## 3 Machine Learning Techniques
### Validation of Machine Learning Model  
To find the optimal design parameters of the bent wire antenna based on the proposed DNN model, a grid search is formed to the parameters \(l_{1}\), \(l_{2}\), \(l_{3}\), \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) with the number of points for each parameter double that of the one used for the training set while maintaining the range of each variable. This search ensures that this grid is denser than the training dataset. In other words, the number of points for each wire section length and each bending angle is 20 and 18, respectively. Therefore, the total number of samples generated by all possible combinations of parameters in this grid is 46,656,000. All the samples in the grid are then put through our well-trained DNN model to obtain the corresponding _Cost_ value. To validate our ML model, we compared their _Costs_ derived from the ML model with those calculated by NEC simulation after extracting several designs of the antenna with _Costs_ of around 60, 70, and 80 from the grid search. Table 5 represents the design parameters and corresponding _Costs_ derived from ML model and NEC simulation. The comparison of _Costs_ in Table 5 reveals that the relative errors between _Costs_ from the ML model and NEC simulation are smaller than 2%, which verifies the accuracy of our ML model. After searching the antenna models with high _AF_ on the entire grid using our ML model, we tabulated the design parameters and _AFs_ and _Costs_ of resulting antennas in comparison with the corresponding _AFs_ and _Costs_ derived by NEC simulations in Table 6. Table 6 shows the design parameters of the ranked antennas in terms of _AF_/_Cost_ when searching the antenna models on the entire grid. The ranked antennas in Table 6 achieve _AF_ over about 77%, which means the _Costs_ are smaller than about 46. In addition, the resulting _AFs_ from our ML model favorably agree with those from NEC simulation in that the maximum deviation between both _AFs_ for the same model is smaller than 1%. This agreement of the _AF_ indicates that our ML model is well-trained and capable of evaluating the impedance matching characteristic of a wire monopole antenna accurately. The performed validation informs that the ML technique can be possibly applied to the antenna design from a practical standpoint. We thus expect that there may be a few advantages if the proposed method using ML technique is applied to antenna design. First, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Mean Squared Error** & **R-Squared Score** \\ \hline Lasso & 753.498 & 0.0533 \\ Linear regression & 521.797 & 0.3444 \\ Ours (DNN) & 0.0353 & 0.999 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison with other machine learning techniques. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Design Parameters**} & \multicolumn{3}{c}{_AF (\%)/Cost_} \\ \hline  & **Unit: mm** & \multicolumn{3}{c}{**Unit: degree**} & \multicolumn{3}{c}{**ML**} & **NEC** \\ \hline \(l_{1}\) & \(l_{2}\) & \(l_{3}\) & \(\theta_{1}\) & \(\theta_{2}\) & \(\theta_{3}\) & **Model** & **Simulation** \\ \hline
23.2 & 16.1 & 27.9 & 75.9 & 71.2 & 19.4 & 70.1/60.0 & 69.6/61.0 \\
32.7 & 18.5 & 18.5 & 71.2 & 75.9 & 42.9 & 65.2/70.0 & 65.6/69.1 \\
35.1 & 18.5 & 18.5 & 47.6 & 90.0 & 85.29 & 60.2/80.0 & 60.1/80.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Design parameters of the ranked antenna models in terms of _AF_ when searching antenna models on the entire grid. we can extract approximated optimal design parameters from the grid search using the ML model when beginning antenna design. Second, we can easily derive the key design parameters by analyzing the change of an evaluation function corresponding to a variation of an individual design parameter. Finally, we can expansively apply the proposed method using the ML technique to the design of antennas with a more complicated structure.