Fractional Kolmogorov-Arnold Network (fKAN)

Fractional Kolmogorov-Arnold Network (fKAN) is a novel neural network that incorporates the distinctive attributes of Kolmogorov-Arnold Networks (KANs) with a trainable adaptive fractional-orthogonal Jacobi function as its basis function. This method offers several advantages, including non-polynomial behavior, activity for both positive and negative input values, faster execution, and better accuracy.

Installation

To install fKAN, use the following command:

$ pip install fkan

Example Usage

The current implementation of fKAN works with both the TensorFlow and PyTorch APIs.

TensorFlow

from tensorflow import keras
from tensorflow.keras import layers
from fkan.tensorflow import FractionalJacobiNeuralBlock as fJNB

model = keras.Sequential(
    [
        layers.InputLayer(input_shape=input_shape),
        layers.Conv2D(32, kernel_size=(3, 3)),
        fJNB(3),
        layers.MaxPooling2D(pool_size=(2, 2)),
        layers.Flatten(),
        layers.Dropout(0.5),
        layers.Dense(16),
        fJNB(2),
        layers.Dense(num_classes, activation="softmax"),
    ]
)

PyTorch

import torch.nn as nn
from fkan.torch import FractionalJacobiNeuralBlock as fJNB

model = nn.Sequential(
    nn.Linear(1, 16),
    fJNB(3),
    nn.Linear(16, 32),
    fJNB(6),
    nn.Linear(32, 1),
)

Experiments

The example folder contains the implementation of the experiments from the paper using fKAN. These experiments include:

Deep Learning Tasks

Synthetic Regression
MNIST Classification
Fashion MNIST Image Denoising
IMDB Sentiment Analysis

Physics Informed Deep Learning

Lane Emden Ordinary Differential Equation
Burgers Partial Differential Equation
Fractional Delay Differential Equation with Caputo definition

Current Limitations

Maximum allowed Jacobi polynomial degree is set to six.
The current library is not compatible with other deep learning frameworks, but it can be converted easily.

Contribution

We encourage the community to contribute by opening issues and submitting pull requests to help address these limitations and improve the overall functionality of fKAN.

Contact

If you have any questions or encounter any issues, please open an issue in this repository (preferred) or reach out to the author directly.

Citation

If you use fKAN in your research, please cite our paper:

@misc{aghaei2024fkan,
      title={fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions},
      author={Alireza Afzal Aghaei},
      year={2024},
      eprint={2406.07456},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

alirezaafzalaghaei / fKAN