Prox-PINNs: A Deep Learning Algorithmic Framework for Elliptic Variational Inequalities
Yu Gao, Yongcun Song, Zhiyu Tan, Hangrui Yue, Shangzhi Zeng
TL;DR
Elliptic variational inequalities ($EVIs$) pose numerical challenges due to non-smoothness and inequality constraints, motivating a mesh-free approach. Prox-PINNs combines proximal operators with physics-informed neural networks to recast $EVI$s as nonlinear equations and solve them with hard-boundary neural surrogates, training on physics-informed residuals. The method yields explicit losses for four proximal-function cases and demonstrates strong accuracy and efficiency across obstacle problems, elasto-plastic torsion, Bingham visco-plastic flows, and friction, often outperforming traditional FEM-based methods at comparable resolutions. This framework offers broad generality and scalability for $EVIs$, with promising avenues for theory, adaptive sampling, uncertainty quantification, and extensions to stochastic or parabolic variational inequalities.
Abstract
Elliptic variational inequalities (EVIs) present significant challenges in numerical computation due to their inherent non-smoothness, nonlinearity, and inequality formulations. Traditional mesh-based methods often struggle with complex geometries and high computational costs, while existing deep learning approaches lack generality for diverse EVIs. To alleviate these issues, this paper introduces Prox-PINNs, a novel deep learning algorithmic framework that integrates proximal operators with physics-informed neural networks (PINNs) to solve a broad class of EVIs. The Prox-PINNs reformulate EVIs as nonlinear equations using proximal operators and then approximate the solutions via neural networks that enforce boundary conditions as hard constraints. Then the neural networks are trained by minimizing physics-informed residuals. The Prox-PINNs framework advances the state-of-the-art by unifying the treatment of diverse EVIs within a mesh-free and scalable computational architecture. The framework is demonstrated on several prototypical applications, including obstacle problems, elasto-plastic torsion, Bingham visco-plastic flows, and simplified friction problems. Numerical experiments validate the method's accuracy, efficiency, robustness, and flexibility across benchmark examples.