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，呈现出非线性、多尺度时空特征。为解决这些问题，我们提出MHDnet作为一种物理信息学习方法，将多模态多尺度特征嵌入多尺度神经网络架构中，通过减轻磁流体耦合在不同频率模态间的相互作用，从而加速神经网络的收敛。研究考虑了三种不同的数学表述，分别命名为原始表述（$B$）、磁矢势表述（$A_1$）以及磁感应强度和速度均无散表述（$A_2$）。将这些表述的残差与初始条件和边界条件共同融入MHDnet的损失函数中。此外，三种表述的压力场作为隐状态，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，将所提出的MHDnet与不同神经网络架构及数值表述进行性能对比，结果表明，采用$A_1$和$A_2$表述的MHDnet能够以高精度给出压力场。