Neural Spectral Element Methods for stiff multiphysics PDEs with electrochemical transport benchmarks
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| Format: | Preprint |
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2026
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| _version_ | 1866916074898849792 |
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| author | Feugmo, Conrard Giresse Tetsassi Pankaczy, David |
| author_facet | Feugmo, Conrard Giresse Tetsassi Pankaczy, David |
| contents | The Neural Spectral Element Method (NSEM) evaluates each network only at fixed Legendre-Gauss-Lobatto quadrature nodes and replaces all derivative calls with precomputed spectral differentiation matrices. The resulting deterministic loss enables limited-memory BFGS (L-BFGS) to reach residuals of 10^-9 to 10^-10. A Kosloff-Tal-Ezer coordinate map resolves electrochemical boundary layers, while a mesh-free neural mortar framework couples multi-element domains. On the four-example Poisson-Nernst-Planck (PNP) benchmark of Huang and co-workers, NSEM attains 10^-4 to 10^-7 relative pointwise error with two orders of magnitude fewer collocation points than the adaptive-resampling PINN baseline. Both a tanh multilayer perceptron (MLP) and a basis-aligned Legendre Kolmogorov-Arnold Network (KAN) backbone attain spectral accuracy within the same NSEM infrastructure, with the KAN requiring roughly half the Adam steps to enter the L-BFGS basin of attraction on the 1D PNP benchmark. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02335 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Neural Spectral Element Methods for stiff multiphysics PDEs with electrochemical transport benchmarks Feugmo, Conrard Giresse Tetsassi Pankaczy, David Materials Science Mathematical Physics Applied Physics Computational Physics The Neural Spectral Element Method (NSEM) evaluates each network only at fixed Legendre-Gauss-Lobatto quadrature nodes and replaces all derivative calls with precomputed spectral differentiation matrices. The resulting deterministic loss enables limited-memory BFGS (L-BFGS) to reach residuals of 10^-9 to 10^-10. A Kosloff-Tal-Ezer coordinate map resolves electrochemical boundary layers, while a mesh-free neural mortar framework couples multi-element domains. On the four-example Poisson-Nernst-Planck (PNP) benchmark of Huang and co-workers, NSEM attains 10^-4 to 10^-7 relative pointwise error with two orders of magnitude fewer collocation points than the adaptive-resampling PINN baseline. Both a tanh multilayer perceptron (MLP) and a basis-aligned Legendre Kolmogorov-Arnold Network (KAN) backbone attain spectral accuracy within the same NSEM infrastructure, with the KAN requiring roughly half the Adam steps to enter the L-BFGS basin of attraction on the 1D PNP benchmark. |
| title | Neural Spectral Element Methods for stiff multiphysics PDEs with electrochemical transport benchmarks |
| topic | Materials Science Mathematical Physics Applied Physics Computational Physics |
| url | https://arxiv.org/abs/2606.02335 |