A neural network approach to learning solutions of a class of elliptic variational inequalities

TL;DR

This paper introduces a neural network-based weak adversarial method for solving elliptic obstacle problems, effectively handling complex cases like biactivity without requiring symmetric operators.

Contribution

It presents a novel neural network approach that reformulates obstacle problems as minmax problems, accommodating non-symmetric elliptic operators and providing error analysis.

Findings

Successfully handles obstacle problems with biactivity

Provides error estimates for discretised solutions

Demonstrates effectiveness through experiments

Abstract

We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to learning. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem and conclude the paper with various examples and experiments. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.