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math.NA2026
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 common…
math.NA2026
GLENN: Neural network-enhanced computation of Ginzburg-Landau energy minimizers
Michael Crocoll, Christian Döding, Benjamin Dörich +1
In this work, we propose a neural network-enhanced finite element strategy to compute the minimizer of the Ginzburg-Landau energy based on an unsupervised deep Ritz-type strategy.…
math.NA2026
Complexity bounds on neural networks for the solution of structured linear systems of equations
Benjamin Dörich, Roland Maier, Lukas Ullmer
We derive upper bounds on the complexity of ReLU neural networks approximating the solution of a linear system given the matrix and the right-hand side. We focus on matrices which…