# KANs need curvature: penalties for compositional smoothness

arXiv: https://arxiv.org/abs/2605.02190 · HTML: https://mlsanity.com/paper/2605.02190/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Kolmogorov-Arnold Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [Kolmogorov-Arnold Transformer](https://mlsanity.com/paper/2409.10594/) (2024)
- [GKAN: Graph Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.06470/) (2024)
- [Smooth Kolmogorov Arnold networks enabling structural knowledge representation](https://mlsanity.com/paper/2405.11318/) (2024)
- [Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Study](https://mlsanity.com/paper/2509.03417/) (2025)
- [CR-SAM: Curvature Regularized Sharpness-Aware Minimization](https://mlsanity.com/paper/2312.13555/) (2023)
- [Data-driven model discovery with Kolmogorov-Arnold networks](https://mlsanity.com/paper/2409.15167/) (2024)
- [Multi-Exit Kolmogorov-Arnold Networks: enhancing accuracy and parsimony](https://mlsanity.com/paper/2506.03302/) (2025)
- [Training Deep Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.23501/) (2025)
- [The Curvature Rate λ: A Scalar Measure of Input-Space Sharpness in Neural Networks](https://mlsanity.com/paper/2511.01438/) (2025)
- [Optimized Architectures for Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2512.12448/) (2025)
- [A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.18672/) (2026)

## Abstract (arXiv)

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.