mlsanityData to 7 Oct 2026

Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

arXiv 1706.04702 ↗First posted 15 Jun 2017math.NA cs.LG cs.NAcited by 188

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Abstract (arXiv)

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.

Key facts. Cited by 188 papers in the map (data through 7 Oct 2026). First posted on arXiv 15 Jun 2017.

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Citing papers per year: 2021: 36; 2022: 33; 2023: 42; 2024: 27; 2025: 25; 2026: 25.

188 papers.

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  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  2. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  3. 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 98
  4. 2021On the approximation of functions by tanh neural networkscited by 81
  5. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  6. 2022Overview frequency principle/spectral bias in deep learningcited by 68
  7. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  8. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  9. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  10. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 55
  11. 2023Stochastic Optimal Control Matchingcited by 39
  12. 2024Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Datacited by 35
38 more
  1. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  2. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 34
  3. 2021Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantitiescited by 31
  4. 2023Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivativescited by 30
  5. 2024PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Lawscited by 30
  6. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  7. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  8. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  9. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  10. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  11. 2022Finite Expression Method for Solving High-Dimensional Partial Differential Equationscited by 21
  12. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  13. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  14. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  15. 2021Deep Learning for Mean Field Games and Mean Field Control with Applications to Financecited by 20
  16. 2021Reinforcement Learning with Dynamic Convex Risk Measurescited by 20
  17. 2022PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equationscited by 20
  18. 2023GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEscited by 20
  19. 2025Comprehensive Review of Neural Differential Equations for Time Series Analysiscited by 20
  20. 2025MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Controlcited by 20
  21. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  22. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  23. 2021Learning the temporal evolution of multivariate densities via normalizing flowscited by 18
  24. 2024FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Modelcited by 18
  25. 2021Solving high-dimensional parabolic PDEs using the tensor train formatcited by 17
  26. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  27. 2022Mean-field neural networks: learning mappings on Wasserstein spacecited by 15
  28. 2023A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networkscited by 15
  29. 2025Recurrent Neural Operators: Stable Long-Term PDE Predictioncited by 15
  30. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  31. 2021Data-driven approaches for predicting spread of infectious diseases through DINNs: Disease Informed Neural Networkscited by 14
  32. 2024Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Losscited by 14
  33. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  34. 2022Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependencecited by 13
  35. 2022BI-GreenNet: Learning Green's functions by boundary integral networkcited by 13
  36. 2022Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learningcited by 13
  37. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  38. 2023CS4ML: A general framework for active learning with arbitrary data based on Christoffel functionscited by 11

50 most cited of 188 All 188 (JSON)