Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs

TL;DR

This paper provides a theoretical foundation for weakly adversarial networks (WAN) in high-dimensional PDE approximation, introducing new stabilized formulas and a pseudo-time neural network class for improved convergence.

Contribution

It establishes existence, stability, and approximation bounds for WAN solutions, introduces stabilized WAN formulas, and proposes a pseudo-time neural network for faster PDE solution convergence.

Findings

Proved existence of discrete WAN solutions with quasi-best approximation bounds.

Introduced two stabilized WAN formulas avoiding direct normalization.

Developed a pseudo-time XNODE neural network class with faster convergence.

Abstract

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.