Modelling Lane–Emden type equations using Physics-Informed Neural Networks

Abstract

The solutions of Lane–Emden (LE) type equations arising in astrophysics for the polytropic gas spheres, the isothermal gas sphere, and the white dwarf stars are revisited. We explore the potentiality of recent methods of deep learning using neural networks constrained by the physics and called Physics-Informed Neural Networks (PINNs). The basics of PINNs is introduced for solving each equation individually. The method consists in constraining the equation residual at some collocation dataset in addition to the boundary data via a minimization procedure. When the training process is complete, a learned differentiable function is obtained that can generate solution at any value of the variable. The novelty of this study is the additional possibility of learning solutions for several equations collectively with the same network, e.g. for the polytropic equations family for all the indices. We demonstrate the performances of PINNs in comparison with classical numerical methods. Advantages and drawbacks are highlighted. Interestingly, PINNs are meshless methods that can quasi-instantaneously generate the solution and its derivative once trained. However, the training procedure and accuracy of the method remain two future points of improvement.