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

TL;DR

Free-RBF-KAN replaces the computationally expensive B-spline basis of Kolmogorov–Arnol'd Networks with trainable radial basis functions and adaptive grids. By learning centroids and kernel sharpness, it achieves universal approximation while reducing training time and improving accuracy on regression, PINN, and operator-learning problems. The work establishes a twofold theoretical foundation: an extended universal approximation theorem for RBF-KAN and an NTK analysis indicating no spectral bias, supported by extensive experiments including heat conduction and Helmholtz PDEs, as well as DeepONet integration. Overall, Free-RBF-KAN offers a practical, scalable architecture that balances expressivity, adaptivity, and efficiency for high-dimensional, structured modeling tasks.

Abstract

Kolmogorov-Arnold Networks (KANs) have shown strong potential for efficiently approximating complex nonlinear functions. However, the original KAN formulation relies on B-spline basis functions, which incur substantial computational overhead due to De Boor's algorithm. To address this limitation, recent work has explored alternative basis functions such as radial basis functions (RBFs) that can improve computational efficiency and flexibility. Yet, standard RBF-KANs often sacrifice accuracy relative to the original KAN design. In this work, we propose Free-RBF-KAN, a RBF-based KAN architecture that incorporates adaptive learning grids and trainable smoothness to close this performance gap. Our method employs freely learnable RBF shapes that dynamically align grid representations with activation patterns, enabling expressive and adaptive function approximation. Additionally, we treat smoothness as a kernel parameter optimized jointly with network weights, without increasing computational complexity. We provide a general universality proof for RBF-KANs, which encompasses our Free-RBF-KAN formulation. Through a broad set of experiments, including multiscale function approximation, physics-informed machine learning, and PDE solution operator learning, Free-RBF-KAN achieves accuracy comparable to the original B-spline-based KAN while delivering faster training and inference. These results highlight Free-RBF-KAN as a compelling balance between computational efficiency and adaptive resolution, particularly for high-dimensional structured modeling tasks.

Figures (8)

• Figure 1: (Left) Analytical solution; (Middle) Prediction by Free-RBF-KAN; (Right) Error residual.
• Figure 2: The approximation of $f$ in \ref{['eq:f']} using MLP, KAN, RBFKAN, and Free-RBFKAN.
• Figure 3: The training loss of approximating $f$ in \ref{['eq:f']} using MLP, KAN, RBF-KAN, and Free-RBF-KAN.
• Figure 4: The NTK analysis on the spectral bias of MLP, KAN, RBF-KAN, Free-RBF-KAN in approximating $f$ in \ref{['eq:f']}
• Figure 5: Validation loss during training on MNIST dataset.

Theorems & Definitions (7)

• Lemma 4.1: Kolmogorov-Arnold Representation Theorem kolmogorov2009representationarnol1957functions
• Theorem 4.2: Pinkus Theorem Pinkus_1999
• Lemma 4.3: Univariate Density leshno1993multilayer
• Remark 4.4
• Theorem 4.5: Universal Approximation of NP-KAN
• proof
• Corollary 4.6: Universal Approximation of RBF-KAN