mlsanityData to 7 Oct 2026

fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions

arXiv 2406.07456 ↗First posted 11 Jun 2024cs.LG cs.NA math.NAcited by 60

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Abstract (arXiv)

Recent advancements in neural network design have given rise to the development of Kolmogorov-Arnold Networks (KANs), which enhance speed, interpretability, and precision. This paper presents the Fractional Kolmogorov-Arnold Network (fKAN), a novel neural network architecture that incorporates the distinctive attributes of KANs with a trainable adaptive fractional-orthogonal Jacobi function as its basis function. By leveraging the unique mathematical properties of fractional Jacobi functions, including simple derivative formulas, non-polynomial behavior, and activity for both positive and negative input values, this approach ensures efficient learning and enhanced accuracy. The proposed architecture is evaluated across a range of tasks in deep learning and physics-informed deep learning. Precision is tested on synthetic regression data, image classification, image denoising, and sentiment analysis. Additionally, the performance is measured on various differential equations, including ordinary, partial, and fractional delay differential equations. The results demonstrate that integrating fractional Jacobi functions into KANs significantly improves training speed and performance across diverse fields and applications.

Key facts. Cited by 60 papers in the map (data through 7 Oct 2026). Builds on 36 of its 104 references that are in the map. First posted on arXiv 11 Jun 2024.

Builds on

36 of 104 references.

References · 36

  1. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  2. 2015Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classificationcited by 3,590
  3. 2016Gaussian Error Linear Units (GELUs)cited by 3,431
  4. 2016Matching Networks for One Shot Learningcited by 2,246
  5. 2015Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)cited by 1,061
  6. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  7. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 318
  8. 2015Unitary Evolution Recurrent Neural Networkscited by 291
  9. 2019Universal Approximation with Deep Narrow Networkscited by 198
  10. 2022How Powerful are Spectral Graph Neural Networkscited by 169
  11. 2018Understanding Batch Normalizationcited by 157
  12. 2024Kolmogorov-Arnold Networks are Radial Basis Function Networkscited by 125
24 more
  1. 2024Wav-KAN: Wavelet Kolmogorov-Arnold Networkscited by 118
  2. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 115
  3. 2020A survey on modern trainable activation functionscited by 89
  4. 2020The Kolmogorov-Arnold representation theorem revisitedcited by 89
  5. 2024DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problemscited by 82
  6. 2024TKAN: Temporal Kolmogorov-Arnold Networkscited by 70
  7. 2024A Temporal Kolmogorov-Arnold Transformer for Time Series Forecastingcited by 50
  8. 2024Kolmogorov-Arnold Network for Satellite Image Classification in Remote Sensingcited by 46
  9. 2023Toeplitz Neural Network for Sequence Modelingcited by 43
  10. 2024Enhancing Graph Collaborative Filtering with FourierKAN Feature Transformationcited by 41
  11. 2023Graph Neural Networks with Learnable and Optimal Polynomial Basescited by 36
  12. 2022ExSpliNet: An interpretable and expressive spline-based neural networkcited by 33
  13. 2022How important are activation functions in regression and classification? A survey, performance comparison, and future directionscited by 29
  14. 2024Smooth Kolmogorov Arnold networks enabling structural knowledge representationcited by 22
  15. 2022Approximation of Lipschitz Functions using Deep Spline Neural Networkscited by 15
  16. 2024iKAN: Global Incremental Learning with KAN for Human Activity Recognition Across Heterogeneous Datasetscited by 14
  17. 2016Learning activation functions from data using cubic spline interpolationcited by 13
  18. 2020Universal Approximation Theorem for Equivariant Maps by Group CNNscited by 12
  19. 2021Universal Approximation Theorem for Neural Networkscited by 7
  20. 2023deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equationscited by 7
  21. 2021Batch Normalization Preconditioning for Neural Network Trainingcited by 5
  22. 2023LON-GNN: Spectral GNNs with Learnable Orthonormal Basiscited by 4
  23. 2021A structured proof of Kolmogorov's Superposition Theoremcited by 3
  24. 2024Accelerating Fractional PINNs using Operational Matrices of Derivativecited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 30; 2025: 19; 2026: 11.

60 papers.

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  1. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 116
  2. 2024A Survey on Kolmogorov-Arnold Networkcited by 67
  3. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 37
  4. 2024KANQAS: Kolmogorov-Arnold Network for Quantum Architecture Searchcited by 36
  5. 2024On the expressiveness and spectral bias of KANscited by 36
  6. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  7. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 31
  8. 2024Reduced Effectiveness of Kolmogorov-Arnold Networks on Functions with Noisecited by 17
  9. 2024SineKAN: Kolmogorov-Arnold Networks Using Sinusoidal Activation Functionscited by 15
  10. 2024DropKAN: Regularizing KANs by masking post-activationscited by 14
  11. 2024Higher-order-ReLU-KANs (HRKANs) for solving physics-informed neural networks (PINNs) more accurately, robustly and fastercited by 13
  12. 2024Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networkscited by 11
38 more
  1. 2025Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networkscited by 10
  2. 2024Data-driven model discovery with Kolmogorov-Arnold networkscited by 9
  3. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
  4. 2024DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networkscited by 7
  5. 2024KAN versus MLP on Irregular or Noisy Functionscited by 7
  6. 2024Are KANs Effective for Multivariate Time Series Forecasting?cited by 6
  7. 2024Architectural Scaling Surpass Basis Complexity? Efficient KANs with Single-Parameter Designcited by 6
  8. 2024Rethinking the Function of Neurons in KANscited by 5
  9. 2024FourierKAN outperforms MLP on Text Classification Head Fine-tuningcited by 5
  10. 2024Exploring Kolmogorov-Arnold networks for realistic image sharpness assessmentcited by 5
  11. 2024KAN we improve on HEP classification tasks? Kolmogorov-Arnold Networks applied to an LHC physics examplecited by 4
  12. 2025KAC: Kolmogorov-Arnold Classifier for Continual Learningcited by 4
  13. 2024KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problemscited by 3
  14. 2024KAN v.s. MLP for Offline Reinforcement Learningcited by 3
  15. 2024Evaluating Federated Kolmogorov-Arnold Networks on Non-IID Datacited by 3
  16. 2024LArctan-SKAN: Simple and Efficient Single-Parameterized Kolmogorov-Arnold Networks using Learnable Trigonometric Functioncited by 3
  17. 2025AF-KAN: Activation Function-Based Kolmogorov-Arnold Networks for Efficient Representation Learningcited by 3
  18. 2025Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometriescited by 3
  19. 2025TabKAN: Advancing Tabular Data Analysis using Kolmogorov-Arnold Networkcited by 3
  20. 2025Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learningcited by 3
  21. 2025Automatic Grid Updates for Kolmogorov-Arnold Networks using Layer Histogramscited by 3
  22. 2026Making Gaussian Kolmogorov-Arnold Networks Reliable and Accuratecited by 3
  23. 2024Kolmogorov Arnold Networks in Fraud Detection: Bridging the Gap Between Theory and Practicecited by 2
  24. 2024MLP-KAN: Unifying Deep Representation and Function Learningcited by 2
  25. 2025Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networkscited by 2
  26. 2025Dynamic Hedging Strategies in Derivatives Markets with LLM-Driven Sentiment and News Analyticscited by 2
  27. 2025Fully analogue in-memory neural computing via quantum tunneling effectcited by 2
  28. 2024Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularitiescited by 1
  29. 2025Low Tensor-Rank Adaptation of Kolmogorov--Arnold Networkscited by 1
  30. 2025Efficient Implicit Neural Compression of Point Clouds via Learnable Activation in Latent Spacecited by 1
  31. 2025Taylor expansion-based Kolmogorov-Arnold network for blind image quality assessmentcited by 1
  32. 2025Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networkscited by 1
  33. 2025INR-Bench: A Unified Benchmark for Implicit Neural Representations in Multi-Domain Regression and Reconstructioncited by 1
  34. 2026Hardware-Oriented Inference Complexity of Kolmogorov-Arnold Networkscited by 1
  35. 2026Partition-of-Unity Gaussian Kolmogorov-Arnold Networkscited by 1
  36. 2026SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functionscited by 1
  37. 2026RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basiscited by 1
  38. 2024On the Convergence of (Stochastic) Gradient Descent for Kolmogorov--Arnold Networkscited by 0

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