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Designing efficient and high-accuracy numerical methods for complex dynamicincompressible magnetohydrodynamics (MHD) equations remains a challengingproblem in various analysis and design tasks. This is mainly due to thenonlinear coupling of the magnetic and velocity fields occurring withconvection and Lorentz forces, and multiple physical constraints, which willlead to the limitations of numerical computation. In this paper, we develop theMHDnet as a physics-preserving learning approach to solve MHD problems, wherethree different mathematical formulations are considered and named $B$formulation, $A_1$ formulation, and $A_2$ formulation. Then the formulationsare embedded into the MHDnet that can preserve the underlying physicalproperties and divergence-free condition. Moreover, MHDnet is designed by themulti-modes feature merging with multiscale neural network architecture, whichcan accelerate the convergence of the neural networks (NN) by alleviating theinteraction of magnetic fluid coupling across different frequency modes.Furthermore, the pressure fields of three formulations, as the hidden state,can be obtained without extra data and computational cost. Several numericalexperiments are presented to demonstrate the performance of the proposed MHDnetcompared with different NN architectures and numerical formulations.

MHDnet: Physics-preserving learning for solving magnetohydrodynamics problems