Unifying neural and finite element PDE solvers with Gauss Newton methods
Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers
Machine Learning
Summary
Solving partial differential equations (PDEs) is a key step in modeling physical systems, but there are different methods to do it. The authors connect two common approaches: neural networks trained to reduce direct errors at points, and finite element methods using a different mathematical formulation. They show how these methods fit into a single framework based on Gauss--Newton optimization and flexible testing strategies. This unified view also leads to new hybrid methods combining neural networks and finite elements, giving more options for accurate PDE solutions.
What this means in practice
- •For engineering simulation teams: Build more flexible PDE solvers combining neural networks and finite elements for better simulation accuracy and efficiency.
- •For computational physics software developers: Create new PDE solver tools based on the proposed unified framework enabling varied discretization choices and hybrid approximations.
Authors
Nilo Schwencke, Roland Maier
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.