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.

求解一般的高维偏微分方程（PDE）是数值数学中的长期挑战。在本文中，我们提出了一种新颖的方法，可以利用它们的弱公式来解决在任意域上定义的高维线性和非线性PDE。我们将发现PDE的弱解的问题转换成由弱公式引起的算子范数最小化问题。然后将弱公式中的弱解和测试函数分别参数化为原始网络和对抗网络，将它们交替更新以逼近最佳网络参数设置。我们的方法被称为弱对抗网络（WAN），它快速，稳定且完全无网状，它特别适用于在不规则域中定义的高维PDE，在这些不规则域中，基于有限差分和有限元的经典数值方法会遇到计算速度慢，不稳定和维数诅咒的问题。我们将我们的方法应用于具有高维PDE的各种测试问题，以证明其有希望的性能。