mlsanityData to 7 Oct 2026

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

arXiv 2601.07760 ↗First posted 12 Jan 2026cs.LG cs.NA math.NAcited by 2

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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.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 21 of its 42 references that are in the map. First posted on arXiv 12 Jan 2026.

Builds on

21 of 42 references.

References · 21

  1. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  2. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  3. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  4. 2024Kolmogorov-Arnold Networks are Radial Basis Function Networkscited by 125
  5. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 119
  6. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 115
  7. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 103
  8. 2024DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problemscited by 82
  9. 2024KAN or MLP: A Fairer Comparisoncited by 71
  10. 2024A Survey on Kolmogorov-Arnold Networkcited by 67
  11. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 64
  12. 2024BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networkscited by 43
9 more
  1. 2024On the expressiveness and spectral bias of KANscited by 36
  2. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  3. 2024KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamicscited by 13
  4. 2024Gaussian Process Kolmogorov-Arnold Networkscited by 6
  5. 2025Free-Knots Kolmogorov-Arnold Network: On the Analysis of Spline Knots and Advancing Stabilitycited by 6
  6. 2024RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networkscited by 5
  7. 2025Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinementcited by 4
  8. 2023On the universal approximation property of radial basis function neural networkscited by 3
  9. 2023Solving multiscale elliptic problems by sparse radial basis function neural networkscited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Making Gaussian Kolmogorov-Arnold Networks Reliable and Accuratecited by 3
  2. 2026Partition-of-Unity Gaussian Kolmogorov-Arnold Networkscited by 1