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Neural networks-based backward scheme for fully nonlinear PDEs

arXiv 1908.00412 ↗First posted 31 Jul 2019math.OC cs.NE math.APcited by 23

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Abstract (arXiv)

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient at previous step. This methodology extends to the fully nonlinear case the approach recently proposed in citeHPW19 for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient, Monge-Amp`ere equation and Hamilton-Jacobi-Bellman equation in portfolio optimization.

Key facts. Cited by 23 papers in the map (data through 14 Oct 2026). First posted on arXiv 31 Jul 2019.

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Citing papers per year: 2021: 4; 2022: 7; 2023: 2; 2024: 3; 2025: 6; 2026: 1.

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  1. 2022Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?cited by 23
  2. 2022Neural Q-learning for solving PDEscited by 8
  3. 2024Deep Backward and Galerkin Methods for the Finite State Master Equationcited by 6
  4. 2021Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEscited by 5
  5. 2021Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematicscited by 5
  6. 2022An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equationscited by 5
  7. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  8. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  9. 2022Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusionscited by 4
  10. 2024FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equationscited by 4
  11. 2025One model to solve them all: 2BSDE families via neural operatorscited by 4
  12. 2022A deep branching solver for fully nonlinear partial differential equationscited by 3
11 more
  1. 2025Integration Matters for Learning PDEs with Backward SDEscited by 3
  2. 2022A deep learning method for solving stochastic optimal control problems driven by fully-coupled FBSDEscited by 2
  3. 2022A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equationscited by 2
  4. 2021Cell-average based neural network method for hyperbolic and parabolic partial differential equationscited by 1
  5. 2021A novel control method for solving high-dimensional Hamiltonian systems through deep neural networkscited by 1
  6. 2023Dynamic Scheduling of a Multiclass Queue in the Halfin-Whitt Regime: A Computational Approach for High-Dimensional Problemscited by 1
  7. 2026Deep Neural networks for solving high-dimensional parabolic partial differential equationscited by 1
  8. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  9. 2025Consensus Based Stochastic Controlcited by 0
  10. 2025A hierarchical resource-efficient deep policy gradient method for continuous-time optimal control problemscited by 0
  11. 2025Rough Path Signatures: Learning Neural RDEs for Portfolio Optimizationcited by 0