Robin Pre-Training for the Deep Ritz Method

TL;DR

This paper compares training strategies for the Deep Ritz Method applied to elliptic equations, proposing a novel pre-training approach with Robin boundary conditions to improve stability and accuracy.

Contribution

It introduces a new pre-training technique using small penalization strength for Robin boundary conditions to enhance Deep Ritz Method training stability.

Findings

Pre-training with small penalization improves stability.

Robin boundary approximation is feasible for complex domains.

Numerical and theoretical evidence supports the method's effectiveness.

Abstract

We compare different training strategies for the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight the problems arising from the boundary values. We distinguish between an exact resolution of the boundary values by introducing a distance function and the approximation through a Robin Boundary Value problem. However, distance functions are difficult to obtain for complex domains. Therefore, it is more feasible to solve a Robin Boundary Value problem which approximates the solution to the Dirichlet Boundary Value problem, yet the na\"ive approach to this problem becomes unstable for large penalizations. A novel method to compensate this problem is proposed using a small penalization strength to pre-train the model before the main training on the target penalization strength is conducted. We present numerical and theoretical evidence that the proposed method is beneficial.