Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs

TL;DR

This paper uses Neural Tangent Kernel theory to analyze the training dynamics and convergence of physics-informed neural solvers, specifically cPIKANs, revealing insights into their spectral properties and optimization strategies.

Contribution

First systematic NTK analysis of cPIKANs, linking kernel behavior to convergence and training efficiency in physics-informed neural PDE solvers.

Findings

NTK exhibits tractable behavior in cPIKANs, unlike standard PINNs.

Spectral properties of NTK relate to convergence rates and domain decomposition.

Optimization strategies influence NTK evolution and learning dynamics.

Abstract

Physics-informed Kolmogorov-Arnold Networks (PIKANs), and in particular their Chebyshev-based variants (cPIKANs), have recently emerged as promising models for solving partial differential equations (PDEs). However, their training dynamics and convergence behavior remain largely unexplored both theoretically and numerically. In this work, we aim to advance the theoretical understanding of cPIKANs by analyzing them using Neural Tangent Kernel (NTK) theory. Our objective is to discern the evolution of kernel structure throughout gradient-based training and its subsequent impact on learning efficiency. We first derive the NTK of standard cKANs in a supervised setting, and then extend the analysis to the physics-informed context. We analyze the spectral properties of NTK matrices, specifically their eigenvalue distributions and spectral bias, for four representative PDEs: the steady-state Helmholtz equation, transient diffusion and Allen-Cahn equations, and forced vibrations governed by the Euler-Bernoulli beam equation. We also conduct an investigation into the impact of various optimization strategies, e.g., first-order, second-order, and hybrid approaches, on the evolution of the NTK and the resulting learning dynamics. Results indicate a tractable behavior for NTK in the context of cPIKANs, which exposes learning dynamics that standard physics-informed neural networks (PINNs) cannot capture. Spectral trends also reveal when domain decomposition improves training, directly linking kernel behavior to convergence rates under different setups. To the best of our knowledge, this is the first systematic NTK study of cPIKANs, providing theoretical insight that clarifies and predicts their empirical performance.