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

Enzo Nicolas Spotorno, Josafat Leal Filho, Antonio Augusto Medeiros Frohlich

TL;DR

This paper examines whether vanilla Kolmogorov-Arnold Networks (KANs) can improve residual discovery in hard-constrained recurrent physics-informed discovery. It combines HRPINN with KANs and tests on the Duffing oscillator with residual $-0.3 x^3$ and the Van der Pol oscillator with residual $(1 - x^2) v$, revealing strong performance for separable univariate terms but poor handling of multiplicative interactions due to optimization fragility. The main contribution is a PoC baseline showing the limitations of the additive bias in vanilla KANs within recurrent constraints and highlighting the bottleneck as optimization stability rather than expressivity, with BPTT providing limited gains. The work also points to future directions such as deeper operator networks, hybrid formulations, SKANODEs, and automatic symbolic extraction via spline pruning to realize KAN-based physics-informed modeling.

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.

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

TL;DR

This paper examines whether vanilla Kolmogorov-Arnold Networks (KANs) can improve residual discovery in hard-constrained recurrent physics-informed discovery. It combines HRPINN with KANs and tests on the Duffing oscillator with residual and the Van der Pol oscillator with residual , revealing strong performance for separable univariate terms but poor handling of multiplicative interactions due to optimization fragility. The main contribution is a PoC baseline showing the limitations of the additive bias in vanilla KANs within recurrent constraints and highlighting the bottleneck as optimization stability rather than expressivity, with BPTT providing limited gains. The work also points to future directions such as deeper operator networks, hybrid formulations, SKANODEs, and automatic symbolic extraction via spline pruning to realize KAN-based physics-informed modeling.

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.
Paper Structure (3 sections, 1 figure, 2 tables)

This paper contains 3 sections, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Experimental Methods and Results
  3. Discussion and Conclusion

Figures (1)

  • Figure 1: Comparison of True Residual Surfaces vs. KAN-Recovered Manifolds (Config A). (a) Duffing: The KAN successfully identifies the univariate cubic manifold ($x^3$), demonstrating high fidelity in separable terms. (b) Van der Pol: The KAN struggles to resolve the multiplicative structure ($(1-x^2)v$), approximating the magnitude (with relevant error given the difference in the colorbars) but failing on the coupled interaction. Results show representative seeds (65 and 82) that align with the median performance reported in Table 1.