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Numerical Robustness of PINNs for Multiscale Transport Equations

Jesser, Alexander ORCID iD icon 1; Krycki, Kai; McClarren, Ryan G.; Frank, Martin ORCID iD icon 1
1 Scientific Computing Center (SCC), Karlsruher Institut für Technologie (KIT)

Abstract:

We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least-Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first-order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.

Zugehörige Institution(en) am KIT Scientific Computing Center (SCC)
Publikationstyp Forschungsbericht/Preprint
Publikationsdatum 19.12.2024
Sprache Englisch
Identifikator KITopen-ID: 1000178537
HGF-Programm 46.21.02 (POF IV, LK 01) Cross-Domain ATMLs and Research Groups
Verlag arxiv
Serie Mathematics : Numerical Analysis ; 14683
Projektinformation RAPID (BMBF, 033RK094B)
Schlagwörter Physics Informed Neural Networks, Diffusive Regime, Multiscale, Transport Equations, Numerical Analysis
Nachgewiesen in arXiv
Dimensions

Volltext §
DOI: 10.5445/IR/1000178537
Veröffentlicht am 29.01.2025
Seitenaufrufe: 11
seit 30.01.2025
Downloads: 6
seit 02.02.2025
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