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. Abdeljawad, Ahmed; Grohs, Philipp: Approximations with deep neural networks in Sobolev time-space (2022)
  2. Al-Aradi, Ali; Correia, Adolfo; Jardim, Gabriel; de Freitas Naiff, Danilo; Saporito, Yuri: Extensions of the deep Galerkin method (2022)
  3. Basir, Shamsulhaq; Senocak, Inanc: Physics and equality constrained artificial neural networks: application to forward and inverse problems with multi-fidelity data fusion (2022)
  4. Biswas, A.; Tian, J.; Ulusoy, S.: Error estimates for deep learning methods in fluid dynamics (2022)
  5. Cai, Zhiqiang; Chen, Jingshuang; Liu, Min: Least-squares ReLU neural network (LSNN) method for scalar nonlinear hyperbolic conservation law (2022)
  6. Cai, Zhiqiang; Chen, Jingshuang; Liu, Min: Self-adaptive deep neural network: numerical approximation to functions and PDEs (2022)
  7. Cheng, Lin; Wagner, Gregory J.: A representative volume element network (RVE-net) for accelerating RVE analysis, microscale material identification, and defect characterization (2022)
  8. Chen, Yuyan; Dong, Bin; Xu, Jinchao: Meta-mgnet: meta multigrid networks for solving parameterized partial differential equations (2022)
  9. Chiu, Pao-Hsiung; Wong, Jian Cheng; Ooi, Chinchun; Dao, My Ha; Ong, Yew-Soon: CAN-PINN: a fast physics-informed neural network based on coupled-automatic-numerical differentiation method (2022)
  10. Cui, Tao; Wang, Ziming; Xiang, Xueshuang: An efficient neural network method with plane wave activation functions for solving Helmholtz equation (2022)
  11. Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas: Optimization with learning-informed differential equation constraints and its applications (2022)
  12. Duan, Chenguang; Jiao, Yuling; Lai, Yanming; Li, Dingwei; Lu, Xiliang; Yang, Jerry Zhijian: Convergence rate analysis for deep Ritz method (2022)
  13. Elbrächter, Dennis; Grohs, Philipp; Jentzen, Arnulf; Schwab, Christoph: DNN expression rate analysis of high-dimensional PDEs: application to option pricing (2022)
  14. E, Weinan; Han, Jiequn; Jentzen, Arnulf: Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning (2022)
  15. Fatone, L.; Funaro, D.; Manzini, G.: A decision-making machine learning approach in Hermite spectral approximations of partial differential equations (2022)
  16. Gao, Yihang; Ng, Michael K.: Wasserstein generative adversarial uncertainty quantification in physics-informed neural networks (2022)
  17. Germain, Maximilien; Pham, Huyên; Warin, Xavier: Approximation error analysis of some deep backward schemes for nonlinear PDEs (2022)
  18. Gim, Daeyung; Park, Hyungbin: A deep learning algorithm for optimal investment strategies under Merton’s framework (2022)
  19. Grohs, Philipp; Jentzen, Arnulf; Salimova, Diyora: Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms (2022)
  20. Guo, Hailong; Yang, Xu: Deep unfitted Nitsche method for elliptic interface problems (2022)

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