DeepSets and their derivative networks for solving symmetric PDEs *

Maximilien Germain (1, 2, 3) , Mathieu Laurière (4) , Huyên Pham (5, 6) , Xavier Warin (1, 6, 3)

1 EDF - EDF
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
3 EDF R&D - EDF R&D
4 ORFE - Department of Operations Research and Financial Engineering
5 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
6 FiME Lab - Laboratoire de Finance des Marchés d'Energie

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.

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

Domaines: Finance quantitative [q-fin.CP]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Maximilien Germain, Mathieu Laurière, Huyên Pham, Xavier Warin. DeepSets and their derivative networks for solving symmetric PDEs *. 2021. ⟨hal-03154116v1⟩