Partition-of-Unity Gaussian Kolmogorov-Arnold Networks

원문 초록

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 \epsilon. 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 \epsilon, 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 Matern RBF bases. PU-GKAN also improves physics-informed neural networks for Helmholtz and wave equations, with validation-error reductions of up to 90%. 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.