DeepSets and their derivative networks for solving symmetric PDEs

TL;DR

This paper introduces neural network architectures based on DeepSets and PointNet to efficiently solve symmetric PDEs, which are invariant under permutations, with applications in physics and finance.

Contribution

It proposes a novel class of neural networks tailored for symmetric PDEs, demonstrating improved approximation of solutions and gradients over traditional methods.

Findings

DeepSet/PointNet networks outperform classical feedforward networks in accuracy.

The approach effectively solves mean-field control problems.

Numerical results validate the method's applicability to complex symmetric PDEs.

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.