WAN Discretization of PDEs: Best Approximation, Stabilization, and Essential Boundary Conditions

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. 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. We also devise a pseudo-time XNODE neural network for static PDE problems, yielding significantly faster convergence results than the classical DNN.