## 4 Results
\[\epsilon=\frac{||\left(\
icefrac{{L}}{{D}}\right)_{ANN}-\left(\
icefrac{{L}}{{D}} \right)_{CFD}||_{2}}{||\left(\
icefrac{{L}}{{D}}\right)_{CFD}||_{2}}. \tag{12}\]  
The values of \(\epsilon\) obtained only with test data sets are used for predicting ANN accuracy, because \(\epsilon\) obtained with the training data sets is not a good measurement for the prediction accuracy due to the possibility of the overfitting. Figure 12 shows the influence of ANN-related parameters on the prediction accuracy. To identify the influence of the regularization constant on ANN accuracy, the prediction error (measured by the averaged value of \(\epsilon\)) of the 80 test cases is plotted with respect to \(\lambda\) in Figure 12a. No clear difference of the error is observed at \(\lambda=0.1\), 0.01, or 0.001, but the best regularization constant is set at 0.001 in the current study because a slightly smaller error is measured at \(\alpha=0.001\) compared with the other cases. Based on this result, the variation of the prediction accuracy is investigated as a function of \(M\) while fixing \(\lambda\) to 0.001. As shown in Figure 12b, the influence of \(M\) on the prediction accuracy is certainly more distinguishable than that of \(\lambda\), and the best value of \(M\) is set to 3 based on the results. Finally, the graph plotted in Figure 12c presents the change of the prediction accuracy with respect to \(N\) when \(\lambda=0.001\) and \(M=3\). As the smallest error is obtained at \(N=32\), the 32 neurons are distributed at each hidden layer when the surrogate model is constructed. With these ANN-related parameters, the training data error is \(\epsilon=8.419\times 10^{-6}\), while the test data error is \(\epsilon=1.204\times 10^{-4}\), demonstrating the high prediction accuracy of the current study's constructed surrogate model (see Figure 13). Figure 11: Structure of the constructed ANN. The number of hidden layers and neurons at each hidden layer are defined as \(M\) and \(N\), respectively. Using the ANN, the value of the lift-to-drag ratio is predicted as a function of \(\omega_{1}\) and \(\omega_{2}\). Figure 12: The influence of the ANN-related parameters on its prediction accuracy. (**a**) Regularization constant, (**b**) number of hidden layers, and (**c**) number of neurons. Figure 13: The accuracy of the ANN in predicting the CFD-computed lift-to-drag ratio. The black dots are the 19 data sets used to train the ANN, while the red dots are four test data sets used to measure the ANN’s accuracy. The R-squared values of the training data sets and test data sets with a 45\({}^{\circ}\) line are 0.9992 and 0.9643, respectively. added to the training data sets; then, the surrogate model is updated. An optimal solution is computed again, and the process is repeated until the loop reaches the maximum iteration number. In the current study, the maximum iteration number for this optimization loop is set to 5. Table 3 shows the changes of the optimal sets of weights during the iterative optimization process. For all the optimization iterations, \(\omega_{1}\) is almost fixed to \(-0.25\), while \(\omega_{2}\) slightly varies from 0.775 to 0.805. The fourth and fifth columns from the left show the ANN-computed and CFD-computed lift-to-drag ratios, respectively, and the rightmost column presents their percentage difference compared with the CFD-computed lift-to-drag ratio. Although the monotonical decrease in the discrepancy between the ANN-computed and CFD-computed lift-to-drag ratio is not observed, the discrepancy tends to reduce as the optimization iteration number increases, indicating that the prediction accuracy of the constructed ANN is improved as the optimization loop repeats. Due to the improvement of the ANN's prediction accuracy, the lift-to-drag ratio of the optimal models become better in the optimization step. When the optimization loop has been completed, the best performing optimal model is found at \(\omega_{1}=-0.2499\) and \(\omega_{2}=0.7751\), for which the lift-to-drag ratio is about 6.703.