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.

💡 Research Summary

The paper introduces Free‑RBF‑KAN, a novel variant of Kolmogorov‑Arnold Networks (KANs) that replaces the computationally expensive B‑spline basis with adaptive radial basis functions (RBFs) while preserving the expressive power of the original architecture. Traditional KANs rely on the De Boor algorithm to evaluate B‑splines, which incurs significant overhead during both training and inference, especially when many knots are required for high‑resolution modeling. Recent RBF‑based KANs mitigate this cost but typically fix the RBF centers and smoothness parameters, leading to a loss of approximation fidelity compared with spline‑based KANs.

Free‑RBF‑KAN addresses these shortcomings through three key design innovations. First, it adopts a “free‑knot” mechanism: RBF centers are treated as learnable parameters that can move freely within a predefined domain. To keep them inside the domain while remaining differentiable, the authors re‑parameterize each raw center ˜c with a smooth monotonic mapping (tanh) that maps ˜c to a valid centroid c. This allows the mesh to adapt dynamically to data‑driven activation patterns, starting from a uniform initialization and evolving during training. Second, the smoothness (bandwidth) σ of each RBF is also learned by parameterizing an unconstrained scalar ˜σ and setting σ = exp(˜σ), guaranteeing positivity and smooth gradients. The architecture supports both Gaussian and Matérn kernels, enabling control over differentiability (e.g., Matérn ν = 5/2 yields C³ continuity). Third, each layer incorporates scaling matrices (W_rbf) and two activation functions: a SiLU nonlinearity for hidden layers and a sigmoid (or identity for the output) to improve training stability, mirroring the successful design of recent KAN variants.

On the theoretical side, the authors extend classical universal approximation results to the RBF‑KAN setting. By combining the Kolmogorov‑Arnold representation theorem with Pinkus’s density theorem (and its univariate specialization by Leshno et al.), they prove that any continuous multivariate function on a compact domain can be approximated arbitrarily well by a network composed of non‑polynomial RBFs arranged according to the KAN superposition structure—what they term “Non‑Polynomial KAN” (NP‑KAN). This is the first formal universal approximation proof for the RBF‑KAN family. Additionally, a Neural Tangent Kernel (NTK) analysis shows that Free‑RBF‑KAN does not suffer from spectral bias; its kernel spectrum remains balanced across frequencies, which is crucial for learning high‑frequency components in physics‑informed problems.

Empirical evaluation spans three representative domains. (1) Multiscale regression tasks demonstrate that Free‑RBF‑KAN matches or surpasses B‑spline KAN accuracy while reducing training time by roughly 2–3× and inference time by 1.5–2×. (2) Physics‑informed neural networks (PINNs) for the heat equation and Helmholtz equation reveal that the baseline PINN fails to converge on the heat problem and yields larger errors on Helmholtz, even with two orders of magnitude more parameters. Free‑RBF‑KAN, by contrast, converges reliably and attains lower residuals with far fewer parameters. (3) Operator learning via DeepONet shows that a Free‑RBF‑KAN trunk network (Free‑RBF‑KAN‑ONet) achieves lower approximation error than standard DeepONet and DeepOKAN while using 30–40 % fewer trainable weights. Across all experiments, the proposed model consistently outperforms both the original spline‑based KAN and prior RBF‑KAN variants.

In summary, Free‑RBF‑KAN delivers (i) adaptive mesh refinement through learnable centroids, (ii) trainable smoothness for each RBF, (iii) rigorous universal approximation guarantees and bias‑free NTK properties, and (iv) demonstrated efficiency and accuracy on high‑dimensional scientific computing tasks. The work positions Free‑RBF‑KAN as a compelling, high‑performance alternative for structured modeling in scientific machine learning, offering a practical balance between computational speed and expressive fidelity.