Partition-of-Unity Gaussian Kolmogorov-Arnold Networks

TL;DR

This paper introduces PU-GKAN, a normalized Gaussian kernel approximation network that enhances stability and accuracy by employing a Shepard-type partition-of-unity normalization, applicable across various architectures and problems.

Contribution

The paper presents a novel partition-of-unity Gaussian KAN (PU-GKAN) that improves stability and accuracy over standard Gaussian KANs through a Shepard-type normalization scheme.

Findings

PU-GKAN reduces sensitivity to the scale parameter 

PU-GKAN improves validation accuracy for smooth and non-smooth targets

PU-GKAN offers more stable training across different architectures and problems

Abstract

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 \(\epsilon\), 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 \(\epsilon\), 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, Mat\'ern 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.