## 3 Machine Learning for the Design Optimization of Electromagnetic Devices
### Machine Learning for Performance Prediction of Electromagnetic Devices  
The performance of electromagnetic devices highly depends on the field analysis results of electromagnetic, mechanical, and thermal analyses. These analyses are usually based on FEA and time-consuming, as different dimension and materials and excitations will affect the results. Moreover, the performance of electromagnetic devices, like the torque and efficiency of a motor, depends on the accurate estimation of the flux linkage and core loss. Typically, those measures feature strongly nonlinear and multi-modal characteristics regarding the input parameters, e.g., due to saturation effects. Consequently, surrogate models based on parametric or semi-parametric approaches might not follow accurate results in general. Through several attempts on machine learning methods, it is found that deep learning algorithms, like CNN and RNN, are good at the distribution estimation of magnetic field and temperature [129; 130; 131], and the prediction of torque and efficiency for motors [132; 133; 134]. These works established a solid foundation for the generalizable data-driven model for the analysis, design, and optimization of electromagnetic devices [129]. An example for the torque prediction of a switched reluctance motor (SRM) based on SVM is considered in the following. Figure 7 shows the machine topology of a segmented-rotor SRM with 16/10 stator/rotor poles. As shown, the motor consists of 8 excited stator poles and 8 auxiliary poles (16 poles in total). The basic operating principle and structural parameters of this motor have been introduced in our previous work [133]. In general, accurate torque modeling of SRM is a difficult problem, as this motor features a double salient structure. Thus, the torque response usually shows a significant ripple, and its modeling is a highly nonlinear problem. In a previous study, two significant factors, phase current and position angle, were investigated as inputs for modeling the torque based on three forms of SVM algorithms. They are a conventional SVM, a least square support vector regression (LSSVR), and a maximum-cornerntropy-criterion-based least squares support vector regression (MCC-LSSVR). Figure 8 illustrates the modeling of phase flux linkage and torque of this motor by using the MCC-LSSVR model. Table 3 lists the mean absolute error (MAE) and root mean square error (RMSE) for all three modeling approaches. As shown, the MCC-LSSVR model appears more effective than the other two techniques. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Mathematical Expression** & **Type** \\ \hline Response surface model (RSM) & \(y=X\beta+\varepsilon\) & Parametric \\  & X: Structure matrix; \(\beta\): Coefficient matrix & Parametric \\ Radial basis function (RBF) & \(y=\sum\limits_{i=1}^{n}\beta_{i}H(||\mathbf{x}-\mathbf{x}_{i}||)\) & Parametric \\  & \(H\): RBF function; \(\beta_{i}\): Coefficient matrix & Parametric \\  & \(y=q(\mathbf{x})^{\prime}\beta+z(\mathbf{x})\) & \\ Kriging & \(q(\mathbf{x})\): Basis function; \(\beta\): Coefficient matrix; z(x): A stochastic process & Semi-parametric \\  & \(y_{j}=f(\sum\limits_{i=1}^{n}w_{ji}x_{i}-\theta_{j})\); & \\ Artificial neural networks (ANN) & Basic artificial neuron model, \(w_{ji}\): Weightings, \(\theta_{j}\): Neuron’s & Non-parametric \\  & activation threshold; \(f\): Transfer function. & \\  & \(y=w\cdot\phi(x)+b\) & \\ Support vector machines (SVM) & \(\phi\): A function maps the input space to a higher dimensional & Non-parametric \\  & feature space, \(w\) is a weighting vector, \(b\): Bias term. & \\  & \(y=\sum\limits_{i=1}^{K}\beta_{i}\beta\left(\mathbf{w}_{i}\mathbf{x}_{j}+b_{i}\right)\) & \\ Extreme learning machines (ELM) & \(g\): Activation function, \(\mathbf{w}\): Weighting vector; \(\mathbf{b}\): Threshold & Non-parametric \\  & [138]. & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of several surrogate models.