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arXiv Machine Learning By Nilo Schwencke, Roland Maier

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

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The paper proposes a unified framework that combines physics‑informed neural networks (PINNs) and finite element methods (FEM) by discretizing functional Gauss–Newton problems using finite families of linear measurements. By interpreting these measurements as test functions, the resulting Gauss–Newton system becomes a Petrov–Galerkin discretization of the linearized functional problem, thereby encompassing pointwise collocation and natural‑gradient approaches as special cases. The framework is specialized to elliptic partial differential equations, yielding weak residual formulations and a hybrid finite‑element–neural architecture that operates on complementary approximation spaces, with numerical experiments confirming its effectiveness.

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