# A PINN-Based Nonlinear PMSM Electromagnetic Model Using Differential Inductance Theory
## 4 Examples of PINN Surrogate Modeling
### Analysis of PINN Modeling Results  
Figure 6 shows the loss function curves of the Fourier coefficients of the flux linkage for the sine and cosine terms, respectively. The results show that for the 0 th-, 6 th-, 18 th-, and 24 th-order Fourier coefficients of \(\tilde{\Psi}_{dd0\_\text{AM}}\), the PINN can converge effectively within 4000 epochs. As the higher order of the Fourier coefficients, the more complex the chain-current relationship is, the lower-order coefficients are fitted best, among which the 0 th-order coefficients reflect the DC component and average torque of the armature flux linkage  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Rated Voltage (DC/V) & 96 & Stator Outer Diameter (mm) & 155 \\ Rated Power (kW) & 12 & Stator Inner Diameter (mm) & 100 \\ Peak Power (kW) & 30 & Number of Pole Pairs & 4 \\ Peak Speed (rpm) & 9000 & Number of Slots & 48 \\ Peak Torque (N-m) & 75 & Core Material & B35 APV1900 \\ Number of Skew Poles & 2 & Permanent Magnet Material & N42 UH\_0/0708 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key parameters and materials of the PMSM.  
Figure 5: The finite element model of a \(1/8\) segment of an interior PMSM.  
\(\widetilde{\Psi}_{dq0\_AM}\), and the minimum mean square error reaches \(7\times 10^{-7}\) at all current operating points. The surfaces with 18 th- and 24 th-order coefficients are very complex, but PINN can still achieve satisfactory results, and the minimum mean square error can reach \(\mathbf{A}_{\text{f}}^{(6)}\) for all of them.  
The fitting performance of the PINN model for flux linkage is exemplified by the Fourier cosine coefficients, as illustrated in Figure 7, where the continuous surfaces represent PINN-generated results and solid red dots denote FEA sample points. The results demonstrate that the PINN framework achieves high-fidelity reconstruction of flux linkage harmonics across all orders while simultaneously satisfying both numerical agreement with FEA benchmarks and rigorous physical constraints--including curl-free field conditions and specified initial/boundary requirements.  
The PINN provides a continuously differentiable model for the PMSM, enabling systematic analysis of their electromagnetic field characteristics. As shown in Figure 8, the Fourier coefficient distribution of the armature magnetic co-energy is visualized alongside the flux linkage vector coefficients (indicated by red arrows). Notably, the flux linkage Fourier coefficient vectors remain perpendicular to the armature co-energy contour lines at all positions, numerically confirming the gradient relationship defined in (42). With increasing harmonic orders, both the armature co-energy and flux linkage distributions exhibit heightened spatial complexity, particularly in saturation-governed weak magnetic regions where localized field variations intensify. These nonlinear saturation effects manifest as sharp transitions in physical quantities and non-orthogonal harmonic interactions, demonstrating the intrinsic nonlinear behavior of PMSM under high saturation conditions.  
Figure 6: Training loss curves of the PINN. (**a**) Cosine term loss; (**b**) Sine term loss. DC, A6, B6,... denote the loss contributions associated with the \(\mathbf{B}_{\text{f}}^{(6)}\), \(\mathbf{A}_{\text{f}}^{(6)}\), \(\mathbf{B}_{\text{f}}^{(6)}\)..., respectively.  
Figure 7: Fitting Fourier coefficients of armature flux linkage by PINN.  
Figure 8: Distribution of the Fourier coefficients of armature flux linkage and magnetic co-energy over the dq-axis current plane. The boundary denotes the current-limit circle. At every point, the armature flux linkage vector (red arrows in the figure) is perpendicular to the co-energy contour lines.