Abstract: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.
From: Conrard Giresse Tetsassi Feugmo [view email]
[v1]
Mon, 1 Jun 2026 14:44:48 UTC (5,561 KB)
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