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。本文提出单位分割高斯KAN（PU-GKAN），即一种谢泼德型归一化高斯KAN，其中每条边上的高斯基值除以固定中心点上的局部和。该操作生成具有可训练系数的单位分割特征映射，同时保持标准边级KAN结构。这一归一化构建在边级实现精确常数复现，并允许显式的有限特征核解释。我们从有限特征与加性核视角系统阐述了标准高斯KAN（GKAN）与PU-GKAN，明确给出诱导层核与经验特征矩阵。以首层特征矩阵为参照对象，我们采用实用的尺度选择区间\(ε\)，其下界由相邻中心重叠程度决定，上界由保守的条件数阈值确定。数值实验表明，PU-GKAN降低了对\(ε\)的敏感性，在大多数光滑及适度非光滑目标上提升了验证精度，并呈现更稳定的训练行为。该优势在样本量与中心数量扫描、高维架构、Matérn径向基函数及涉及亥姆霍兹方程与波动方程的物理信息示例中持续存在。这些结果表明，谢泼德型单位分割归一化是RBF类KAN的一种简单有效的稳定性机制。