Neural Network Dual Norms for Minimal Residual Finite Element Methods
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912596765966336 |
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| author | Alsobhi, Hamd Benny-Chacko, Emin Brevis, Ignacio van der Zee, Kristoffer G. |
| author_facet | Alsobhi, Hamd Benny-Chacko, Emin Brevis, Ignacio van der Zee, Kristoffer G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural Network Dual Norms for Minimal Residual Finite Element Methods Alsobhi, Hamd Benny-Chacko, Emin Brevis, Ignacio van der Zee, Kristoffer G. Numerical Analysis 65N30, 65N12, 65N22, 65F10, 68T07 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. |
| title | Neural Network Dual Norms for Minimal Residual Finite Element Methods |
| topic | Numerical Analysis 65N30, 65N12, 65N22, 65F10, 68T07 |
| url | https://arxiv.org/abs/2509.16961 |