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Deep Solution Operators for Variational Inequalities via Proximal Neural Networks

by Ch. Schwab and A. Stein

(Report number 2021-37)

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 real, 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 projected Jacobi and projected SOR methods.

Keywords: Deep Neural Networks, Deep Solution Operators, Variational Inequalities, Proximity Operators, ReLU

@Techreport{SS21_979,
  author = {Ch. Schwab and A. Stein},
  title = {Deep Solution Operators for Variational Inequalities via Proximal Neural Networks},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-37.pdf },
  year = {2021}
}

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