Kolmogorov-Arnold网络中网格适应的动态框架
A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks
AI总结：
本文提出基于曲率的适应策略，通过重要密度函数进行网格分配，提升KAN在函数拟合、费曼数据集回归和Helmholtz偏微分方程求解中的性能，减少误差达25.3%。
摘要：
本文提出了一种基于曲率的适应策略，通过重要密度函数进行网格分配，提升KAN在函数拟合、费曼数据集回归和Helmholtz偏微分方程求解中的性能，减少误差达25.3%。
英文摘要：
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.
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