MHDnet: Multi-modes multiscale physics informed neural networks for solving magnetohydrodynamics problems

Modeling and control of the magnetohydrodynamics (MHD) system remain a challenging problem, which involves the coupling between fluid dynamics and electromagnetism with the nonlinear, multiscale spatiotemporal features. To address these issues, we develop the MHDnet as a physics-informed learning approach to MHD problems with the multi-modes multiscale feature embedding into multiscale neural network architecture, which can accelerate the convergence of the neural networks (NN) by alleviating the interaction of magnetic fluid coupling across different frequency modes. Three different mathematical formulations are considered and named the original formulation ($B$), magnetic vector potential formulation ($A_1$), and divergence-free both magnetic induction and velocity formulation ($A_2$). The residual of them, together with the initial and boundary conditions, are emerged into the loss function of MHDnet. Moreover, the pressure fields of three formulations, as the hidden state, can be obtained without extra data and computational cost. Several numerical experiments are presented to demonstrate the performance of the proposed MHDnet compared with different NN architectures and numerical formulations, and the pressure fields can also be given by MHDnet with $A_1$ and $A_2$ formulations with high accuracy.