rKAN: Rational Kolmogorov-Arnold Networks This paper introduces rational Kolmogorov-Arnold Networks (rKANs) as a novel approach to function approximation using rational basis functions. Traditional KANs use B-spline curves, but rKANs employ rational functions, specifically Padé approximations and rational Jacobi functions, as trainable basis functions. The paper proposes two approaches for rKANs: one based on Padé approximation and another using rationalized Jacobi functions. These approaches are evaluated on various deep learning and physics-informed tasks to demonstrate their effectiveness in function approximation. The paper begins by discussing the importance of basis functions in function approximation, highlighting the limitations of polynomial and spline methods. It then introduces Jacobi polynomials, which are used to construct rational approximations. The paper describes the two approaches for rKANs, explaining how they utilize rational functions to approximate complex functions. The first approach uses Padé approximations, while the second maps Jacobi polynomials into a rational space using nonlinear mappings. The paper then presents experiments on deep learning tasks, including regression and classification. In the regression task, rKANs are tested on synthetic data with asymptotic behavior, showing improved accuracy compared to traditional methods. In the classification task, rKANs are applied to the MNIST dataset, demonstrating superior performance compared to other activation functions. The paper also evaluates rKANs in physics-informed deep learning tasks, including solving ordinary and partial differential equations. The results show that rKANs can accurately approximate solutions to these equations, outperforming other methods in some cases. The paper concludes by discussing the potential of rKANs in various applications, including image processing and physics-informed learning. Future work includes exploring the use of rational versions of B-spline curves and further evaluating fractional rational KANs for solving physics-informed problems on semi-infinite domains.rKAN: Rational Kolmogorov-Arnold Networks This paper introduces rational Kolmogorov-Arnold Networks (rKANs) as a novel approach to function approximation using rational basis functions. Traditional KANs use B-spline curves, but rKANs employ rational functions, specifically Padé approximations and rational Jacobi functions, as trainable basis functions. The paper proposes two approaches for rKANs: one based on Padé approximation and another using rationalized Jacobi functions. These approaches are evaluated on various deep learning and physics-informed tasks to demonstrate their effectiveness in function approximation. The paper begins by discussing the importance of basis functions in function approximation, highlighting the limitations of polynomial and spline methods. It then introduces Jacobi polynomials, which are used to construct rational approximations. The paper describes the two approaches for rKANs, explaining how they utilize rational functions to approximate complex functions. The first approach uses Padé approximations, while the second maps Jacobi polynomials into a rational space using nonlinear mappings. The paper then presents experiments on deep learning tasks, including regression and classification. In the regression task, rKANs are tested on synthetic data with asymptotic behavior, showing improved accuracy compared to traditional methods. In the classification task, rKANs are applied to the MNIST dataset, demonstrating superior performance compared to other activation functions. The paper also evaluates rKANs in physics-informed deep learning tasks, including solving ordinary and partial differential equations. The results show that rKANs can accurately approximate solutions to these equations, outperforming other methods in some cases. The paper concludes by discussing the potential of rKANs in various applications, including image processing and physics-informed learning. Future work includes exploring the use of rational versions of B-spline curves and further evaluating fractional rational KANs for solving physics-informed problems on semi-infinite domains.