A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks

2026-01-26

AI Summary

This work proposes a novel adaptive grid refinement mechanism for Kolmogorov–Arnold Networks (KANs) that overcomes the limitation of existing strategies, which rely solely on input data density while neglecting the geometric complexity of the target function and training dynamics. By introducing curvature information into KANs for the first time, the method employs an importance density function (IDF) to guide grid node allocation, integrating dynamic feedback from training with local curvature estimates to align grid refinement with the intrinsic geometry of the target function. Empirical evaluations demonstrate consistent improvements across diverse tasks: average relative errors are reduced by 25.3% on synthetic functions, 9.4% on the Feynman dataset, and 23.3% on Helmholtz partial differential equation problems. Statistical significance is confirmed via Wilcoxon signed-rank tests, establishing clear superiority over baseline approaches.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Kolmogorov-Arnold Networks
• grid adaptation
• geometric complexity
• curvature-based adaptation
• scientific machine learning

Innovation

• Kolmogorov-Arnold Networks
• grid adaptation
• curvature-based adaptation
• importance density functions
• scientific machine learning