Inexact Uzawa-Double Deep Ritz Method for Weak Adversarial Neural Networks

TL;DR

This paper introduces an inexact Uzawa-Double Deep Ritz method for weak adversarial neural networks, providing a mesh-free PDE solver with proven convergence and demonstrated robustness through numerical experiments.

Contribution

It develops a novel inexact Uzawa iterative scheme for neural network-based PDE solvers, ensuring stability and convergence in a mesh-free deep learning framework.

Findings

The method converges stably under inexact updates.

Numerical experiments confirm robustness and accuracy.

The approach is mesh-free and theoretically sound.

Abstract

The emergence of deep learning has stimulated a new class of PDE solvers in which the unknown solution is represented by a neural network. Within this framework, residual minimization in dual norms -- central to weak adversarial neural network approaches -- naturally leads to saddle-point problems whose stability depends on the underlying iterative scheme. Motivated by this structure, we develop an inexact Uzawa methodology in which both trial and test functions are represented by neural networks and updated only approximately. We introduce the Uzawa Deep Double Ritz method, a mesh-free deep PDE solver equipped with a continuous level convergence showing that the overall iteration remains stable and convergent provided the inexact inner updates move in the correct descent direction. Numerical experiments validate the theoretical findings and demonstrate the practical robustness and accuracy of the proposed approach.