Solutions to ideal magnetohydrodynamics equations by deep learning neural networks

We investigate solutions to ideal magnetohydrodynamic (MHD) equations using the idea of Physics-Informed Neural Networks (PINNs). The loss function is formulated into two parts: first, labeled training data such as the initial/boundary conditions; and the second, unlabelled data (inside the domain) monitored by the ideal MHD equations. We also provide additional labeled data from probes to the neural networks, which successfully reconstruct spatial-time solutions to classical one-dimensional shock tube problems, such as Brio-Wu and Sod shock tubes. We extend this to more complex two-dimensional (2D) problems that permit self-similar solutions, such as the MHD shock refraction problem. We inform the 2D self-similar MHD equations to the neural network. Some preliminary results indicate that the neural network is able to obtain self-similar solutions to some complex 2D shock refraction problems in MHD.