Deep Solution Operators for Variational Inequalities via Proximal Neural Networks

Abstract

We introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g. in valuation of American-style contracts in Black-Scholes financial market models. In the finite-dimensional setting, the VIs reduce to matrix VIs in Euclidean space, and ProxNets emulate classical projected matrix iterations, such as PSOR and semi-smooth Newton iterations which are realized as primal-dual active set strategies, which we encode in the novel PDASNet.