• Wav-KAN: Wavelet Kolmogorov-Arnold Networks
• KAN or MLP: A Fairer Comparison - spline activation function. | [code](https://github.com/yu-rp/KANbeFair) ｜ 
• KAN: Kolmogorov-Arnold Networks - Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability. For accuracy, much smaller KANs can achieve comparable or better accuracy than much larger MLPs in data fitting and PDE solving. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful collaborators helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today's deep learning models which rely heavily on MLPs.
• KAN 2.0: Kolmogorov-Arnold Networks Meet Science
• KAN: Kolmogorov-Arnold Networks - Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability. For accuracy, much smaller KANs can achieve comparable or better accuracy than much larger MLPs in data fitting and PDE solving. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful collaborators helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today's deep learning models which rely heavily on MLPs.
• KAN or MLP: A Fairer Comparison - spline activation function. | [code](https://github.com/yu-rp/KANbeFair) ｜ 
• DropKAN: Regularizing KANs by masking post-activations - Arnold Networks) is a regularization method that prevents co-adaptation of activation function weights in Kolmogorov-Arnold Networks (KANs). DropKAN operates by randomly masking some of the post-activations within the KANs computation graph, while scaling-up the retained post-activations. We show that this simple procedure that require minimal coding effort has a regularizing effect and consistently lead to better generalization of KANs. | [code](https://github.com/Ghaith81/dropkan) ｜ 
• Rethinking the Function of Neurons in KANs - Arnold Networks (KANs) perform a simple summation motivated by the Kolmogorov-Arnold representation theorem, Our findings indicate that substituting the sum with the average function in KAN neurons results in significant performance enhancements compared to traditional KANs. Our study demonstrates that this minor modification contributes to the stability of training by confining the input to the spline within the effective range of the activation function. | [code](https://github.com/Ghaith81/dropkan) ｜ 
• DropKAN: Regularizing KANs by masking post-activations - Arnold Networks) is a regularization method that prevents co-adaptation of activation function weights in Kolmogorov-Arnold Networks (KANs). DropKAN operates by randomly masking some of the post-activations within the KANs computation graph, while scaling-up the retained post-activations. We show that this simple procedure that require minimal coding effort has a regularizing effect and consistently lead to better generalization of KANs. | [code](https://github.com/Ghaith81/dropkan) ｜ 
• SigKAN: Signature-Weighted Kolmogorov-Arnold Networks for Time Series
• Demonstrating the Efficacy of Kolmogorov-Arnold Networks in Vision Tasks - in-VIsion) ｜ 
• Gaussian Process Kolmogorov-Arnold Networks
• Kolmogorov--Arnold networks in molecular dynamics
• KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems
• GNN-SKAN: Harnessing the Power of SwallowKAN to Advance Molecular Representation Learning with GNNs
• SigKAN: Signature-Weighted Kolmogorov-Arnold Networks for Time Series
• Demonstrating the Efficacy of Kolmogorov-Arnold Networks in Vision Tasks - in-VIsion) ｜ 
• CoxKAN: Kolmogorov-Arnold Networks for Interpretable, High-Performance Survival Analysis - Arnold Networks, which combines both interpretability and high performance. CoxKAN outperforms traditional models like the Cox proportional hazards model and rivals deep learning-based models, but with the advantage of interpretability, making it more useful in medical settings where understanding the underlying risk factors and relationships is essential. We find that CoxKAN extracts complex interactions between predictor variables and identifies the precise effect of important biomarkers on patient survival. | [code](https://github.com/knottwill/coxkan) ｜ 
• Kolmogorov-Arnold Transformer
• Chebyshev Polynomial-Based Kolmogorov-Arnold Networks
• Convolutional Kolmogorov-Arnold Networks - KANs) ｜ 
• Kolmogorov-Arnold Convolutions: Design Principles and Empirical Studies - conv-kan) ｜ 
• Smooth Kolmogorov Arnold networks enabling structural knowledge representation
• TKAN: Temporal Kolmogorov-Arnold Networks
• ReLU-KAN: New Kolmogorov-Arnold Networks that Only Need Matrix Addition, Dot Multiplication, and ReLU
• CoxKAN: Kolmogorov-Arnold Networks for Interpretable, High-Performance Survival Analysis - Arnold Networks, which combines both interpretability and high performance. CoxKAN outperforms traditional models like the Cox proportional hazards model and rivals deep learning-based models, but with the advantage of interpretability, making it more useful in medical settings where understanding the underlying risk factors and relationships is essential. We find that CoxKAN extracts complex interactions between predictor variables and identifies the precise effect of important biomarkers on patient survival. | [code](https://github.com/knottwill/coxkan) ｜ 
• RKAN: Residual Kolmogorov-Arnold Network - Arnold Network (RKAN) is designed to enhance the performance of classic CNNs by incorporating RKAN blocks into existing architectures. | [code](https://github.com/withray/residualKAN) 
• U-KAN Makes Strong Backbone for Medical Image Segmentation and Generation - AIM-Group/U-KAN) ｜ 
• Kolmogorov-Arnold Networks (KANs) for Time Series Analysis
• A Temporal Kolmogorov-Arnold Transformer for Time Series Forecasting
• Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networks
• Effective Integration of KAN for Keyword Spotting
• RKAN: Residual Kolmogorov-Arnold Network - Arnold Network (RKAN) is designed to enhance the performance of classic CNNs by incorporating RKAN blocks into existing architectures. | [code](https://github.com/withray/residualKAN) 
• Chebyshev Polynomial-Based Kolmogorov-Arnold Networks
• Kolmogorov-Arnold Transformer
• Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving PDEs based on Kolmogorov Arnold Networks - wang/research-on-solving-partial-differential-equations-of-solid-mechanics-based-on-pinn) ｜ 
• Kolmogorov-Arnold Convolutions: Design Principles and Empirical Studies - conv-kan) ｜ 
• Smooth Kolmogorov Arnold networks enabling structural knowledge representation
• TKAN: Temporal Kolmogorov-Arnold Networks
• Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving PDEs based on Kolmogorov Arnold Networks - wang/research-on-solving-partial-differential-equations-of-solid-mechanics-based-on-pinn) ｜ 
• Convolutional Kolmogorov-Arnold Networks - KANs) ｜ 
• DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems
• ReLU-KAN: New Kolmogorov-Arnold Networks that Only Need Matrix Addition, Dot Multiplication, and ReLU
• U-KAN Makes Strong Backbone for Medical Image Segmentation and Generation - AIM-Group/U-KAN) ｜ 
• Kolmogorov-Arnold Networks (KANs) for Time Series Analysis
• Wav-KAN: Wavelet Kolmogorov-Arnold Networks
• fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions
• BSRBF-KAN: A combination of B-splines and Radial Basic Functions in Kolmogorov-Arnold Networks
• GraphKAN: Enhancing Feature Extraction with Graph Kolmogorov Arnold Networks
• BSRBF-KAN: A combination of B-splines and Radial Basic Functions in Kolmogorov-Arnold Networks
• GraphKAN: Enhancing Feature Extraction with Graph Kolmogorov Arnold Networks
• A First Look at Kolmogorov-Arnold Networks in Surrogate-assisted Evolutionary Algorithms - EA)｜ 
• fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions
• Gaussian Process Kolmogorov-Arnold Networks
• Kolmogorov--Arnold networks in molecular dynamics
• Kolmogorov-Arnold Network for Online Reinforcement Learning - PPO) | 
• FourierKAN-GCF: Fourier Kolmogorov-Arnold Network--An Effective and Efficient Feature Transformation for Graph Collaborative Filtering - Xu/FKAN-GCF) ｜ 
• KANQAS: Kolmogorov-Arnold Network for Quantum Architecture Search
• A Comprehensive Survey on Kolmogorov Arnold Networks (KAN)
• Sparks of Quantum Advantage and Rapid Retraining in Machine Learning - KAN) | 
• Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks
• rKAN: Rational Kolmogorov-Arnold Networks
• A deep machine learning algorithm for construction of the Kolmogorov–Arnold representation
• Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networks
• A Comprehensive Survey on Kolmogorov Arnold Networks (KAN)
• A deep machine learning algorithm for construction of the Kolmogorov–Arnold representation
• KANQAS: Kolmogorov-Arnold Network for Quantum Architecture Search
• Sparks of Quantum Advantage and Rapid Retraining in Machine Learning - KAN) | 
• Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks
• TC-KANRecon: High-Quality and Accelerated MRI Reconstruction via Adaptive KAN Mechanisms and Intelligent Feature Scaling - kanrecon) | 
• Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretability
• KAN4TSF: Are KAN and KAN-based models Effective for Time Series Forecasting?
• Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems
• FC-KAN: Function Combinations in Kolmogorov-Arnold Networks
• KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems
• GNN-SKAN: Harnessing the Power of SwallowKAN to Advance Molecular Representation Learning with GNNs
• On the expressiveness and spectral bias of KANs - KANs can represent MLPs of similar size. While MLPs can represent KANs, the number of parameters in an MLP increases significantly with KAN grid size. In addition, KANs have a lower spectral bias for low-frequency patterns.
• P1-KAN an effective Kolmogorov Arnold Network for function approximation - lab.org/warin-xavier/)
• CF-KAN: Kolmogorov-Arnold Network-based Collaborative Filtering to Mitigate Catastrophic Forgetting in Recommender Systems - KAN) | 
• KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
• Kolmogorov-Arnold Network for Online Reinforcement Learning - PPO) | 
• A Gated Residual Kolmogorov-Arnold Networks for Mixtures of Experts - code](https://github.com/remigenet/kamoe) | 
• TC-KANRecon: High-Quality and Accelerated MRI Reconstruction via Adaptive KAN Mechanisms and Intelligent Feature Scaling - kanrecon) | 
• Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretability
• KAN4TSF: Are KAN and KAN-based models Effective for Time Series Forecasting?
• FC-KAN: Function Combinations in Kolmogorov-Arnold Networks
• A Gated Residual Kolmogorov-Arnold Networks for Mixtures of Experts - code](https://github.com/remigenet/kamoe) | 
• Implicit Neural Representations with Fourier Kolmogorov-Arnold Networks - Layer Learnable Activation for Implicit Neural Representation (SL2A-INR)](https://arxiv.org/abs/2409.10836)
• On the expressiveness and spectral bias of KANs - KANs can represent MLPs of similar size. While MLPs can represent KANs, the number of parameters in an MLP increases significantly with KAN grid size. In addition, KANs have a lower spectral bias for low-frequency patterns.
• P1-KAN an effective Kolmogorov Arnold Network for function approximation - lab.org/warin-xavier/)

Theorem

• On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables
• On functions of three variables
• On a constructive proof of Kolmogorov’s superposition theorem
• The Kolmogorov-Arnold representation theorem revisited
• The Kolmogorov Superposition Theorem can Break the Curse of Dimension When Approximating High Dimensional Functions
• On functions of three variables
• On a constructive proof of Kolmogorov’s superposition theorem
• The Kolmogorov-Arnold representation theorem revisited
• The Kolmogorov Superposition Theorem can Break the Curse of Dimension When Approximating High Dimensional Functions

Theorem

• FourierKAN - linear activation | 
• Vision-KAN - 5), 80% on ImageNet1000 (training in progress) | 
• ChebyKAN - Arnold Networks (KAN) using Chebyshev polynomials instead of B-splines. ｜ 
• BSRBF_KAN - Spline (BS) and Radial Basic Function (RBF) in Kolmogorov-Arnold Networks (KANs) | 
• TaylorKAN - Arnold Networks (KAN) using Taylor series instead of Fourier | 
• pykan
• efficient-kan - PyTorch implementation of Kolmogorov-Arnold Network (KAN). ｜ 
• FastKAN - Arnold Networks (KAN) ｜ 
• FasterKAN - kan.svg)
• FourierKAN - linear activation | 
• Vision-KAN - 5), 80% on ImageNet1000 (training in progress) | 
• Large Kolmogorov-Arnold Networks - Arnold Networks (including CUDA-supported KAN convolutions) ｜ 
• xKAN - Arnold Networks with various basis functions like B-Splines, Fourier, Chebyshev, Wavelets etc ｜ 
• GraphKAN - Graph-Kolmogorov-Arnold-Networks.svg)
• FCN-KAN - KAN.svg)
• X-KANeRF - Splines, Fourier, Radial Basis Functions, Polynomials, etc ｜ 
• OrthogPolyKAN - Arnold Networks (KAN) using orthogonal polynomials instead of B-splines. ｜ 
• Deep-KAN - KAN.svg)
• xKAN - Arnold Networks with various basis functions like B-Splines, Fourier, Chebyshev, Wavelets etc ｜ 
• GraphKAN - Graph-Kolmogorov-Arnold-Networks.svg)
• FCN-KAN - KAN.svg)
• X-KANeRF - Splines, Fourier, Radial Basis Functions, Polynomials, etc ｜ 
• ChebyKAN - Arnold Networks (KAN) using Chebyshev polynomials instead of B-splines. ｜ 
• Large Kolmogorov-Arnold Networks - Arnold Networks (including CUDA-supported KAN convolutions) ｜ 
• JacobiKAN - Arnold Networks (KAN) using Jacobi polynomials instead of B-splines. ｜ 
• kansformers
• RBF-KAN - KAN is a PyTorch module that implements a Radial Basis Function Kolmogorov-Arnold Network | 
• JacobiKAN - Arnold Networks (KAN) using Jacobi polynomials instead of B-splines. ｜ 
• Wav-KAN - KAN: Wavelet Kolmogorov-Arnold Networks | 
• KANX - Arnold Network in JAX | 
• Wav-KAN - KAN: Wavelet Kolmogorov-Arnold Networks | 
• Initial Investigation of Kolmogorov-Arnold Networks (KANs) as Feature Extractors for IMU Based Human Activity Recognition
• TKAN - Arnold Networks Keras3 layer implementations multibackend (Jax, Tensorflow, Torch) | 
• SigKAN - Weighted Kolmogorov-Arnold Networks tensorflow 2.x layer implementations, based on iisignature | 
• fKAN - Arnold Networks with trainable Jacobi basis functions | 
• TaylorKAN - Arnold Networks (KAN) using Taylor series instead of Fourier | 
• fKAN - Arnold Networks with trainable Jacobi basis functions | 
• Initial Investigation of Kolmogorov-Arnold Networks (KANs) as Feature Extractors for IMU Based Human Activity Recognition
• FlashKAN - independent computation of Kolmogorov Arnold networks | 
• rKAN - Arnold Networks | 
• HiPPO-KAN: Efficient KAN Model for Time Series Analysis
• KAN-SGAN - supervised learning with Generative Adversarial Networks (GANs) using Kolmogorov-Arnold Network Layers (KANLs) | 
• TKAN - Arnold Networks Keras3 layer implementations multibackend (Jax, Tensorflow, Torch) | 
• TKAT - Arnold Transformer Tensorflow 2.x model implementation | 
• KAN-SGAN - supervised learning with Generative Adversarial Networks (GANs) using Kolmogorov-Arnold Network Layers (KANLs) | 
• HiPPO-KAN: Efficient KAN Model for Time Series Analysis
• FourierKAN - linear activation | 
• FCN-KAN - KAN.svg)
• FlashKAN - independent computation of Kolmogorov Arnold networks | 
• BSRBF_KAN - Spline (BS) and Radial Basic Function (RBF) in Kolmogorov-Arnold Networks (KANs) | 
• TaylorKAN - Arnold Networks (KAN) using Taylor series instead of Fourier | 
• fKAN - Arnold Networks with trainable Jacobi basis functions | 
• rKAN - Arnold Networks | 
• KAN-SGAN - supervised learning with Generative Adversarial Networks (GANs) using Kolmogorov-Arnold Network Layers (KANLs) | 

Library-based

• Quantum KAN - KAN.svg)
• KAN: Kolmogorov–Arnold Networks in MLX for Apple silicon - Guelmez/mlx-kan.svg)
• TorchKAN
• jaxKAN
• keras_efficient_kan
• efficient-kan-jax - kan | 
• cuda-Wavelet-KAN - Wavelet-KAN.svg)

ConvKANs

• Convolutional-KANs - Arnold Networks (KAN) to the Convolutional Layers, changing the classic linear transformation of the convolution to non linear activations in each pixel. ｜ 
• Torch Conv KAN - Arnold Layers with various basis functions. The repository includes implementations of 1D, 2D, and 3D convolutions with different kernels, ResNet-like, Unet-like, and DenseNet-like models, training code based on accelerate/PyTorch, and scripts for experiments with CIFAR-10/100, Tiny ImageNet and ImageNet1k. Pretrained weights on ImageNet1k are also available ｜ 
• convkan - in replacement of Conv2d) ｜ 
• ConvKAN3D - kan implementation (importable Python package from PyPi), drop-in replacement of Conv3d.
• KA-Conv - Arnold Convolutional Networks with Various Basis Functions (Optimization for Efficiency and GPU memory usage) | 
• KAN-Conv2D - in Convolutional KAN built on multiple implementations ([Original pykan](https://github.com/KindXiaoming/pykan) / [efficient-kan](https://github.com/Blealtan/efficient-kan) / [FastKAN](https://github.com/ZiyaoLi/fast-kan)) to support the original paper hyperparameters. | 
• CNN-KAN - Arnold Networks | 
• Convolutional-KANs - Arnold Networks (KAN) to the Convolutional Layers, changing the classic linear transformation of the convolution to non linear activations in each pixel. ｜ 
• KAN-Conv2D - in Convolutional KAN built on multiple implementations ([Original pykan](https://github.com/KindXiaoming/pykan) / [efficient-kan](https://github.com/Blealtan/efficient-kan) / [FastKAN](https://github.com/ZiyaoLi/fast-kan)) to support the original paper hyperparameters. | 
• convkan - in replacement of Conv2d) ｜ 
• KA-Conv - Arnold Convolutional Networks with Various Basis Functions (Optimization for Efficiency and GPU memory usage) | 
• ConvKAN3D - kan implementation (importable Python package from PyPi), drop-in replacement of Conv3d.
• Torch Conv KAN - Arnold Layers with various basis functions. The repository includes implementations of 1D, 2D, and 3D convolutions with different kernels, ResNet-like, Unet-like, and DenseNet-like models, training code based on accelerate/PyTorch, and scripts for experiments with CIFAR-10/100, Tiny ImageNet and ImageNet1k. Pretrained weights on ImageNet1k are also available ｜ 

Benchmark

• seydi1370/Basis_Functions - based KANs on the MNIST dataset for handwritten digit classification. | 
• KAN-benchmarking - Master/KAN-benchmarking.svg)
• seydi1370/Basis_Functions - based KANs on the MNIST dataset for handwritten digit classification. | 

Non-Python

• KolmogorovArnold.jl
• KolmogorovArnold.jl
• kan-polar - Arnold Networks in MATLAB ｜ 
• kan-polar - Arnold Networks in MATLAB ｜ 
• FluxKAN.jl
• kamo - Arnold Networks in Mojo ｜ 
• Julia-Wav-KAN - Arnold Networks. ｜ 
• Building a Kolmogorov-Arnold Neural Network in C
• C# and C++ implementations, benchmarks, tutorials
• kamo - Arnold Networks in Mojo ｜ 
• Julia-Wav-KAN - Arnold Networks. ｜ 
• Building a Kolmogorov-Arnold Neural Network in C
• Julia-Wav-KAN - Arnold Networks. ｜ 

Alternative

• high-order-layers-torch - order-layers-torch.svg)