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：本文对近期提出的弱对抗网络（WAN）方法进行了理论分析，该方法用于逼近高维偏微分方程的解。我们研究了解的存在性、稳定性以及逼近误差界。具体而言，我们证明了在适当弱意义下离散解的存在性，并给出了类似于有限元方法中Cea引理的拟最佳逼近估计。我们还提出了两种新的基于WAN的稳定化公式，从而避免了显式归一化的需求。此外，我们分析了该方法在采用几何隐式表示的Dirichlet边值问题中的有效性。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，该条件本质上要求测试网络集合相对于试探空间足够大。然而，方法的精度仅由试探网络空间决定。我们还针对静态偏微分方程问题设计了一类伪时间XNODE神经网络，相较于经典深度神经网络（DNN），该方法实现了显著更快的收敛速度。