DeepSets and their derivative networks for solving symmetric PDEs

Maximilien Germain, Mathieu Lauri\`ere, Huy\^en Pham and Xavier Warin
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Maximilien Germain: EDF, LPSM
Mathieu Lauri\`ere: ORFE
Huy\^en Pham: LPSM
Xavier Warin: EDF, FiME Lab, EDF R&D, EDF R&D OSIRIS

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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.

Date: 2021-03, Revised 2022-01
New Economics Papers: this item is included in nep-big, nep-cmp and nep-ore
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Citations: View citations in EconPapers (4)

Published in Journal of Scientific Computing, Springer Verlag, In press

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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2103.00838

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