Partition-of-Unity Gaussian Kolmogorov-Arnold Networks

• URL: http://arxiv.org/abs/2604.23599v2
• Date: Sun, 03 May 2026 07:38:23 GMT
• Title: Partition-of-Unity Gaussian Kolmogorov-Arnold Networks
• Abstract summary: We introduce the partition-of-unity Gaussian KAN (PU-GKAN) in which the Gaussian basis values on each edge are divided by their local sum over fixed centers.<n>This produces a partition-of-unity feature map with trainable coefficients, while preserving the standard edge-based KAN structure.<n> Numerical experiments show that PU-GKAN reduces sensitivity to (), improves validation accuracy for most smooth and moderately non-smooth targets, and gives more stable training behavior.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 \(ε\), 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.

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