用曲率动态调整网格，让KAN模型更精准地拟合复杂函数。
A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks
- 将网格节点分配视为由重要性密度函数控制的密度估计任务
- 在合成函数、费曼数据集和泊松方程上平均误差降低超20%
- 适合需要高精度拟合的科学机器学习场景
Kolmogorov-Arnold Networks（KANs）在科学机器学习中展现出巨大潜力，部分原因在于其训练过程中可进行网格自适应。然而，现有方法仅依赖输入数据密度进行网格调整，未能考虑目标函数的几何复杂度或训练过程中的度量信息。本文提出一种通用框架，将节点分配视为由重要性密度函数（IDFs）控制的密度估计任务，使训练动态决定网格分辨率。引入基于曲率的自适应策略，并在合成函数拟合、费曼数据集子集及不同实例的Helmholtz PDE上进行评估，结果表明该方法显著优于传统的基于输入的基线。具体而言，在合成函数上平均相对误差降低25.3%，在费曼数据集上降低9.4%，在PDE基准上降低23.3%。通过Wilcoxon符号秩检验验证了统计显著性，证实曲率自适应是一种鲁棒且计算高效的KAN训练替代方案。
原文摘要 · Abstract (English)
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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