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Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations

arXiv 1709.05963 ↗First posted 18 Sep 2017math.NA cs.LG cs.NAcited by 65

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Abstract (arXiv)

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using rm Tsmall ENSORFsmall LOW in rm Psmall YTHON illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black-Scholes-Barenblatt equation, a 100-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a 100 -dimensional G -Brownian motion.

Key facts. Cited by 65 papers in the map (data through 14 Oct 2026). First posted on arXiv 18 Sep 2017.

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Citing papers per year: 2021: 12; 2022: 13; 2023: 13; 2024: 8; 2025: 14; 2026: 5.

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  2. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
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  6. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
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  8. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
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  10. 2022Convergence of a robust deep FBSDE method for stochastic controlcited by 12
  11. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  12. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
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  5. 2022Temporal Difference Learning with Continuous Time and State in the Stochastic Settingcited by 5
  6. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited 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. 2021dNNsolve: an efficient NN-based PDE solvercited by 4
  10. 2022Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusionscited by 4
  11. 2023Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problemscited by 4
  12. 2024FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equationscited by 4
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  16. 2022Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditionscited 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. 2025Integration Matters for Learning PDEs with Backward SDEscited by 3
  19. 2025Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studiescited by 3
  20. 2021Approximation properties of Residual Neural Networks for Kolmogorov PDEscited by 2
  21. 2021Multigoal-oriented dual-weighted-residual error estimation using deep neural networkscited by 2
  22. 2022A deep learning method for solving stochastic optimal control problems driven by fully-coupled FBSDEscited by 2
  23. 2022A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equationscited by 2
  24. 2023A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equationscited by 2
  25. 2021Cell-average based neural network method for hyperbolic and parabolic partial differential equationscited by 1
  26. 2022Quantum-Inspired Tensor Neural Networks for Option Pricingcited by 1
  27. 2023Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Timecited by 1
  28. 2025A deep solver for backward stochastic Volterra integral equationscited by 1
  29. 2025Statistical Learning Guarantees for Group-Invariant Barron Functionscited by 1
  30. 2021Accuracy and Architecture Studies of Residual Neural Network solving Ordinary Differential Equationscited by 0
  31. 2021Deep neural networks for solving forward and inverse problems of (2+1)-dimensional nonlinear wave equations with rational solitonscited by 0
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  34. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  35. 2023Application of Tensor Neural Networks to Pricing Bermudan Swaptionscited by 0
  36. 2023Efficient Learning of PDEs via Taylor Expansion and Sparse Decomposition into Value and Fourier Domainscited by 0
  37. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  38. 2024Neural Networks-based Random Vortex Methods for Modelling Incompressible Flowscited by 0

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