A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks

TL;DR

This paper introduces a curvature-based grid adaptation framework for Kolmogorov-Arnold Networks, improving their accuracy by dynamically adjusting grid resolution based on training dynamics rather than just input data density.

Contribution

It proposes a generalized importance density function approach and demonstrates significant error reductions across multiple scientific machine learning tasks.

Findings

Achieved 25.3% error reduction on synthetic functions.

Reduced error by 9.4% on the Feynman dataset.

Lowered error by 23.3% on PDE benchmarks.

Abstract

Kolmogorov-Arnold Networks (KANs) have recently demonstrated promising potential in scientific machine learning, partly due to their capacity for grid adaptation during training. However, existing adaptation strategies rely solely on input data density, failing to account for the geometric complexity of the target function or metrics calculated during network training. In this work, we propose a generalized framework that treats knot allocation as a density estimation task governed by Importance Density Functions (IDFs), allowing training dynamics to determine grid resolution. We introduce a curvature-based adaptation strategy and evaluate it across synthetic function fitting, regression on a subset of the Feynman dataset and different instances of the Helmholtz PDE, demonstrating that it significantly outperforms the standard input-based baseline. Specifically, our method yields average relative error reductions of 25.3% on synthetic functions, 9.4% on the Feynman dataset, and 23.3% on the PDE benchmark. Statistical significance is confirmed via Wilcoxon signed-rank tests, establishing curvature-based adaptation as a robust and computationally efficient alternative for KAN training.