Designing efficient and high-accuracy numerical methods for complex dynamic incompressible magnetohydrodynamics (MHD) equations remains a challenging problem in various analysis and design tasks. This is mainly due to the nonlinear coupling of the magnetic and velocity fields occurring with convection and Lorentz forces, and multiple physical constraints, which will lead to the limitations of numerical computation. In this paper, we develop the MHDnet as a physics-preserving learning approach to solve MHD problems, where three different mathematical formulations are considered and named $B$ formulation, $A_1$ formulation, and $A_2$ formulation. Then the formulations are embedded into the MHDnet that can preserve the underlying physical properties and divergence-free condition. Moreover, MHDnet is designed by the multi-modes feature merging with 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. Furthermore, 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.

翻译：设计高效高精度的数值方法来求解复杂动态不可压缩磁流体动力学（MHD）方程，仍是众多分析与设计任务中的挑战性问题。这主要源于对流和洛伦兹力作用下磁场与速度场的非线性耦合，以及多重物理约束带来的数值计算局限性。本文提出MHDnet作为一种物理保持学习方法来解决MHD问题，其中考虑了三种不同的数学公式形式，分别命名为$B$公式、$A_1$公式和$A_2$公式。随后将这些公式嵌入MHDnet中，使其能够保持潜在物理特性与无散条件。此外，MHDnet采用多模态特征融合与多尺度神经网络架构设计，通过缓解不同频率模态间的磁流体耦合相互作用，加速神经网络（NN）收敛。同时，作为隐变量的三种公式的压力场可在无需额外数据和计算成本的情况下获得。通过多个数值实验，展示了所提出的MHDnet相较于不同神经网络架构与数值公式的性能优势。