Newton

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Period ending 2026-09-21

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80 papers

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Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers
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Error whitening: Why Gauss-Newton outperforms Newton
May 11, 2026Maricela Best McKay, Nathan P. Lawrence, Brian Wetton +1Newton Physics-Informed Machine Learning
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May 11, 2026Shourya Bose, Helgi Hilmarsson, Dhruv SuriAc Optimal Power Flow Newton

Sep 17, 2026 · math.NAJ/K move · Enter open · S save

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

Nilo Schwencke, Roland Maier

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.

Bayesian Quadrature Partial Differential Equation Benchmarks Finite Element Method Newton Neural-Network Approximation

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