Prox-PINNs: A Deep Learning Algorithmic Framework for Elliptic Variational Inequalities

TL;DR

Prox-PINNs is a new deep learning framework that combines proximal operators with physics-informed neural networks to efficiently and flexibly solve a wide range of elliptic variational inequalities, overcoming traditional computational challenges.

Contribution

It introduces a unified, mesh-free deep learning approach that reformulates EVIs as nonlinear equations and enforces boundary conditions as hard constraints, broadening applicability and improving computational efficiency.

Findings

Demonstrated high accuracy across multiple benchmark problems.

Showed robustness and efficiency in complex geometries.

Validated flexibility for diverse applications.

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.