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.

翻译：本文为近年来提出的弱对抗网络方法（Weakly Adversarial Networks, WAN）——一种用于高维偏微分方程逼近的方法——提供理论分析。我们探讨了解的存在性、稳定性以及逼近界。具体而言，我们证明了在适当弱意义下离散解的存在性，并为此类解建立了类似Céa引理的拟最佳逼近估计（该引理常见于有限元方法）。同时，我们提出两种基于WAN的新稳定化公式，避免了直接归一化的需求。此外，我们分析了该方法对采用几何隐式表示的Dirichlet边界问题的有效性。实现最佳逼近结果的关键在于确保测试网络空间满足特定条件（即inf-sup条件），本质上要求测试网络集相对于试验空间足够大。然而，方法的精度仅由试验网络空间决定。针对静态偏微分方程问题，我们设计了一类伪时间XNODE神经网络，其收敛速度显著优于经典深度神经网络（DNN）。