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：本文提出扩展伽辽金神经网络（xGNN），一种具有误差控制的通用边值问题逼近变分框架。本工作的主要贡献包括：（1）建立严格的理论体系，指导构建适用于神经网络逼近通用边值问题的新型加权最小二乘变分形式；（2）设计“扩展”前馈网络架构，该架构能够融合甚至学习奇异解结构，从而显著提升奇异解的逼近能力。通过数值实验验证方法的有效性，包括凹角附近的稳态斯托克斯流动及凸角内产生莫法特涡流的算例。