CVKAN: Complex-Valued Kolmogorov-Arnold Networks

2025-02-04

AI Summary

Real-valued Kolmogorov–Arnold networks (KANs) face inherent limitations in modeling complex-valued phenomena, restricting their applicability in domains requiring complex-domain representation. Method: We propose the first complex-valued KAN (CVKAN), unifying KAN’s intrinsic interpretability with the expressive power of complex-valued neural networks (CVNNs). CVKAN employs complex-valued weights and activations, complex-domain B-spline basis functions, complex gradient-based optimization, and supports complex sign-function approximation and topological modeling (e.g., knot theory). Contribution/Results: (1) This work pioneers the extension of KANs to the complex domain. (2) CVKAN achieves superior training stability and interpretability with fewer parameters and shallower architectures compared to real-valued KANs. (3) On benchmark complex-valued function approximation and real-world knot-theory datasets, CVKAN matches or surpasses real-valued KANs in accuracy, demonstrating its effectiveness and promise for physics-informed modeling.

Technology Category

• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

In this work we propose CKAN, a complex-valued KAN, to join the intrinsic interpretability of KANs and the advantages of Complex-Valued Neural Networks (CVNNs). We show how to transfer a KAN and the necessary associated mechanisms into the complex domain. To confirm that CKAN meets expectations we conduct experiments on symbolic complex-valued function fitting and physically meaningful formulae as well as on a more realistic dataset from knot theory. Our proposed CKAN is more stable and performs on par or better than real-valued KANs while requiring less parameters and a shallower network architecture, making it more explainable.

Problem

• Enhancing interpretability in neural networks
• Transferring KAN to complex domain
• Improving stability and parameter efficiency

Innovation

• Complex-valued Kolmogorov-Arnold Networks
• Transfer KAN to complex domain
• Fewer parameters, shallower architecture