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Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations

arXiv 1804.07010 ↗First posted 19 Apr 2018stat.ML cs.LG cs.SYcited by 54

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Abstract (arXiv)

Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.

Key facts. Cited by 54 papers in the map (data through 7 Oct 2026). First posted on arXiv 19 Apr 2018.

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Citing papers per year: 2021: 11; 2022: 8; 2023: 12; 2024: 10; 2025: 7; 2026: 6.

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  1. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  2. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  3. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  4. 2021Physics-Informed Neural Nets for Control of Dynamical Systemscited by 31
  5. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  6. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  7. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  8. 2024Stochastic Taylor Derivative Estimator: Efficient amortization for arbitrary differential operatorscited by 20
  9. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  10. 2025Recurrent Neural Operators: Stable Long-Term PDE Predictioncited by 15
  11. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  12. 2021Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equationscited by 13
38 more
  1. 2022Transfer Learning with Physics-Informed Neural Networks for Efficient Simulation of Branched Flowscited by 12
  2. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  3. 2022Convergence of a robust deep FBSDE method for stochastic controlcited by 11
  4. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  5. 2022Quantum-Inspired Tensor Neural Networks for Partial Differential Equationscited by 9
  6. 2021Distributional Offline Continuous-Time Reinforcement Learning with Neural Physics-Informed PDEs (SciPhy RL for DOCTR-L)cited by 6
  7. 2021Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEscited by 5
  8. 2023Temporal Difference Learning for High-Dimensional PIDEs with Jumpscited by 5
  9. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  10. 2022Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusionscited by 4
  11. 2022Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networkscited by 4
  12. 2024A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equationscited by 4
  13. 2024FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equationscited by 4
  14. 2025A PyTorch-Enabled Tool for Synthetic Event Camera Data Generation and Algorithm Developmentcited by 4
  15. 2021Study of Drug Assimilation in Human System using Physics Informed Neural Networkscited by 3
  16. 2021Uncertainty Quantification in Neural Differential Equationscited by 3
  17. 2022DEQGAN: Learning the Loss Function for PINNs with Generative Adversarial Networkscited by 3
  18. 2023Simultaneous upper and lower bounds of American-style option prices with hedging via neural networkscited by 3
  19. 2025FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formulacited by 3
  20. 2025Integration Matters for Learning PDEs with Backward SDEscited by 3
  21. 2023A Neural RDE-based model for solving path-dependent PDEscited by 2
  22. 2024G-Adaptivity: optimised graph-based mesh relocation for finite element methodscited by 2
  23. 2021On Theory-training Neural Networks to Infer the Solution of Highly Coupled Differential Equationscited by 1
  24. 2021A novel control method for solving high-dimensional Hamiltonian systems through deep neural networkscited by 1
  25. 2022Quantum-Inspired Tensor Neural Networks for Option Pricingcited by 1
  26. 2023Neural Partial Differential Equations with Functional Convolutioncited by 1
  27. 2023Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equationscited by 1
  28. 2025A deep solver for backward stochastic Volterra integral equationscited by 1
  29. 2022Deep learning for gradient flows using the Brezis-Ekeland principlecited by 0
  30. 2023Deep Generative Modeling with Backward Stochastic Differential Equationscited by 0
  31. 2023Application of Tensor Neural Networks to Pricing Bermudan Swaptionscited by 0
  32. 2024Reservoir History Matching of the Norne field with generative exotic priors and a coupled Mixture of Experts -- Physics Informed Neural Operator Forward Modelcited by 0
  33. 2024Optimal Control of Agent-Based Dynamics under Deep Galerkin Feedback Lawscited by 0
  34. 2024A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equationscited by 0
  35. 2024Deep Feature Response Discriminative Calibrationcited by 0
  36. 2025Neural Network approximation power on homogeneous and heterogeneous reaction-diffusion equationscited by 0
  37. 2026Deep Neural networks for solving high-dimensional parabolic partial differential equationscited by 0
  38. 2026Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domainscited by 0

50 most cited of 54 All 54 (JSON)