Empirical Stability Analysis of Kolmogorov-Arnold Networks in Hard-Constrained Recurrent Physics-Informed Discovery

TL;DR

This paper empirically evaluates Kolmogorov-Arnold Networks within physics-informed recurrent architectures, revealing their limitations in stability and handling complex residuals compared to standard MLPs.

Contribution

The study provides the first empirical analysis of KANs in hard-constrained recurrent physics-informed models, highlighting their fragility and limitations in complex systems.

Findings

Small KANs perform well on simple residuals but are fragile.

Deeper KAN configurations are unstable and often fail.

Standard MLPs outperform KANs in complex residual scenarios.

Abstract

We investigate the integration of Kolmogorov-Arnold Networks (KANs) into hard-constrained recurrent physics-informed architectures (HRPINN) to evaluate the fidelity of learned residual manifolds in oscillatory systems. Motivated by the Kolmogorov-Arnold representation theorem and preliminary gray-box results, we hypothesized that KANs would enable efficient recovery of unknown terms compared to MLPs. Through initial sensitivity analysis on configuration sensitivity, parameter scale, and training paradigm, we found that while small KANs are competitive on univariate polynomial residuals (Duffing), they exhibit severe hyperparameter fragility, instability in deeper configurations, and consistent failure on multiplicative terms (Van der Pol), generally outperformed by standard MLPs. These empirical challenges highlight limitations of the additive inductive bias in the original KAN formulation for state coupling and provide preliminary empirical evidence of inductive bias limitations for future hybrid modeling.