The Gaussian scale parameter \(ε\) is central to the behavior of Gaussian Kolmogorov--Arnold Networks (KANs), yet its role in deep edge-based architectures has not been studied systematically. In this paper, we investigate how \(ε\) affects Gaussian KANs through first-layer feature geometry, conditioning, and approximation behavior. Our central observation is that scale selection is governed primarily by the first layer, since it is the only layer constructed directly on the input domain and any loss of distinguishability introduced there cannot be recovered by later layers. From this viewpoint, we analyze the first-layer feature matrix and identify a practical operating interval, \[ ε\in \left[\frac{1}{G-1},\frac{2}{G-1}\right], \] where \(G\) denotes the number of Gaussian centers. For the standard shared-center Gaussian KAN used in current practice, we interpret this interval not as a universal optimality result, but as a stable and effective design rule, and validate it through brute-force sweeps over \(ε\) across function-approximation problems with different collocation densities, grid resolutions, network architectures, and input dimensions, as well as a physics-informed Helmholtz problem. We further show that this range is useful for fixed-scale selection, variable-scale constructions, constrained training of \(ε\), and efficient scale search using early training MSE. Finally, using a matched Chebyshev reference, we show that a properly scaled Gaussian KAN can already be competitive in accuracy relative to another standard KAN basis. In this way, the paper positions scale selection as a practical design principle for Gaussian KANs rather than as an ad hoc hyperparameter choice.

翻译：高斯尺度参数 \(ε\) 是高斯柯尔莫戈洛夫–阿诺德网络（KANs）行为的关键，但其在深层基于边的架构中的作用尚未被系统研究。本文通过第一层特征几何、条件数及逼近行为，探究 \(ε\) 对高斯KAN的影响。核心发现是：尺度选择主要由第一层决定——因为该层是唯一直接构建于输入域的层级，且该层引入的任何区分性损失均无法被后续层恢复。基于此视角，我们分析了第一层特征矩阵，并确定了一个实用操作区间：\[ ε\in \left[\frac{1}{G-1},\frac{2}{G-1}\right] \]，其中 \(G\) 表示高斯中心数量。对于当前实践中使用的标准共享中心高斯KAN，我们将该区间解读为一种稳定有效的设计准则（而非通用最优性结论），并通过在具有不同配点密度、网格分辨率、网络架构及输入维度的函数逼近问题（以及物理信息驱动的亥姆霍兹问题）上对 \(ε\) 进行暴力扫描加以验证。进一步研究表明，该区间适用于固定尺度选择、变尺度构造、\(ε\) 的约束训练，以及利用早期训练均方误差进行高效尺度搜索。最后，通过匹配的切比雪夫参考方法，我们证明：适当缩放的高斯KAN在精度上已能与其他标准KAN基函数相竞争。由此，本文将尺度选择定位为高斯KAN的一种实用设计原则，而非临时性的超参数选择。