgers’ equation. The deep learning algorithm approximates the general solution to the Burgers’ equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a “Deep Galerkin method (DGM)” since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs.

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  1. Li, Yixin; Hu, Xianliang: Artificial neural network approximations of Cauchy inverse problem for linear PDEs (2022)
  2. Ainsworth, Mark; Dong, Justin: Galerkin neural networks: a framework for approximating variational equations with error control (2021)
  3. Amini Niaki, Sina; Haghighat, Ehsan; Campbell, Trevor; Poursartip, Anoush; Vaziri, Reza: Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture (2021)
  4. Bar, Leah; Sochen, Nir: Strong solutions for PDE-based tomography by unsupervised learning (2021)
  5. Beck, Christian; Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Neufeld, Ariel: Deep splitting method for parabolic PDEs (2021)
  6. Calabrò, Francesco; Fabiani, Gianluca; Siettos, Constantinos: Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients (2021)
  7. Carmona, René; Laurière, Mathieu: Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games. I: The ergodic case (2021)
  8. Chen, Xinyang; Yang, Gengchao; Yao, Qinghe; Nie, Zisen; Jiang, Zichao: A compressed lattice Boltzmann method based on ConvLSTM and resnet (2021)
  9. Choi, So Eun; Jang, Hyun Jin; Lee, Kyungsub; Zheng, Harry: Optimal market-making strategies under synchronised order arrivals with deep neural networks (2021)
  10. Cho, Sung Woong; Hwang, Hyung Ju; Son, Hwijae: Traveling wave solutions of partial differential equations via neural networks (2021)
  11. Dolgov, Sergey; Kalise, Dante; Kunisch, Karl K.: Tensor decomposition methods for high-dimensional Hamilton-Jacobi-Bellman equations (2021)
  12. Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold; Kutyniok, Gitta: Numerical solution of the parametric diffusion equation by deep neural networks (2021)
  13. Grüne, Lars: Computing Lyapunov functions using deep neural networks (2021)
  14. Ito, Kazufumi; Reisinger, Christoph; Zhang, Yufei: A neural network-based policy iteration algorithm with global (H^2)-superlinear convergence for stochastic games on domains (2021)
  15. Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M.: \textithp-VPINNs: variational physics-informed neural networks with domain decomposition (2021)
  16. Laakmann, Fabian; Petersen, Philipp: Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs (2021)
  17. Lee, Jae Yong; Jang, Jin Woo; Hwang, Hyung Ju: The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the deep neural network approach (2021)
  18. Lee, Junbeom; Yu, Xiang; Zhou, Chao: Lifetime ruin under high-water mark fees and drift uncertainty (2021)
  19. Lefebvre, William; Miller, Enzo: Linear-quadratic stochastic delayed control and deep learning resolution (2021)
  20. Liao, Yulei; Ming, Pingbing: Deep Nitsche method: deep Ritz method with essential boundary conditions (2021)

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