A deep first-order system least squares method for the obstacle problem

TL;DR

This paper introduces a deep learning-based least-squares method for the obstacle problem, leveraging the FOSLS framework to achieve a mesh-free, scalable, and theoretically sound solution approach suitable for high-dimensional problems.

Contribution

It develops a novel deep FOSLS method that reformulates the obstacle problem as a convex minimization, with proven convergence and scalability to high dimensions.

Findings

Method scales to 20 dimensions

Robust on non-Lipschitz domains

Convergence guaranteed under mild assumptions

Abstract

We propose a deep learning approach to the obstacle problem inspired by the first-order system least-squares (FOSLS) framework. This method reformulates the problem as a convex minimization task; by simultaneously approximating the solution, gradient, and Lagrange multiplier, our approach provides a flexible, mesh-free alternative that scales efficiently to high-dimensional settings. Key theoretical contributions include the coercivity and local Lipschitz continuity of the proposed least-squares functional, along with convergence guarantees via $\Gamma$-convergence theory under mild regularity assumptions. Numerical experiments in dimensions up to 20 demonstrate the method's robustness and scalability, even on non-Lipschitz domains.