Partition-of-unity Gaussian Kolmogorov–Arnold networks

Abstract

Gaussian basis functions provide an efficient and flexible alternative to spline activations in Kolmogorov–Arnold networks (KANs), but their performance is sensitive to the Gaussian scale parameter . We introduce the partition-of-unity Gaussian KAN (PU-GKAN), a kernel-based normalization framework that transfers the Shepard partition-of-unity principle to trainable KAN edge functions. Each Gaussian edge basis is normalized by its local sum over fixed centers, preserving the standard edge-based architecture and essentially the same trainable coefficient structure as GKAN. The resulting feature map reproduces constants exactly at the edge level and admits an explicit normalized finite-feature kernel interpretation. Numerical experiments show that PU-GKAN reduces sensitivity to , improves validation accuracy for most smooth and moderately non-smooth targets, and provides more stable training behavior. These benefits persist across variations in sample size and number of centers, higher-dimensional architectures, and Matérn RBF bases. PU-GKAN also improves physics-informed neural networks for Helmholtz and wave equations, with validation-error reductions of up to . These results establish partition-of-unity normalization as a mathematically grounded and broadly applicable framework for improving the reliability and accuracy of radial basis function KANs. The implementation is available at https://github.com/AmirNoori68/PU-GKAN .

The authors' abstract, as published at the source. Neurocomputing, 2026 · DOI ↗