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

2026-09-17

AI Summary

本文提出一种结合Gauss-Newton和Petrov-Galerkin方法的框架，用于解决偏微分方程数值解问题，通过线性测量离散化并选择测试函数来优化神经网络和混合有限元求解器。

Abstract

Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.

Problem

• Physics-informed neural networks
• Finite element methods
• Partial differential equations
• Gauss-Newton
• Petrov-Galerkin

Innovation

• Gauss-Newton framework
• Petrov-Galerkin discretization
• weak residual formulations
• hybrid finite element-neural construction