Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear Problems

TL;DR

This paper establishes uniform convergence guarantees for the Deep Ritz Method applied to nonlinear variational problems, including the p-Laplace and Modica-Mortola energies, under certain assumptions.

Contribution

It provides the first convergence guarantees for the Deep Ritz Method on nonlinear variational problems, extending theoretical understanding of its effectiveness.

Findings

Convergence guarantees are established for the Deep Ritz Method on nonlinear problems.

Uniform convergence across bounded families of right-hand sides is demonstrated.

Applicable to problems like the p-Laplace equation and Modica-Mortola energy.

Abstract

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover non-linear variational problems such as the $p$-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across % bounded families of right-hand sides.