Extended Galerkin Neural Network Approximation of Singular Variational Problems with Error Control

This work presents extended Galerkin neural networks (xGNN), a variational framework for approximating general boundary value problems (BVPs) with error control. The key contributions are: (1) a rigorous theory for constructing new weighted least squares variational formulations suitable for neural network approximation of general BVPs, and (2) an "extended" feedforward network architecture which can incorporate and learn singular solution structures, greatly improving approximability of singular solutions.