A Neural Network Approach to Learning Solutions of a Class of Elliptic Variational Inequalities

We develop a weak adversarial approach to solving obstacle problems using neural networks. By using (generalized) regularized 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 a learning approach. 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.