Free-RBF-KAN: Kolmogorov-Arnold Networks with Adaptive Radial Basis Functions for Efficient Function Learning

2026-01-12

AI Summary

This work proposes Free-RBF-KAN, a novel Kolmogorov–Arnold Network (KAN) architecture that addresses the high computational cost of conventional B-spline-based KANs and the accuracy–efficiency trade-off in existing RBF-KAN variants. By introducing learnable radial basis functions, adaptive grids, and trainable smoothness parameters, Free-RBF-KAN achieves significantly improved computational efficiency while preserving high expressive power. We establish, for the first time, a general universal approximation theorem for RBF-KANs, providing theoretical grounding for their representational capacity. Empirical evaluations across diverse tasks—including multiscale function approximation, physics-informed learning, and operator learning for partial differential equations—demonstrate that Free-RBF-KAN matches the accuracy of the original KAN while substantially accelerating both training and inference.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Kolmogorov-Arnold Networks (KANs) offer a promising framework for approximating complex nonlinear functions, yet the original B-spline formulation suffers from significant computational overhead due to De Boor algorithm. While recent RBF-based variants improve efficiency, they often sacrifice the approximation accuracy inherent in the original spline-based design. To bridge this gap, we propose Free-RBF-KAN, an architecture that integrates adaptive learning grids and trainable smoothness parameters to enable expressive, high-resolution function approximation. Our method utilizes learnable RBF shapes that dynamically align with activation patterns, and we provide the first formal universal approximation proof for the RBF-KAN family. Empirical evaluations across multiscale regression, physics-informed PDEs, and operator learning demonstrate that Free-RBF-KAN can achieve accuracy comparable to its B-spline counterparts while delivering significantly faster training and inference. These results establish Free-RBF-KAN as an efficient and adaptive alternative for high-dimensional structured modeling tasks.

Problem

• Kolmogorov-Arnold Networks
• Radial Basis Functions
• Function Approximation
• Computational Efficiency
• Accuracy

Innovation

• Kolmogorov-Arnold Networks
• Radial Basis Functions
• Adaptive Grids
• Trainable Smoothness
• Function Approximation