KANs need curvature: penalties for compositional smoothness

arXiv stat.ML / 5/5/2026

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Key Points

The paper argues that Kolmogorov-Arnold Networks (KANs) that fit well often develop highly pathological high-curvature oscillations in their learned activations, hurting interpretability.
It shows that common regularization penalties are insufficient to suppress these curvature oscillations.
The authors derive a basis-agnostic curvature penalty that encourages substantially smoother activation functions while preserving model accuracy.
By analyzing how function composition affects curvature, the paper proves an upper bound linking the model’s overall curvature to the curvature penalty and uses this to motivate more expressive penalty formulations.
The work aims to reduce the accuracy–interpretability bottleneck for KANs, strengthening their usefulness for both prediction and scientific insight.

Abstract

Kolmogorov-Arnold networks (KANs) offer a potent combination of accuracy and interpretability, thanks to their compositions of learnable univariate activation functions. However, the activations of well-fitting KANs tend to exhibit pathologically high-curvature oscillations, making them difficult to interpret, and standard regularization penalties do not prevent this. Here we derive a basis-agnostic curvature penalty and show that penalized models can maintain accuracy while achieving substantially smoother activations. Accounting for how function composition shapes curvature, we prove an upper bound on the full model's curvature relative to the curvature penalty, and use this to motivate richer forms of penalties. Scientific machine learning is increasingly bottlenecked by the trade-off between accuracy and interpretability. Results such as ours that improve interpretability without sacrificing accuracy will further strengthen KANs as a practical tool for both prediction and insight.