rKAN: Rational Kolmogorov-Arnold networks

The development of Kolmogorov-Arnold networks (KANs) marks a significant shift from traditional multi-layer perceptrons in deep learning. Initially, KANs employed learnable B-spline curves as their primary basis function, but their inherent complexity posed implementation challenges. Consequently, researchers have explored alternative basis functions such as wavelets, polynomials, and fractional functions. This paper explores the use of rational functions as a novel basis function for KANs. Two different approaches based on Padé approximation and rational Jacobi functions as trainable basis functions are proposed to define the rational KAN (rKAN). The rKAN's performance has been evaluated across various domains, including deep learning and physics-informed deep learning, where it demonstrates its superiority in comparison to its alternatives with 99.29% accuracy on MNIST classification, 86.6% accuracy in text sentiment analysis, and a solution to the CartPole reinforcement learning task in approximately 200 episodes.