A hands-on introduction to Physics-Informed Neural Networks for solving partial differential equations with benchmark tests taken from astrophysics and plasma physics

TL;DR

This paper introduces physics-informed neural networks (PINNs) as a flexible deep learning approach for solving complex PDEs from astrophysics and plasma physics, demonstrating their effectiveness through benchmark tests and boundary condition handling.

Contribution

The paper provides a practical introduction to PINNs, showcasing their application to diverse PDEs and boundary conditions in astrophysics and plasma physics, with implementation code.

Findings

PINNs effectively solve PDEs from astrophysics and plasma physics.

The method handles various boundary conditions and parametric problems.

Python implementations demonstrate ease of use and flexibility.

Abstract

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the equation evaluated at various points within the domain. Boundary conditions are incorporated either by introducing soft constraints with corresponding boundary data values in the minimization process or by strictly enforcing the solution with hard constraints. PINNs are tested on diverse PDEs extracted from two-dimensional physical/astrophysical problems. Specifically, we explore Grad-Shafranov-like equations that capture magnetohydrodynamic equilibria in magnetically dominated plasmas. Lane-Emden equations that model internal structure of stars in sef-gravitating hydrostatic equilibrium are also considered. The flexibility of the method to handle various boundary conditions is illustrated through various examples, as well as its ease in solving parametric and inverse problems. The corresponding Python codes based on PyTorch/TensorFlow libraries are made available.