🤖 machine learning

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

Free-RBF-KAN introduces an efficient Kolmogorov-Arnold Network variant that replaces computationally expensive B-splines with adaptive, learnable Radial Basis Functions to achieve high-accuracy function approximation with significantly faster training and inference speeds.

Original authors: Shao-Ting Chiu, Siu Wun Cheung, Ulisses Braga-Neto, Chak Shing Lee, Rui Peng Li

Published 2026-02-10

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Shao-Ting Chiu, Siu Wun Cheung, Ulisses Braga-Neto, Chak Shing Lee, Rui Peng Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to draw a highly detailed, wavy landscape on a piece of paper. To do this perfectly, you have two main choices: you can use a ruler and a set of pre-set templates (like stencils), or you can use a set of flexible, adjustable brushes.

This paper introduces a new "digital brush" called Free-RBF-KAN. To understand why it’s special, we first need to look at the two "artists" that came before it.

1. The Old Artist: The "Stencil" Method (Original KAN)

The original Kolmogorov-Arnold Network (KAN) is like an artist who uses very precise, mathematical stencils (called B-splines) to draw curves.

The Good: It is incredibly accurate and follows mathematical rules beautifully.
The Bad: It is very "stiff." Because the stencils are pre-set and mathematically complex, the artist spends a huge amount of time and energy adjusting the stencils just to get a single curve right. It’s slow, heavy, and computationally "expensive."

2. The Second Artist: The "Fixed Brush" Method (RBF-KAN)

To speed things up, someone suggested using Radial Basis Functions (RBFs). Think of these as soft, circular ink blots. Instead of rigid stencils, you just dab these ink blots on the paper to create shapes.

The Good: It is much faster! You aren't fighting with complex stencils; you’re just dabbing ink.
The Bad: Because the ink blots are "fixed" (you can't easily change their shape or where they land), the artist often misses the fine details. It’s fast, but it’s a bit "blurry."

3. The New Hero: The "Smart Brush" (Free-RBF-KAN)

The authors of this paper created Free-RBF-KAN. This is like giving the artist a "Smart Brush" that can do two amazing things:

Adaptive Meshing (The "Where"): The brush can decide exactly where to land on the paper. If there is a sudden, sharp mountain in the landscape, the brush moves more points there to capture the detail. If the area is a flat plain, it spreads out to save time.
Adaptive Smoothness (The "How"): The brush can change its own texture. It can be a fine-tipped needle for sharp edges or a wide, soft sponge for smooth hills.

In short: It is as fast as the "Fixed Brush" method, but as accurate as the "Stencil" method.

Why does this matter in the real world?

The researchers didn't just play with digital ink; they tested this "Smart Brush" on some very difficult "blueprints":

Predicting Physics (The Weather/Heat): They used it to solve complex equations about how heat moves. While older AI models struggled or were too slow, Free-RBF-KAN zoomed through, capturing the "waves" of heat accurately.
Learning Patterns (MNIST): They tested it on recognizing handwritten numbers. While it’s still a work in progress compared to standard AI, it showed it could learn much more efficiently than previous versions of this specific technology.
Complex Operators (DeepONet): They used it to learn how entire systems change over time (like how a chemical reaction spreads). It outperformed the old methods while using much less "brainpower" (fewer parameters).

The "Big Idea" Summary

If traditional AI is like a massive, heavy encyclopedia that is hard to move, Free-RBF-KAN is like a high-tech, lightweight tablet. It’s small, it’s fast, it’s incredibly smart, and it can reshape itself to fit whatever problem you throw at it—whether you're drawing a simple line or simulating the complex physics of the universe.

Technical Summary: Free-RBF-KAN

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

1. Problem Statement

The original Kolmogorov-Arnold Network (KAN) architecture, while theoretically powerful due to its use of univariate B-spline basis functions, faces two primary practical limitations:

Computational Overhead: The use of B-splines requires De Boor’s algorithm for computation and complex domain rescaling, making training and inference significantly slower than traditional Multi-Layer Perceptrons (MLPs).
Rigidity of Basis Functions: While recent variants have attempted to use Radial Basis Functions (RBFs) to improve speed, they often suffer from a loss in approximation accuracy because they use fixed grid centers and fixed smoothness parameters, failing to adapt to the underlying data patterns.

2. Methodology

The authors propose Free-RBF-KAN, an architecture that replaces B-splines with adaptive Gaussian RBFs within the Kolmogorov-Arnold superposition framework. The core technical innovations include:

Adaptive Meshing (Free Knots): Unlike B-splines, which require strictly ordered grid points, RBFs allow for "free" centroids. The model learns the optimal placement of these centroids ( $c$ ) via a reparameterization technique (using a bounded monotonic function like tanh) to ensure they remain within a prescribed domain during gradient-based optimization.
Adaptive Smoothness: The model treats the RBF smoothness parameter ( $\sigma$ ) as a trainable variable. By using an unconstrained parameter $\tilde{\sigma}$ and mapping it via $\sigma = \exp(\tilde{\sigma})$ , the network can dynamically adjust the "sharpness" of its activation functions to match the target function's complexity.
Hierarchical Structure: It maintains the KAN structure where multivariate functions are constructed through the superposition of univariate RBF components, allowing it to scale to high-dimensional problems more effectively than traditional RBF networks.
Residual Connections: The architecture incorporates optional scaling matrices and nonlinear activations (SiLU/Sigmoid) to improve optimization landscapes.

3. Key Contributions

Architectural Innovation: Developed a "free" RBF formulation that combines adaptive meshing with tunable shape parameters, decoupling the mesh from fixed uniform structures.
Theoretical Foundation:
- Provided the first formal universal approximation proof for the RBF-KAN family.
- Conducted a Neural Tangent Kernel (NTK) analysis, proving that Free-RBF-KAN does not exhibit "spectral bias" (the tendency of neural networks to learn low-frequency components before high-frequency ones), similar to the original KAN.
Efficiency: Demonstrated a method that achieves B-spline-level accuracy with significantly faster training and inference speeds.

4. Empirical Results

The authors evaluated the model across three major domains:

Functional Approximation & Regression: In non-smooth function approximation and high-dimensional tasks (MNIST), Free-RBF-KAN outperformed standard KAN and RBF-KAN. On MNIST, it significantly narrowed the performance gap between KAN-based models and MLPs.
Physics-Informed Machine Learning (PINNs):
- 2D Heat Conduction: Free-RBF-KAN successfully learned high-frequency forcing terms where standard PINNs failed to converge.
- 2D Helmholtz Equation: It outperformed both the original KAN (which struggled with smoothness demands) and the MLP baseline in capturing the solution.
Operator Learning (DeepONet): When used as a "trunk network" in DeepONet for reaction-diffusion PDEs, Free-RBF-KAN achieved the highest accuracy with fewer parameters than standard KAN or MLP-based architectures.

5. Significance

Free-RBF-KAN represents a major step forward in making Kolmogorov-Arnold architectures practical for scientific computing. By bridging the gap between the high expressivity of splines and the computational efficiency of RBFs, it provides a robust tool for solving complex partial differential equations (PDEs) and learning operators. Its ability to adapt its own "resolution" (via centroids and smoothness) makes it uniquely suited for multiscale physical phenomena where traditional fixed-grid methods struggle.