Beyond PINNs: A Unified Gauss-Newton and Petrov-Galerkin Framework for Neural and Hybrid PDE Solvers

This paper introduces a unified framework that bridges physics-informed neural networks (PINNs) and finite element methods (FEM) for numerical approximation of partial differential equations. The approach is based on discretizing functional Gauss-Newton problems using Petrov-Galerkin formulations, enabling hybrid neural and traditional PDE solvers. The framework provides a common mathematical ground for understanding both strong-residual minimization (typical of PINNs) and weak variational methods (typical of FEM).