mlsanityData to 7 Oct 2026

Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

arXiv 2008.13333 ↗First posted 31 Aug 2020math.NA cs.NA math.OCcited by 77

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Abstract (arXiv)

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

Key facts. Cited by 77 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 Aug 2020.

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Citing papers per year: 2021: 9; 2022: 16; 2023: 21; 2024: 11; 2025: 11; 2026: 9.

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  3. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
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  5. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
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  9. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
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  11. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  12. 2022SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decompositioncited by 13
38 more
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  7. 2021Distributional Offline Continuous-Time Reinforcement Learning with Neural Physics-Informed PDEs (SciPhy RL for DOCTR-L)cited by 6
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  9. 2022Towards Size-Independent Generalization Bounds for Deep Operator Netscited by 6
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  11. 2022A Curriculum-Training-Based Strategy for Distributing Collocation Points during Physics-Informed Neural Network Trainingcited by 5
  12. 2023Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEscited by 5
  13. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  14. 2023Residual Multi-Fidelity Neural Network Computingcited by 5
  15. 2024A Matrix Product State Model for Simultaneous Classification and Generationcited by 5
  16. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  17. 2022Bayesian Deep Learning with Multilevel Trace-class Neural Networkscited by 4
  18. 2022Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Methodcited by 4
  19. 2022BASiS: Batch Aligned Spectral Embedding Spacecited by 4
  20. 2023Q-Flow: Generative Modeling for Differential Equations of Open Quantum Dynamics with Normalizing Flowscited by 4
  21. 2024Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equationcited by 4
  22. 2022Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditionscited by 3
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  25. 2024Pontryagin Neural Operator for Solving Parametric General-Sum Differential Gamescited by 3
  26. 2024TENG: Time-Evolving Natural Gradient for Solving PDEs With Deep Neural Nets Toward Machine Precisioncited by 3
  27. 2024Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in Lp-sensecited by 3
  28. 2025FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formulacited by 3
  29. 2022Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditionscited by 2
  30. 2022A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equationscited by 2
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