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Deep splitting method for parabolic PDEs

arXiv 1907.03452 ↗First posted 8 Jul 2019math.NA cs.LG cs.NAcited by 53

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Abstract (arXiv)

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

Key facts. Cited by 53 papers in the map (data through 7 Oct 2026). First posted on arXiv 8 Jul 2019.

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

53 papers.

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  1. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  2. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  3. 2022Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?cited by 23
  4. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  5. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  6. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  7. 2024Stochastic Taylor Derivative Estimator: Efficient amortization for arbitrary differential operatorscited by 20
  8. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  9. 2021Solving high-dimensional parabolic PDEs using the tensor train formatcited by 17
  10. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  11. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  12. 2022Convergence of a robust deep FBSDE method for stochastic controlcited by 11
38 more
  1. 2022An energy-based deep splitting method for the nonlinear filtering problemcited by 7
  2. 2022Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?cited by 6
  3. 2024Solving stochastic partial differential equations using neural networks in the Wiener chaos expansioncited by 6
  4. 2025Deep Learning for Continuous-Time Stochastic Control with Jumpscited by 6
  5. 2021Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEscited by 5
  6. 2021Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematicscited by 5
  7. 2022An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equationscited by 5
  8. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  9. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  10. 2025Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programscited by 5
  11. 2023Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron modelscited 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. 2022A deep branching solver for fully nonlinear partial differential equationscited by 3
  15. 2022Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditionscited by 3
  16. 2022Computing non-equilibrium trajectories by a deep learning approachcited by 3
  17. 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
  18. 2025FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formulacited by 3
  19. 2025Integration Matters for Learning PDEs with Backward SDEscited by 3
  20. 2021Computation of conditional expectations with guaranteescited by 2
  21. 2022EgPDE-Net: Building Continuous Neural Networks for Time Series Prediction with Exogenous Variablescited by 2
  22. 2022A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equationscited by 2
  23. 2023Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Studycited by 2
  24. 2023A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equationscited by 2
  25. 2024A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splittingcited by 2
  26. 2023Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equationscited by 1
  27. 2024Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equationscited by 1
  28. 2025Nonlinear filtering based on density approximation and deep BSDE predictioncited by 1
  29. 2024A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equationscited by 0
  30. 2024Practical Aspects on Solving Differential Equations Using Deep Learning: A Primercited by 0
  31. 2025Error analysis for the deep Kolmogorov methodcited by 0
  32. 2025Robust Exploratory Stopping under Ambiguity in Reinforcement Learningcited by 0
  33. 2025Rough Path Signatures: Learning Neural RDEs for Portfolio Optimizationcited by 0
  34. 2025High-dimensional Bayesian filtering through deep density approximationcited by 0
  35. 2025Parametric Numerical Integration with (Differential) Machine Learningcited by 0
  36. 2026Deep Neural networks for solving high-dimensional parabolic partial differential equationscited by 0
  37. 2026Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domainscited by 0
  38. 2026INEUS: Iterative Neural Solver for High-Dimensional PIDEscited by 0

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