mlsanityData to 7 Oct 2026

An overview on deep learning-based approximation methods for partial differential equations

arXiv 2012.12348 ↗First posted 22 Dec 2020math.NA cs.LG cs.NAcited by 65

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Abstract (arXiv)

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

Key facts. Cited by 65 papers in the map (data through 7 Oct 2026). First posted on arXiv 22 Dec 2020.

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Citing papers per year: 2021: 6; 2022: 13; 2023: 17; 2024: 12; 2025: 11; 2026: 6.

65 papers.

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  2. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 43
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  9. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  10. 2022Convergence of a robust deep FBSDE method for stochastic controlcited by 11
  11. 2022DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splittingcited by 11
  12. 2023Achieving High Accuracy with PINNs via Energy Natural Gradientscited by 10
38 more
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  6. 2021Distributional Offline Continuous-Time Reinforcement Learning with Neural Physics-Informed PDEs (SciPhy RL for DOCTR-L)cited by 6
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  11. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  12. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  13. 2022Temporal Difference Learning with Continuous Time and State in the Stochastic Settingcited by 4
  14. 2022Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditionscited by 3
  15. 2022From Monte Carlo to neural networks approximations of boundary value problemscited by 3
  16. 2022r-Adaptive Deep Learning Method for Solving Partial Differential Equationscited by 3
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  18. 2023Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONetscited by 3
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  21. 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
  22. 2025FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formulacited by 3
  23. 2025Universal Fourier Neural Operators for periodic homogenization problems in linear elasticitycited by 3
  24. 2023Drift Control of High-Dimensional RBM: A Computational Method Based on Neural Networkscited by 2
  25. 2023Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Studycited by 2
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  27. 2024META-ANOVA: Screening interactions for interpretable machine learningcited by 2
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  29. 2025Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equationcited by 2
  30. 2026Shallow ReLUs Networks in Lp-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalizationcited by 2
  31. 2023Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equationscited by 1
  32. 2023Quantile and moment neural networks for learning functionals of distributionscited by 1
  33. 2023Probing optimisation in physics-informed neural networkscited by 1
  34. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  35. 2023Dynamic Scheduling of a Multiclass Queue in the Halfin-Whitt Regime: A Computational Approach for High-Dimensional Problemscited by 1
  36. 2024A PNP ion channel deep learning solver with local neural network and finite element input datacited by 1
  37. 2024Gradient Flow Based Phase-Field Modeling Using Separable Neural Networkscited by 1
  38. 2025An Iterative Deep Ritz Method for Monotone Elliptic Problemscited by 1

50 most cited of 65 All 65 (JSON)