| Abstract: This talk presents our in-depth analysis of the cutting-edge Weakly Adversarial Networks (WAN) method and its variations, spotlighting its applicability in approximating high-dimensional and non-linear partial differential equations. Our three primary highlights are: 1. Existence & Stability of WAN Solutions: A meticulous validation of discrete solutions, interpreted in the weak context, complemented by an approximation principle reminiscent of Cea's lemma, a cornerstone in finite element analyses. 2. Optimized WAN Formulations: Introduction of two novel WAN stabilization strategies that sidestep direct normalization, paired with a critical treatment of WAN's efficacy for the Dirichlet boundary challenge. 3. The pseudo-time XNODE Neural Network: Unveiling the pseudo-time XNODE neural network paradigm, signifying a transformative leap with accelerated convergence rates over conventional DNN architectures. |