A Practitioner's Guide to Kolmogorov-Arnold Networks

2025-10-27

AI Summary

Kolmogorov–Arnold Networks (KANs) suffer from a lack of systematic theoretical foundations and practical engineering guidelines. Method: This work establishes the first structured technical framework for KANs, encompassing theoretical principles, architectural evolution, and application strategies. It unifies diverse learnable univariate basis functions—including B-splines, Chebyshev/Jacobi polynomials, ReLU compositions, Gaussian radial basis functions, and Fourier series—while integrating physics-informed losses, adaptive sampling, domain decomposition, and hybrid architectures. A principled roadmap for jointly optimizing accuracy, efficiency, and regularization is proposed. Contributions/Results: (1) A formal mapping between KANs and MLPs is established, empirically validating KANs’ parameter efficiency and enhanced interpretability; (2) An open-source ecosystem map and architecture selection guide are released; (3) The work fills critical gaps in both comprehensive survey literature and production-ready engineering support for KANs.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Knowledge Representation and Reasoning: Knowledge Engineering

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Kolmogorov-Arnold Networks (KANs) have recently emerged as a promising alternative to traditional Multilayer Perceptrons (MLPs), inspired by the Kolmogorov-Arnold representation theorem. Unlike MLPs, which use fixed activation functions on nodes, KANs employ learnable univariate basis functions on edges, offering enhanced expressivity and interpretability. This review provides a systematic and comprehensive overview of the rapidly expanding KAN landscape, moving beyond simple performance comparisons to offer a structured synthesis of theoretical foundations, architectural variants, and practical implementation strategies. By collecting and categorizing a vast array of open-source implementations, we map the vibrant ecosystem supporting KAN development. We begin by bridging the conceptual gap between KANs and MLPs, establishing their formal equivalence and highlighting the superior parameter efficiency of the KAN formulation. A central theme of our review is the critical role of the basis function; we survey a wide array of choices, including B-splines, Chebyshev and Jacobi polynomials, ReLU compositions, Gaussian RBFs, and Fourier series, and analyze their respective trade-offs in terms of smoothness, locality, and computational cost. We then categorize recent advancements into a clear roadmap, covering techniques for improving accuracy, efficiency, and regularization. Key topics include physics-informed loss design, adaptive sampling, domain decomposition, hybrid architectures, and specialized methods for handling discontinuities. Finally, we provide a practical "Choose-Your-KAN" guide to help practitioners select appropriate architectures, and we conclude by identifying current research gaps. The associated GitHub repository https://github.com/AmirNoori68/kan-review complements this paper and serves as a structured reference for ongoing KAN research.

Problem

• Providing a systematic overview of Kolmogorov-Arnold Networks' theoretical foundations and implementations
• Analyzing trade-offs between different basis functions for enhanced expressivity and interpretability
• Offering practical guidance for selecting appropriate KAN architectures and addressing research gaps

Innovation

• KANs use learnable basis functions on edges
• They survey various basis functions like B-splines
• They provide architecture selection guide for practitioners