## 3 Design Optimization
### Metamodeling  
Based on the DOE results, metamodels for the multi-objective functions and constraints were generated. Metamodels were generated using the proposed MLP and the conventional Kriging model. These metamodels can be classified as a regression model and an interpolation model, respectively. Regression models, e.g., polynomial regression (PR), radial basis function regression (RBFr), ensemble of decision trees (EDT), and MLPs, smoothen the noise in the data because they do not pass through the test points exactly. Therefore, this type of model is useful for real experiments with random errors. MLP is a type of deep learning algorithm and has the advantage of being able to represent nonlinear relationships between the input and output variables [20]. Neural networks can learn complex patterns using neurons that mathematically transform data. A neuron, also called a perceptron, is a mathematical function. One or more inputs are multiplied by weights, and the results are then passed to a nonlinear function known as the activation function for output. One layer that consists of several neurons is called a hidden layer. The number of hidden layers and the number of neurons that make up each hidden layer are determined by the design problem, as shown in Figure 6.  
In contrast, interpolation models, e.g., Kriging model and radial basis function regression interpolation (RBFi), are well suited for function approximation using analytical results without random errors because these models pass through the test points exactly. The estimated equation of the Kriging model was defined to eliminate bias and thereby minimize error variance [21]. The Kriging model is a weighted linear combination as follows:  
\[z\star=\sum_{i=1}^{n}\lambda_{i}z_{i} \tag{4}\]  
where \(z^{\star}\) is an estimated point using the Kriging model, \(n\) is the total number of experiments, \(\lambda_{i}\) is the weight value function, and \(z_{i}\) is the experimental point. Metamodeling was carried out using a software package for optimal design called PIAnO. Metamodeling using MLP requires the selection of hyperparameters such as the number of hidden layers and neurons, activation functions, and learning rates. Hyperparameters can also be selected manually, but it is difficult to find the optimal value and these values do not reflect any prior knowledge gained during the investigation. Therefore, hyperparameters were automatically selected to maximize the predictive performance of the MLP using the 'hyperparameter optimization' function provided by PIAnO [22]. Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) was used as the optimization algorithm for the automatic selection of hyperparameters, as this algorithm is suitable for large-scale numerical optimization [23]. The metamodeling results for the multi-objective function  
Figure 6: Structure of MLP. and constraints using MLP are shown in Figure 7. The metamodels for the torque and THD of the back EMF set by the multi-objective function are shown in Figure 7a,b. The number of hidden layers of the MLP for the torque was seven and the number of neurons of each layer was between two and five. The number of hidden layers for the THD of the back EMF was one, and the number of neurons was three. The activation function of neurons was also automatically selected by the 'hyperparameter optimization' function provided by PIAnO, and the icons of the neurons in Figure 7 represent the selected activation function [22]. The activation functions of neurons for the torque and THD of the back EMF were hyperbolic tangent and sigmoid functions, respectively. Torque ripple is calculated from the average and the peak-to-peak values of the torque. Efficiency is also calculated from the mechanical output and losses. Because the torque ripple and efficiency were calculated using the FEA results, they are not suitable for direct metamodeling. Therefore, metamodels for the peak-to-peak value of the torque and core loss were generated, as shown in Figure 7c,d.