Application of a Neural-Network Algorithm Based on Orthogonal Laguerre Polynomials to the Solution of the Cauchy Problem for Differential Equations with a Variable-Order Fractional Derivative on the Example of the Fractional Lane–Emden Equation

This paper proposes, implements, and tests a neural-network method based on orthogonal Laguerre polynomials for the numerical solution of the Cauchy problem for a singular nonlinear Lane–Emden equation with a variable-order fractional derivative. Such equations rarely admit analytical solutions, while classical finite-difference schemes struggle with the singularity near zero and the nonlocal character of the fractional derivative. The approximate solution is expanded in modified Laguerre polynomials shifted to the initial point, so the Cauchy initial conditions are satisfied exactly. The network has one trainable linear layer; nonlinearity is provided by the polynomial basis. The weights are found by minimizing a loss function containing the equation residual at collocation points, a weight factor improving accuracy near the singularity, and regularization limiting weight growth. Training uses Adam with an adaptive learning-rate scheduler and early stopping. Implementation is in Python 3.13.1 with PyTorch 2.7.1. A vectorized tensor version accelerates computations by about thirty times compared with the loop version. On test problems with known solutions, the maximum absolute error reaches about five ten-thousandths; for a numerical reference, about eight thousandths; in a stress test, about three hundredths. It is promising for singular variable-order fractional differential equations and can be parallelized on central and graphics processors.