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

arXiv: https://arxiv.org/abs/2505.14430 · HTML: https://mlsanity.com/paper/2505.14430/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2025.

## Built on by (most cited first, 4 of 4)

- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs](https://mlsanity.com/paper/2602.12273/) (2026)
- [A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems](https://mlsanity.com/paper/2601.04120/) (2026)
- [Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps](https://mlsanity.com/paper/2606.29436/) (2026)

## Abstract (arXiv)

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.