# Free-Knots Kolmogorov-Arnold Network: On the Analysis of Spline Knots and Advancing Stability

arXiv: https://arxiv.org/abs/2501.09283 · HTML: https://mlsanity.com/paper/2501.09283/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 25 references that are in the map.
- First posted on arXiv 16 Jan 2025.

## Builds on (most cited first, 16 of 16)

- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [Multitask learning and benchmarking with clinical time series data](https://mlsanity.com/paper/1703.07771/) (2017)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [U-KAN Makes Strong Backbone for Medical Image Segmentation and Generation](https://mlsanity.com/paper/2406.02918/) (2024)
- [MultiBench: Multiscale Benchmarks for Multimodal Representation Learning](https://mlsanity.com/paper/2107.07502/) (2021)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Convolutional Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.13155/) (2024)
- [Kolmogorov-Arnold Transformer](https://mlsanity.com/paper/2409.10594/) (2024)
- [Mad Max: Affine Spline Insights into Deep Learning](https://mlsanity.com/paper/1805.06576/) (2018)
- [A Temporal Kolmogorov-Arnold Transformer for Time Series Forecasting](https://mlsanity.com/paper/2406.02486/) (2024)
- [Enhancing Graph Collaborative Filtering with FourierKAN Feature Transformation](https://mlsanity.com/paper/2406.01034/) (2024)
- [GraphKAN: Enhancing Feature Extraction with Graph Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.13597/) (2024)
- [The Upper Bound on Knots in Neural Networks](https://mlsanity.com/paper/1611.09448/) (2016)

## Built on by (most cited first, 6 of 6)

- [MatrixKAN: Parallelized Kolmogorov-Arnold Network](https://mlsanity.com/paper/2502.07176/) (2025)
- [Free-RBF-KAN: Kolmogorov-Arnold Networks with Adaptive Radial Basis Functions for Efficient Function Learning](https://mlsanity.com/paper/2601.07760/) (2026)
- [Uncertainty Quantification for Scientific Machine Learning using Sparse Variational Gaussian Process Kolmogorov-Arnold Networks (SVGP KAN)](https://mlsanity.com/paper/2512.05306/) (2025)
- [Clifford Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2602.05977/) (2026)
- [From Uniform to Learned Knots: A Study of Spline-Based Numerical Encodings for Tabular Deep Learning](https://mlsanity.com/paper/2604.05635/) (2026)
- [SW-KAN: Kolmogorov-Arnold Networks with Stieltjes-Wigert q-Orthogonal Polynomials](https://mlsanity.com/paper/2610.00050/) (2026)

## Abstract (arXiv)

Kolmogorov-Arnold Neural Networks (KANs) have gained significant attention in the machine learning community. However, their implementation often suffers from poor training stability and heavy trainable parameter. Furthermore, there is limited understanding of the behavior of the learned activation functions derived from B-splines. In this work, we analyze the behavior of KANs through the lens of spline knots and derive the lower and upper bound for the number of knots in B-spline-based KANs. To address existing limitations, we propose a novel Free Knots KAN that enhances the performance of the original KAN while reducing the number of trainable parameters to match the trainable parameter scale of standard Multi-Layer Perceptrons (MLPs). Additionally, we introduce new a training strategy to ensure C2 continuity of the learnable spline, resulting in smoother activation compared to the original KAN and improve the training stability by range expansion. The proposed method is comprehensively evaluated on 8 datasets spanning various domains, including image, text, time series, multimodal, and function approximation tasks. The promising results demonstrates the feasibility of KAN-based network and the effectiveness of proposed method.