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：针对复杂动态不可压缩磁流体动力学（MHD）方程设计高效高精度数值方法，仍是各类分析与设计任务中的挑战性问题。这主要源于对流与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，导致数值计算面临局限性。本文提出MHDnet作为物理守恒学习方法来求解MHD问题，其中考虑了三种不同数学表述，分别命名为$B$表述、$A_1$表述和$A_2$表述。随后将这些表述嵌入MHDnet，使其能够保持底层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磁流体耦合的相互作用，加速神经网络（NN）的收敛。值得一提的是，三种表述中的压力场作为隐藏状态，无需额外数据与计算开销即可获取。通过多项数值实验，验证了所提MHDnet相较于不同神经网络架构与数值表述的性能表现。