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Neural networks-based backward scheme for fully nonlinear PDEs

Author

Listed:
  • Huyên Pham

    (LPSM, Université de Paris, CREST-ENSAE & FiME)

  • Xavier Warin

    (EDF R&D & FiME)

  • Maximilien Germain

    (EDF R&D, LPSM, Université de Paris)

Abstract

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 Huré et al. (Math Comput 89(324):1547–1579, 2020) for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with non-linearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient, Monge-Ampère equation and Hamilton–Jacobi–Bellman equation in portfolio optimization.

Suggested Citation

  • Huyên Pham & Xavier Warin & Maximilien Germain, 2021. "Neural networks-based backward scheme for fully nonlinear PDEs," Partial Differential Equations and Applications, Springer, vol. 2(1), pages 1-24, February.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:1:d:10.1007_s42985-020-00062-8
    DOI: 10.1007/s42985-020-00062-8
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    References listed on IDEAS

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    1. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    2. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    3. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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    Citations

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    Cited by:

    1. Teng, Long, 2022. "Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    2. Jiang Yu Nguwi & Nicolas Privault, 2023. "A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations," Partial Differential Equations and Applications, Springer, vol. 4(4), pages 1-20, August.
    3. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.
    4. William Lefebvre & Gr'egoire Loeper & Huy^en Pham, 2022. "Differential learning methods for solving fully nonlinear PDEs," Papers 2205.09815, arXiv.org.
    5. Jiang, Weixin & Wang, Junfang & Varbanov, Petar Sabev & Yuan, Qing & Chen, Yujie & Wang, Bohong & Yu, Bo, 2024. "Hybrid data-mechanism-driven model of the unsteady soil temperature field for long-buried crude oil pipelines with non-isothermal batch transportation," Energy, Elsevier, vol. 292(C).
    6. Maximilien Germain & Joseph Mikael & Xavier Warin, 2022. "Numerical Resolution of McKean-Vlasov FBSDEs Using Neural Networks," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2557-2586, December.
    7. William Lefebvre & Grégoire Loeper & Huyên Pham, 2023. "Differential learning methods for solving fully nonlinear PDEs," Digital Finance, Springer, vol. 5(1), pages 183-229, March.
    8. Maximilien Germain & Mathieu Laurière & Huyên Pham & Xavier Warin, 2022. "DeepSets and their derivative networks for solving symmetric PDEs ," Post-Print hal-03154116, HAL.
    9. Lorenc Kapllani & Long Teng, 2024. "A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations," Papers 2404.08456, arXiv.org.

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