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DeepSets and their derivative networks for solving symmetric PDEs

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  • Maximilien Germain

    (EDF - EDF, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité, EDF R&D - EDF R&D - EDF - EDF, EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie - EDF R&D - EDF R&D - EDF - EDF)

  • Mathieu Laurière

    (ORFE - Department of Operations Research and Financial Engineering - Princeton University)

  • Huyên Pham

    (FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CREST - EDF R&D - EDF R&D - EDF - EDF, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

  • Xavier Warin

    (EDF - EDF, FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CREST - EDF R&D - EDF R&D - EDF - EDF, EDF R&D - EDF R&D - EDF - EDF, EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie - EDF R&D - EDF R&D - EDF - EDF)

Abstract

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we introduce a class of PDEs that are invariant to permutations, and called symmetric PDEs. Such problems are widespread, ranging from cosmology to quantum mechanics, and option pricing/hedging in multi-asset market with exchangeable payoff. Our main application comes actually from the particles approximation of mean-field control problems. We design deep learning algorithms based on certain types of neural networks, named PointNet and DeepSet (and their associated derivative networks), for computing simultaneously an approximation of the solution and its gradient to symmetric PDEs. We illustrate the performance and accuracy of the PointNet/DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, mean-variance problem and a min/max linear quadratic McKean-Vlasov control problem.

Suggested Citation

  • Maximilien Germain & Mathieu Laurière & Huyên Pham & Xavier Warin, 2022. "DeepSets and their derivative networks for solving symmetric PDEs ," Post-Print hal-03154116, HAL.
    • Handle: RePEc:hal:journl:hal-03154116
    DOI: 10.1007/s10915-022-01796-w
    Note: View the original document on HAL open archive server: https://hal.science/hal-03154116v2
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    References listed on IDEAS

    as
    1. Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse & Tuan Anh Nguyen, 2020. "A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-34, April.
    2. Matteo Basei & Huyên Pham, 2019. "A Weak Martingale Approach to Linear-Quadratic McKean–Vlasov Stochastic Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 347-382, May.
    3. Amine Ismail & Huyên Pham, 2019. "Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 174-207, January.
    4. 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.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Permutation-invariant PDEs; symmetric neural networks; exchangeability; deep backward scheme; mean-field control;
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