We present extended Galerkin neural networks (xGNN), a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs (2) an ``extended'' feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Numerical results are presented for several problems including steady Stokes flow around re-entrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

翻译：本文提出了扩展Galerkin神经网络（xGNN），一种用于逼近一般边值问题（BVP）并具有误差控制的变分框架。本研究的主要贡献包括：（1）一套严格的理论，指导构建适用于神经网络逼近一般BVP的新型加权最小二乘变分形式；（2）一种“扩展”前馈网络架构，该架构能够融入甚至学习奇异解结构，从而显著提升对奇异解的逼近能力。我们给出了多个数值算例，包括绕拐角处及具有Moffatt涡的凸角处的稳态Stokes流动，以验证该方法的有效性。