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

arXiv: https://arxiv.org/abs/2601.07760 · HTML: https://mlsanity.com/paper/2601.07760/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 21 of 21)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [KAN or MLP: A Fairer Comparison](https://mlsanity.com/paper/2407.16674/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.11173/) (2024)
- [On the expressiveness and spectral bias of KANs](https://mlsanity.com/paper/2410.01803/) (2024)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamics](https://mlsanity.com/paper/2412.16738/) (2024)
- [Gaussian Process Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.18397/) (2024)
- [Free-Knots Kolmogorov-Arnold Network: On the Analysis of Spline Knots and Advancing Stability](https://mlsanity.com/paper/2501.09283/) (2025)
- [RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.08367/) (2024)
- [Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement](https://mlsanity.com/paper/2505.18131/) (2025)
- [On the universal approximation property of radial basis function neural networks](https://mlsanity.com/paper/2304.02220/) (2023)
- [Solving multiscale elliptic problems by sparse radial basis function neural networks](https://mlsanity.com/paper/2309.03107/) (2023)

## Built on by (most cited first, 4 of 4)

- [Making Gaussian Kolmogorov-Arnold Networks Reliable and Accurate](https://mlsanity.com/paper/2604.21174/) (2026)
- [Partition-of-Unity Gaussian Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2604.23599/) (2026)
- [Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks](https://mlsanity.com/paper/2604.23765/) (2026)
- [Hybrid Parallel Bessel KAN Model: a Kolmogorov Arnold Network: integrated framework for developing a novel architecture to reconstruct the fused-basis approach via a parameter free quadratic interaction](https://mlsanity.com/paper/2610.17779/) (2026)

## Abstract (arXiv)

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.