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

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.