  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.
