# Weak Adversarial Networks for High-dimensional Partial Differential Equations

## TL;DR

This paper introduces Weak Adversarial Networks (WAN), a mesh-free deep learning method that efficiently solves high-dimensional PDEs on irregular domains by converting the problem into an operator norm minimization and using adversarial training.

## Contribution

The paper proposes a novel weak adversarial network framework that leverages weak formulations and adversarial training to solve high-dimensional PDEs more efficiently and stably than traditional methods.

## Key findings

- WAN is fast and stable for high-dimensional PDEs.
- WAN handles irregular domains effectively.
- Experimental results show promising performance of WAN.

## Abstract

Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.