Solving the Orszag-Tang vortex magnetohydrodynamics problem with physics-constrained convolutional neural networks

We study the 2D Orszag-Tang vortex magnetohydrodynamics (MHD) problem through the use of physics-constrained convolutional neural networks (PCNNs) for forecasting the density, rho, and the magnetic field, B, as well as the prediction of B given the velocity field v of the fluid. In addition to translation equivariance from the convolutional architecture, other physics constraints were embedded: absence of magnetic monopoles, non-negativity of rho, use of only relevant variables, and the periodic boundary conditions of the problem. The use of only relevant variables and the hard constraint of non-negative rho were found to facilitate learning greatly. The divergenceless condition del <middle dot> B = 0 was implemented as a hard constraint up to machine precision through the use of a magnetic potential to define B = del x A. Residual networks and data augmentation were also used to improve performance. This allowed for some of the residual models to function as surrogate models and provide reasonably accurate simulations. For the prediction task, the PCNNs were evaluated against a physics-informed neural network, which had the ideal MHD induction equation as a soft constraint. Several models were able to generate highly accurate fields, which are visually almost indistinguishable and have low mean squared error. Only methods with built-in hard constraints produced physical fields with del <middle dot> B = 0. The use of PCNNs for MHD has the potential to produce physically consistent real-time simulations to serve as virtual diagnostics in cases where inferences must be made with limited observables. (c) 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/)