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Neural networks-based backward scheme for fully nonlinear PDEs

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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.

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Authors and Affiliations

  1. LPSM, Université de Paris, CREST-ENSAE & FiME, Paris, France

    Huyên Pham

  2. EDF R&D & FiME, 7 boulevard Gaspard Monge, 91120, Palaiseau, France

    Xavier Warin

  3. EDF R&D, LPSM, Université de Paris, Paris, France

    Maximilien Germain

Authors
  1. Huyên Pham
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  2. Xavier Warin
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  3. Maximilien Germain
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Corresponding author

Correspondence to Huyên Pham.

Additional information

This work is supported by FiME, Laboratoire de Finance des Marchés de l’Energie, and the ”Finance and Sustainable Development” EDF–CACIB Chair.

This article is part of the topical collection “Deep learning and PDEs” edited by Arnulf Jentzen, Lin Lin, Siddhartha Mishra, and Lexing Ying.

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Pham, H., Warin, X. & Germain, M. Neural networks-based backward scheme for fully nonlinear PDEs. SN Partial Differ. Equ. Appl. 2, 16 (2021). https://doi.org/10.1007/s42985-020-00062-8

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  • DOI: https://doi.org/10.1007/s42985-020-00062-8

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