DeepSets and their derivative networks for solving symmetric PDEs * - ENSAE Paris Accéder directement au contenu
Article Dans Une Revue Journal of Scientific Computing Année : 2022

DeepSets and their derivative networks for solving symmetric PDEs *

Résumé

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.
Fichier principal
Vignette du fichier
PDEsymNNRevision.pdf (1.12 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03154116 , version 1 (27-02-2021)
hal-03154116 , version 2 (03-01-2022)

Identifiants

Citer

Maximilien Germain, Mathieu Laurière, Huyên Pham, Xavier Warin. DeepSets and their derivative networks for solving symmetric PDEs *. Journal of Scientific Computing, 2022, ⟨10.1007/s10915-022-01796-w⟩. ⟨hal-03154116v2⟩
420 Consultations
170 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More