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Solving high-dimensional partial differential equations using deep learning

arXiv 1707.02568 ↗First posted 9 Jul 2017math.NA cs.LG cs.NAcited by 549

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Abstract (arXiv)

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships.

Key facts. Cited by 549 papers in the map (data through 7 Oct 2026). First posted on arXiv 9 Jul 2017.

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Citing papers per year: 2021: 90; 2022: 127; 2023: 95; 2024: 84; 2025: 85; 2026: 68.

549 papers.

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  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  2. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  3. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  4. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  5. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  6. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  7. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  8. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  9. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
  10. 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 98
  11. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  12. 2022Clifford Neural Layers for PDE Modelingcited by 91
38 more
  1. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
  2. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  3. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  4. 2023Solving High-Dimensional PDEs with Latent Spectral Modelscited by 84
  5. 2021On the approximation of functions by tanh neural networkscited by 81
  6. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  7. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  8. 2022Unified Implicit Neural Stylizationcited by 67
  9. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  10. 2021Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approachcited by 62
  11. 2023PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networkscited by 62
  12. 2021Encoding physics to learn reaction-diffusion processescited by 61
  13. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 55
  14. 2021The Power of Contrast for Feature Learning: A Theoretical Analysiscited by 53
  15. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 50
  16. 2021One-Shot Transfer Learning of Physics-Informed Neural Networkscited by 49
  17. 2021Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progressioncited by 48
  18. 2022Deep Generalized Schr"odinger Bridgecited by 47
  19. 2022Probability flow solution of the Fokker-Planck equationcited by 46
  20. 2022Signal Processing for Implicit Neural Representationscited by 46
  21. 2022Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamicscited by 44
  22. 2021Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networkscited by 43
  23. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 43
  24. 2021Solving PDE-constrained Control Problems Using Operator Learningcited by 41
  25. 2021Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modelingcited by 40
  26. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  27. 2023Stochastic Optimal Control Matchingcited by 39
  28. 2021Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusioncited by 38
  29. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  30. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  31. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  32. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  33. 2024Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learningcited by 32
  34. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  35. 2024PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Lawscited by 30
  36. 2021Latent Space Data Assimilation by using Deep Learningcited by 29
  37. 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activationscited by 29
  38. 2021Neural Delay Differential Equationscited by 26

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