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.

翻译：磁流体动力学（MHD）系统的建模与控制仍是一个具有挑战性的问题，该问题涉及流体动力学与电磁学的耦合，且具有非线性、多尺度时空特征。为解决这些问题，我们开发了MHDnet，这是一种将多模态多尺度特征嵌入多尺度神经网络架构的物理信息学习方法，通过缓解不同频率模态下磁流体耦合的相互作用，可加速神经网络（NN）的收敛。本文考虑了三种不同的数学公式，分别命名为原始公式（$B$）、磁矢势公式（$A_1$）以及磁场与速度均无散度公式（$A_2$）。这些公式的残差与初始条件和边界条件一起被整合到MHDnet的损失函数中。此外，三种公式的压力场作为隐状态，无需额外数据和计算成本即可获得。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和数值公式的性能，同时MHDnet在$A_1$和$A_2$公式下也能以高精度给出压力场。