A Physics-Informed Neural Network model combined Pell-Lucas polynomials for solving the Lane-Emden type equation

Aiming at the Lane-Emden type differential equations owing singular initial value problem, an improved Physics-Informed Neural Network method based on Pell-Lucas polynomials is proposed. The method adopts feedforward neural network model and error back propagation principle. The analytical and numerical solutions of linear homogeneous, linear non-homogeneous, and nonlinear homogeneous Lane-Emden equations are compared with the approximate solutions of Chebyshev neural network proposed by Mall. Numerical examples show that the proposed method has high accuracy, which proves the effectiveness of the model.