Gaussian basis functions provide an efficient and flexible alternative to spline activations in KANs. In this work, we introduce the partition-of-unity Gaussian KAN (PU-GKAN), a Shepard-type normalized Gaussian KAN in which the Gaussian basis values on each edge are divided by their local sum over fixed centers. This produces a partition-of-unity feature map with trainable coefficients, while preserving the standard edge-based KAN structure. The normalized construction gives exact constant reproduction at the edge level and admits an explicit finite-feature kernel interpretation. We formulate both the standard Gaussian KAN (GKAN) and PU-GKAN from a finite-feature and additive-kernel viewpoint, making the induced layer kernels and empirical feature matrices explicit. Using the first-layer feature matrix as the reference object, we adopt a practical scale-selection interval for \(ε\), with the lower endpoint determined by adjacent-center overlap and the upper endpoint determined by a conservative conditioning threshold. Numerical experiments show that PU-GKAN reduces sensitivity to \(ε\), improves validation accuracy for most smooth and moderately non-smooth targets, and gives more stable training behavior. The benefit persists across sample-size and center-number sweeps, higher-dimensional architectures, Matérn RBF bases, and physics-informed examples involving Helmholtz and wave equations. These results indicate that Shepard-type partition-of-unity normalization is a simple and effective stabilization mechanism for RBF-based KANs.

翻译：高斯基函数为KAN中的样条激活函数提供了高效且灵活的替代方案。本文提出了单位分解高斯KAN（PU-GKAN），这是一种Shepard型归一化高斯KAN，其中每条边上的高斯基值除以其在固定中心上的局部和。这产生了一个具有可训练系数的单位分解特征映射，同时保留了标准的基于边的KAN结构。该归一化构造在边层面实现了精确的常数再现，并允许显式的有限特征核解释。我们从有限特征和加法核视角阐述了标准高斯KAN（GKAN）和PU-GKAN，使诱导的层核与经验特征矩阵显式化。以第一层特征矩阵作为参考对象，我们为ε采用了实用尺度选择区间，其下界由相邻中心重叠决定，上界由保守的条件数阈值决定。数值实验表明，PU-GKAN降低了对ε的敏感性，提高了大多数光滑和中等非光滑目标的验证精度，并实现了更稳定的训练行为。该优势在样本量与中心数扫描、高维架构、Matérn径向基函数以及涉及亥姆霍兹方程和波动方程的物理信息示例中持续存在。这些结果表明，Shepard型单位分解归一化是RBF基KAN的一种简单有效的稳定化机制。