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Exploring the Potential of Polynomial Basis Functions in Kolmogorov-Arnold Networks: A Comparative Study of Different Groups of Polynomials

arXiv 2406.02583 ↗First posted 30 May 2024cs.LG cs.AI cs.CVcited by 33

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Abstract (arXiv)

This paper presents a comprehensive survey of 18 distinct polynomials and their potential applications in Kolmogorov-Arnold Network (KAN) models as an alternative to traditional spline-based methods. The polynomials are classified into various groups based on their mathematical properties, such as orthogonal polynomials, hypergeometric polynomials, q-polynomials, Fibonacci-related polynomials, combinatorial polynomials, and number-theoretic polynomials. The study aims to investigate the suitability of these polynomials as basis functions in KAN models for complex tasks like handwritten digit classification on the MNIST dataset. The performance metrics of the KAN models, including overall accuracy, Kappa, and F1 score, are evaluated and compared. The Gottlieb-KAN model achieves the highest performance across all metrics, suggesting its potential as a suitable choice for the given task. However, further analysis and tuning of these polynomials on more complex datasets are necessary to fully understand their capabilities in KAN models. The source code for the implementation of these KAN models is available at https://github.com/seydi1370/Basis_Functions .

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). Builds on 4 of its 40 references that are in the map. First posted on arXiv 30 May 2024.

Builds on

4 of 40 references.

References · 4

  1. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  2. 2024Wav-KAN: Wavelet Kolmogorov-Arnold Networkscited by 118
  3. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 115
  4. 2024Smooth Kolmogorov Arnold networks enabling structural knowledge representationcited by 22

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 12; 2025: 12; 2026: 9.

33 papers.

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  1. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 116
  2. 2024BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networkscited by 43
  3. 2024On the expressiveness and spectral bias of KANscited by 36
  4. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 31
  5. 2024FC-KAN: Function Combinations in Kolmogorov-Arnold Networkscited by 18
  6. 2024PointNet with KAN versus PointNet with MLP for 3D Classification and Segmentation of Point Setscited by 15
  7. 2024Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networkscited by 11
  8. 2025Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networkscited by 10
  9. 2024Data-driven model discovery with Kolmogorov-Arnold networkscited by 9
  10. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
  11. 2024Residual Kolmogorov-Arnold Network for Enhanced Deep Learningcited by 5
  12. 2025Medical Image Classification with KAN-Integrated Transformers and Dilated Neighborhood Attentioncited by 5
21 more
  1. 2025AF-KAN: Activation Function-Based Kolmogorov-Arnold Networks for Efficient Representation Learningcited by 3
  2. 2025Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometriescited by 3
  3. 2026Making Gaussian Kolmogorov-Arnold Networks Reliable and Accuratecited by 3
  4. 2025Fully analogue in-memory neural computing via quantum tunneling effectcited by 2
  5. 2024Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularitiescited by 1
  6. 2025Low Tensor-Rank Adaptation of Kolmogorov--Arnold Networkscited by 1
  7. 2025FunKAN: Functional Kolmogorov-Arnold Network for Medical Image Enhancement and Segmentationcited by 1
  8. 2025INR-Bench: A Unified Benchmark for Implicit Neural Representations in Multi-Domain Regression and Reconstructioncited by 1
  9. 2026Partition-of-Unity Gaussian Kolmogorov-Arnold Networkscited by 1
  10. 2026SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functionscited by 1
  11. 2026RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basiscited by 1
  12. 2024On the Convergence of (Stochastic) Gradient Descent for Kolmogorov--Arnold Networkscited by 0
  13. 2025Polynomial, trigonometric, and tropical activationscited by 0
  14. 2025Degree-Optimized Cumulative Polynomial Kolmogorov-Arnold Networkscited by 0
  15. 2025Improving KAN with CDF normalization to quantilescited by 0
  16. 2025Combinations of Fast Activation and Trigonometric Functions in Kolmogorov-Arnold Networkscited by 0
  17. 2026LTBs-KAN: Linear-Time B-splines Kolmogorov-Arnold Networkscited by 0
  18. 2026Adaptive RBF-KAN: A Comparative Evaluation of Dynamic Shape Parameters in Kolmogorov-Arnold Networkscited by 0
  19. 2026Hardware-Aware Functional Kolmogorov-Arnold Networks for Efficient Medical Image Enhancement and Segmentationcited by 0
  20. 2026SW-KAN: Kolmogorov-Arnold Networks with Stieltjes-Wigert q-Orthogonal Polynomialscited by 0
  21. 2026Neural scaling laws and evolution of learnable activation functions of Kolmogorov-Arnold networkscited by 0