DeepSets and their derivative networks for solving symmetric PDEs *

Maximilien Germain (), Mathieu Laurière, Huyên Pham () and Xavier Warin
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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 - IP Paris - Institut Polytechnique de Paris - 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

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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.

Keywords: Permutation-invariant PDEs; symmetric neural networks; exchangeability; deep backward scheme; mean-field control (search for similar items in EconPapers)
Date: 2022-04-08
New Economics Papers: this item is included in nep-big, nep-cmp and nep-his
Note: View the original document on HAL open archive server: https://hal.science/hal-03154116v2
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Published in Journal of Scientific Computing, 2022, ⟨10.1007/s10915-022-01796-w⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-03154116

DOI: 10.1007/s10915-022-01796-w

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