Neural spectral element methods for stiff multiphysics PDEs with electrochemical transport benchmarks

Conrard Giresse Tetsassi Feugmo, David Pankaczy

Abstract Physics-informed neural networks (PINNs) face an empirical accuracy floor of approximately O(10^-2) when applied to stiff multiphysics problems like electrochemical transport. This limitation stems from the "PINN trilemma": Monte Carlo collocation, serial automatic differentiation, and a resulting stochastic loss landscape that prevents efficient quasi-Newton optimization.

To overcome these bottlenecks, we introduce the Neural Spectral Element Method (NSEM), a framework that replaces random collocation pipelines with a deterministic spectral approach. NSEM evaluates neural networks exclusively at fixed Legendre-Gauss-Lobatto (LGL) quadrature nodes and substitutes all automatic differentiation calls with precomputed spectral differentiation matrices. This transition to a deterministic loss landscape enables limited-memory BFGS (L-BFGS) optimization to achieve loss residuals between 10^-9 and 10^-10. To handle complex geometries and multi-scale features, NSEM integrates a Kosloff-Tal-Ezer coordinate map to resolve sharp electrochemical boundary layers without additional node costs, alongside a mesh-free neural mortar framework to couple multi-element domains.

We validate NSEM against a comprehensive Poisson-Nernst-Planck (PNP) benchmark suite spanning 1D, 2D, and 3D steady and time-dependent problems, as well as canonical Helmholtz, Allen-Cahn, and convection-diffusion equations. NSEM achieves a relative pointwise error of 10^-4 to 10^-7, utilizing two orders of magnitude fewer collocation points than adaptive-resampling PINN baselines. Furthermore, we demonstrate that NSEM accommodates interchangeable network backbones, comparing a standard tanh multilayer perceptron (MLP) against a basis-aligned Legendre Kolmogorov-Arnold Network (KAN). Both architectures achieve true spectral accuracy within the NSEM infrastructure, with the KAN backbone requiring roughly half the Adam optimization steps to enter the L-BFGS basin of attraction. NSEM thus provides a high-precision path forward for quantitative parameter inference in stiff, coupled PDE systems.