# A PINN-Based Nonlinear PMSM Electromagnetic Model Using Differential Inductance Theory
## 1 Introduction
[14] proposed an advanced modeling method for interior synchronous motors, developing a dynamic motor model that considers magnetic saturation effects and the rotational-angle dependence of electrical, magnetic, and mechanical quantities, which can be integrated into multidisciplinary system simulations. Based on RBF interpolation technology, this model is constructed using a set of magnetic field finite element method (FEM) solution data. The proposed motor model can simulate arbitrary time-stepping operating conditions, such as field-weakening control research, sensorless motor control algorithm analysis, or short-circuit characteristic simulation, without the need for real-time magnetic field FEA during operation. High-precision PMSM models depend not only on the accuracy of parameters such as flux linkage and inductance but also require the establishment of new motor mathematical models to achieve more accurate torque prediction. [PERSON] and [PERSON] [15] developed a Global Linearization Model (GLM), assuming linearization within the PMSM's operating range and considering the effects of non-sinusoidal distortion in inductances and flux linkage, as well as magnetic energy fluctuations under open-circuit conditions, on torque. However, the linearity assumption fails in the saturation region, limiting the model's application scope. [PERSON] et al. [16] used the frozen permeability method (FPM) to establish a torque model and analyzed the torque of PMSMs under steady-state current excitation. However, since the model still uses apparent inductances, the torque model only includes derivatives of flux linkage and magnetic energy, unable to account for torque ripple under non-steady-state excitation. [PERSON], [PERSON] [17] proposed a new algorithm combining the virtual work principle and the FPM, enabling appropriate on-load torque ripple calculation in both linear and nonlinear scenarios. [PERSON] et al. [18] proposed a nonlinear PMSM model based on differential inductance, establishing differential equations for flux linkage, magnetic energy,and torque with current and electrical angle as independent variables. By using FEA to obtain parameters such as motor differential inductance, flux linkage, and magnetic energy, and calculating torque through the introduction of saturated flux linkage differences, this approach achieves accurate prediction of torque ripple. However, the prediction accuracy of this method depends on the fine grid division of FEA operating points, and the local linearization assumption under non-steady-state excitation can introduce errors, reducing prediction accuracy. In recent years, approaches synergistically integrating physics-informed knowledge with data have gained significant traction for control and modeling in power electronics and motor drives. This trend highlights the potential of physics-informed neural networks (PINNs) to overcome challenges in electromagnetic modeling, such as the precise calculation of torque in permanent magnet synchronous motors (PMSMs). Conventional methods like FEA, while accurate, suffer from the absence of explicit functional relationships and significant computational expense. PINNs emerge as a promising solution, grounded in the universal approximation theorem. Their core methodology involves embedding the governing physical equations and boundary conditions directly into the neural network's loss function. This ensures the back-propagation process inherently respects the underlying physics, enabling the network to iteratively approximate the system's true physical structure through training [19; 20; 21]. Crucially, PINNs bypass the need for complex discretization schemes and numerical algorithms typical of traditional PDE solvers. Instead, they leverage automatic differentiation (AD) techniques during backpropagation to accurately characterize differential operators. This AD approach offers robustness against noisy data, a distinct advantage over numerical differentiation methods like finite differences.