## 3 Machine Learning for the Design Optimization of Electromagnetic Devices
### Machine Learning for Optimization of Electromagnetic Devices  
Overall, there is more research carried out on machine learning for improving the runtime of the optimization of electromagnetic devices, as was also shown in Table 2. Different machine learning algorithms, such as SVM, multi-layer perceptron (MLP), K-nearest neighbor (KNN), and CNN have been investigated to optimize transformers, antennas, and motors (motors are the majority applications) [146; 147; 148; 149; 150; 143; 144; 145; 146; 147; 148; 149; 151; 142]. It is noted that deep learning follows promising results when applied for topology optimization of electromagnetic devices, and this topic has attracted much attention recently [143; 144; 145]. The presented studies confirmed that good optimization results can be obtained by using different machine learning models for optimization.  
To illustrate the effectiveness of these models, as an example, a single-objective optimization problem of a SMES is investigated below. Three types of surrogate models, RBF (a parametric model), Kriging (a semi-parametric model), and ANN (a non-parametric model), are compared. Meanwhile, an optimization strategy, sequential optimization method (SOM), is investigated to decrease the computation cost of FEM.  
In the optimization, the dimensions of the outer superconducting coil, \([R_{2},h_{2}/2,d_{2}]\) as shown in Figure 1b, are optimized to minimize the mean stray fields (\(B_{stray}\)) while keeping the stored energy (\(E\)) close to 180 MJ and guaranteeing the requirements for achieving superconductivity. Other parameters are fixed for this case study. Detailed information about the parameters and objective can be found in [98; 99].  
Figure 9 illustrates the optimization results of SOM by using these three models. As shown, RBF model requires 5 optimization loops to output the final optimum, while the other two models only need 4 loops to converge [6]. Table 4 lists the final optimal results. For the purpose of a sound comparison, the direct optimization results of DEA with FEM are listed in the table as well.  
As shown in the Figure, though the RBF model has the smallest optimum for the first loop of SOM, the differences among the optimal results of three models are small. In the first optimization loop of SOM, the same samples are used to derive the models, then DEA is employed for optimization to find the ideal result. After the convergence of the SOM, the difference among the considered approaches becomes relatively small, and the overall best results are achieved for the ANN-based approach. Regarding the required number of samples evaluated through FEA, the combination of SOM and any of these modeling approaches necessitates approximately 200 samples, which are less than 10% of that required by the direct DEA optimization (2310 evaluations). Therefore, such approaches are effective for optimization and facilitate minimizing the computational cost and the corresponding runtime.  
More importantly, there is no big difference between different models with SOM. Therefore, it can be concluded that the optimization strategy may be more important than the particular modeling approach for the optimization of electromagnetic devices. For high-dimensional problems, this conclusion has been confirmed by further studies [47; 48].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Modeling Method** & \multicolumn{2}{c}{**Flux Linkage (mWb)**} & \multicolumn{2}{c}{**Torque (Nm)**} \\  & **MAE** & **RMSE** & **MAE** & **RMSE** \\ \hline SVM & 0.846 & 0.756 & 0.1008 & 0.0925 \\ LSSVR & 0.424 & 0.306 & 0.0525 & 0.0494 \\ MCC-LSSVR & 0.086 & 0.073 & 0.0252 & 0.0189 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of modeling accuracy of several methods.