## 1 Introduction  
With the development of digital twin technology, the speed of conventional numerical calculation methods is far lower than people's expectations, and solving physical field problems quickly has become the most urgent need, especially for the optimal design of electric machines [1; 2; 3]. During the process of optimizations, the magnetic field approximation for the simulation of electric machines is computationally expensive since the models need a high resolution to yield accurate results. Besides the optimized solutions, most solved candidate models are useless. For example, many studies currently use the evolutionary strategy as the optimization algorithm for motor design. Evolutionary algorithms require many iterations, resulting in thousands of models with magnetic fields that need to be solved. However, only one model is needed, and the rest of the models that consume a lot of computational resources are wasted.  
Therefore, the computational results can be reused to train the neural network, and many data-driven deep learning networks are already leading to multi-physics simulation calculations, such as fluid field [4], stress [5], sound [6], and medicine [7]. However, the application of AI in the magnetic field is underestimated. Consequently, a fast electric motor simulation method using AI is expected to be developed.  
This study proposes a physics-informed generative adversarial network (PIGAN). The PIGAN employs several physics-informed loss functions to fit the magnetic field distribution. The generation process of the physics-informed dataset is presented. The dataset, LiM2D, represents linear machines' two-dimensional (2-D) magnetic field, i.e., the flux density distribution (B) of the linear machines in operation. It consists of material distribution, current density, the magnetization of permanent magnets (PMs), motion area, and flux density on the x-axis and y-axis.  
The specific details are in Section 4. The definitions of multiple channels for the input are presented. Unlike the previous work, multiple channels with separated properties are defined, such as material, vectorized magnetization field, current density direction, and motion band, to achieve the scalability of the dataset. Different channels of input can interact in a way that influences the prediction. Therefore, the magnetic field problems for a given model with appropriate channels of materials, motion area, and source conditions over a finite region of space are solved.  
Based on this dataset, some key challenges are also pointed out in Section 6. In particular, the feasibility of dealing with large-scale problems such as three-dimensional (3-D) problems is discussed first. Second, the nonlinear material representation to make the magnetic field approximation meet more engineering problems is explored. Third, the setup of boundary conditions in magnetic field approximation is discussed. Lastly, the integration error of prediction is evaluated. Note that this paper does not aim to thoroughly tackle these challenges but rather expose them to the community for future research.  
Overall, our main contributions are: (1) a physics-informed loss function, (2) a linear motor magnetic field dataset, and (3) an in-depth study of the channel definition of input and output metrics. We also aspire to highlight the challenges faced in the magnetic field approximation for linear motors, sparking innovations in applications such as real-time electromagnetic simulation, digital twins, and optimal design for the renewable energy system.