A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods

TL;DR

This paper presents the Deep Uzawa algorithm, a novel deep learning framework that effectively enforces boundary conditions in PDE approximations, improving convergence and accuracy in complex and high-dimensional problems.

Contribution

It introduces the Deep Uzawa algorithm with Lagrange multipliers for boundary conditions, enhancing convergence and accuracy in PINNs and Deep Ritz methods.

Findings

Enhanced convergence properties demonstrated.

Effective boundary condition enforcement in high-dimensional problems.

Validated performance on singularly perturbed and non-convex domain problems.

Abstract

We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems. We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains.