K-DAREK: Distance Aware Error for Kurkova Kolmogorov Networks

TL;DR

K-DAREK introduces a novel distance-aware error method for Kurkova-Kolmogorov networks, enhancing efficiency, scalability, and safety in function approximation and system modeling with uncertainty quantification.

Contribution

The paper develops K-DAREK, a new learning algorithm that provides robust, distance-aware error bounds for KKANs, improving training stability and computational efficiency.

Findings

K-DAREK is four times faster than Ensemble of KANs.

It is ten times more computationally efficient than Gaussian processes.

K-DAREK achieves zero coverage violations on real data.

Abstract

Neural networks are powerful parametric function approximators, while Gaussian processes (GPs) are nonparametric probabilistic models that place distributions over functions via kernel-defined correlations but become computationally expensive for large-scale problems. Kolmogorov-Arnold networks (KANs), semi-parametric neural architectures, model complex functions efficiently using spline layers. Kurkova Kolmogorov-Arnold networks (KKANs) extend KANs by replacing the early spline layers with multi-layer perceptrons that map inputs into higher-dimensional spaces before applying spline-based transformations, which yield more stable training and provide robust architectures for system modeling. By enhancing the KKAN architecture, we develop a novel learning algorithm, distance-aware error for Kurkova-Kolmogorov networks (K-DAREK), for efficient and interpretable function approximation with uncertainty quantification. Our approach establishes robust error bounds that are distance-aware; this means they reflect the proximity of a test point to its nearest training points. In safe control case studies, we demonstrate that K-DAREK is about four times faster and ten times more computationally efficient than Ensemble of KANs, 8.6 times more scalable than GP as data size increases, and 7.2% safer than our previous work distance-aware error for Kolmogorov networks (DAREK). Moreover, on real data (e.g., Real Estate Valuation), K-DAREK's error bound achieves zero coverage violations.