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DeepSets and their derivative networks for solving symmetric PDEs

Author

Listed:
  • Maximilien Germain

    (EDF, LPSM)

  • Mathieu Lauri`ere

    (ORFE)

  • Huy^en Pham

    (LPSM)

  • Xavier Warin

    (EDF, FiME Lab, EDF R&D, EDF R&D OSIRIS)

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 Lauri`ere & Huy^en Pham & Xavier Warin, 2021. "DeepSets and their derivative networks for solving symmetric PDEs," Papers 2103.00838, arXiv.org, revised Jan 2022.
    • Handle: RePEc:arx:papers:2103.00838
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    File URL: http://arxiv.org/pdf/2103.00838
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    References listed on IDEAS

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    1. Maximilien Germain & Huy^en Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance," Papers 2101.08068, arXiv.org, revised Apr 2021.
    2. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Working Papers hal-03115503, HAL.
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    Cited by:

    1. Xavier Warin, 2021. "Reservoir optimization and Machine Learning methods," Papers 2106.08097, arXiv.org, revised May 2023.
    2. Maximilien Germain & Huy^en Pham & Xavier Warin, 2021. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Papers 2112.11059, arXiv.org, revised Nov 2022.
    3. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Working Papers hal-03498263, HAL.

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