Solving the Dirichlet problem for the Monge-Amp\`ere equation using neural networks

TL;DR

This paper introduces a neural network-based method to solve the Dirichlet problem for the Monge-Ampère equation, demonstrating effectiveness in handling singularities, noise, and high-dimensional cases with convergence analysis.

Contribution

It proposes using deep input convex neural networks to find unique convex solutions to the Monge-Ampère equation, advancing neural PDE solving techniques.

Findings

Method effectively handles singularities and noise.

Converges in high-dimensional settings.

Outperforms standard networks with convexity penalties.

Abstract

The Monge-Amp\`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge-Amp\`ere equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We investigate the convergence numerically and present error estimates based on a stability result. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.