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arXiv · 2509.16961

Neural Network Dual Norms for Minimal Residual Finite Element Methods

Hamd Alsobhi·Emin Benny-Chacko·Ignacio Brevis·Kristoffer G. van der Zee
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Abstract

Minimal-residual methods for PDEs with a residual in a dual space are non-trivial to guarantee stability. We present a minimal-residual finite element method in which the solution space is a standard finite element space, but neural networks are used as test functions for the evaluation of residual dual norms. The use of a neural network improves the approximation of the residual representer, and thereby improves the stability of the method. Our hybrid approach is implemented through a deep residual Uzawa algorithm that alternates finite element updates with neural network training. We prove consistency and convergence results for the Uzawa methodology. We also prove an a priori error estimate that relies on a suitable Fortin compatibility condition. Numerical experiments on advection-reaction problems with singular or discontinuous data show that the proposed framework delivers robust and accurate approximations.

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Hamd Alsobhi, Emin Benny-Chacko, Ignacio Brevis, Kristoffer G. van der Zee. 2025-09-21. Neural Network Dual Norms for Minimal Residual Finite Element Methods. https://arxiv.org/abs/2509.16961

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