X-KAN: Optimizing Local Kolmogorov-Arnold Networks via Evolutionary Rule-Based Machine Learning

TL;DR

X-KAN introduces a novel evolutionary rule-based framework that optimizes multiple local Kolmogorov-Arnold Networks, significantly improving function approximation for complex and discontinuous functions over traditional neural network methods.

Contribution

The paper presents X-KAN, a new method combining local KAN models with XCSF's adaptive partitioning, enabling better approximation of complex functions with fewer rules.

Findings

X-KAN outperforms conventional methods in approximation accuracy.

X-KAN effectively handles locally complex and discontinuous functions.

X-KAN uses a compact rule set averaging 7.2 rules.

Abstract

Function approximation is a critical task in various fields. However, existing neural network approaches struggle with locally complex or discontinuous functions due to their reliance on a single global model covering the entire problem space. We propose X-KAN, a novel method that optimizes multiple local Kolmogorov-Arnold Networks (KANs) through an evolutionary rule-based machine learning framework called XCSF. X-KAN combines KAN's high expressiveness with XCSF's adaptive partitioning capability by implementing local KAN models as rule consequents and defining local regions via rule antecedents. Our experimental results on artificial test functions and real-world datasets demonstrate that X-KAN significantly outperforms conventional methods, including XCSF, Multi-Layer Perceptron, and KAN, in terms of approximation accuracy. Notably, X-KAN effectively handles functions with locally complex or discontinuous structures that are challenging for conventional KAN, using a compact set of rules (average 7.2 $\pm$ 2.3 rules). These results validate the effectiveness of using KAN as a local model in XCSF, which evaluates the rule fitness based on both accuracy and generality. Our X-KAN implementation is available at https://github.com/YNU-NakataLab/X-KAN.