## 3 Machine Learning Techniques  
In this approach, in order to figure out the optimal antenna design which has the lowest _Cost_ corresponding to a set of design parameters, we employed a deep neural network (DNN) model to handle this regression task [33]. The DNN model is a subfield of machine learning with algorithms inspired by the idea of simulating the human brain [34]. The DNN model has recently gained increasing research interests due to their capability of overcoming the drawback of traditional algorithms dependent on hand-designed programs, and is investigated in many different domains, such as speech recognition, computer vision, pattern recognition, and regression [35]. Figure 2 shows the structure of a DNN model, which consists of an input layer, hidden layers, and an output layer. The input layer contains all visible information about the object of interest (known as \"data\" in machine learning) and a hidden layer executes complicated mathematical functions to extract valuable features from the input layer. Based on that, the appropriate information is calculated for the output. Besides, regression analysis is a statistical measurement for estimating the relationships between one dependent variable (the factor we want to understand or predict) and a series of independent variables (the factors that influence the dependent variable) [33]. Here, we propose a DNN model based on regression for the design of a bent wire monopole antenna. In this problem, the input variables are assigned as a set of six design parameters \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Range** & **Number of Points** \\ \hline Frequency & 950–1050 MHz & 201 \\ Length of wire-section (\(l_{n}\)) & 0.03–0.18 \(\lambda_{0}\) (9–54 mm) & 10 \\ Bending angle (\(\theta_{n}\)) at bending & 10–90\({}^{\circ}\) & 9 \\ point \(P_{n}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Conditions of design parameters and interesting frequency for training data acquisition. Figure 2: DNN structure. \(X=\{l_{1},l_{2},l_{3},\,\theta_{1},\,\theta_{2},\,\theta_{3}\}\) (expressed in Section 2), and the output of the regression task is the corresponding output \(y=Cost\) as defined in (1). Therefore, the relation between the input and the output is formulated as:  
\[y=F(aX_{i}+b), \tag{3}\]  
where \(F\) represents the DNN regression model including \(a\) and \(b\) as unknown parameters; the simple denotation for the complexity of the problem will be defined during the training process. In the training phase, mean squared error (\(MSE\)) is utilized as the loss function [36]. The \(MSE\) is computed as the average of the squared differences between the predicted and actual \(Cost\) values as:  
\[MSE=\frac{1}{n}\sum\limits_{i=1}^{n}(y_{i}\ -\ \hat{y}_{i}), \tag{4}\]  
where \(n\) is represented for the number of data points, as well as \(y_{i}\) and \(\hat{y}_{i}\) are the actual and the corresponding predicted \(Cost\) values to the given design parameters \(X_{i}\), respectively. Minimizing the \(MSE\) to zero is key for training the model because the \(MSE\) approaching zero implies that the predicted \(Cost\) is substantially similar to the actual \(Cost\) computed from NEC simulation. In addition, R-squared score (\(R^{2}\)) was chosen as a performance metrics to evaluate the model performance by comparing the trained model predictions with the observed data [37; 38]. It is a statistical measure of how close the data are to the fitted regression line. The R-squared score is computed as:  
\[R^{2}=1-\frac{\text{Unexplained variation}}{\text{Total variation}}=1-\frac{\sum \limits_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}}{\sum\limits_{i=1}^{n} \left(y_{i}-\overline{y}_{i}\right)^{2}}, \tag{5}\]  
where the unexplained variance is the sum of the square of the difference between the actual and predicted \(Cost\) values and the total variance is calculated in a similar way by replacing the predicted value by the average actual one \(\overline{y}_{i}\).