## 3 Results
### Prediction of the Flow Fields of Airfoils with Different Shapes with Pix2 pix  
Dataset 1 was trained with a batch size of 2. After the training process of 250 epochs, the mean absolute errors (MAEs) for the training and validation datasets were 0.06602 and 0.1162, respectively. The total learning time was 2 h 22 min. The MAE of the test dataset was 0.1369. The pix2 pix model was trained with the adaptive moment (ADAM) optimizer by setting \(\beta\)1 = 0.5, \(\beta\)2 = 0.999, and \(\varepsilon\) = 10\({}^{-8}\). The initial learning rate was set to 0.0002. Figure 6 shows the pix2 pix test results when 19 coordinate images of airfoil shapes were used as input. The left image shows the input and the middle image the flow field obtained through CFD. The right image shows a flow field prediction image obtained through pix2 pix. The mean square error (MSE) was applied to quantitatively evaluate the predictive performance. The more similar the image was, the smaller the MSE; for the same image, the MSE was 0. The MSE was obtained using Equation (2). The MSE for the test data was 0.02168. The minimum MSE is usually 0.00551 and the maximum MSE usually 0.09644. The MSEs shown in Figure 6 are: (a) MSE = 0.00702, (b) MSE = 0.00805, and (c) MSE = 0.00874.  
Figure 6: Prediction results of pix2 pix for the flow fields of three airfoil shapes with a constant angle of attack (10\({}^{\circ}\)) and Reynolds number (1.5 \(\times\) 10\({}^{6}\)): (**a**) DU 97-W-300, (**b**) DU 91-W2-250, and (**c**) DU 91-W2-250.  
Prediction of the Flow Fields of Airfoils with Different Shapes, Angles of Attack, and Reynolds Numbers with Pix2 Pix  
Dataset 2 was trained with a batch size of 10. After the training process of 50 epochs, the MAEs for the training and validation datasets were 0.06425 and 0.06245, respectively. The total learning time was 7 h and 10 min. The MAE for the test dataset was 0.0764. The pix2 pix model was trained with the ADAM optimizer by setting \(\beta\)1 = 0.5, \(\beta\)2 = 0.999, and \(\varepsilon\) = 10\({}^{-8}\). The initial learning rate was set to 0.0002. Figure 7 shows the pix2 pix test results when 19 coordinates of airfoil images with different angles of attack and Reynold's numbers were used as input. The left image shows the input and the middle image the flow field obtained through CFD. The right image shows a flow field prediction image obtained through pix2 pix. The MSE was also determined for dataset 2 for quantitative evaluation. The MSE of the test data was 0.00853. The minimum MSE is usually 0.00251 and the maximum MSE 0.12108. The MSEs shown in Figure 7 are: (a) 0.00531, (b) 0.00358, and (c) 0.00852.  
Overall, all the results (shown in Figures 4 and 5) were in good agreement with the ground-truth simulation results across the entire ranges of angles of attacks and Reynolds numbers for the three different airfoils.  
Prediction of the Aerodynamic Performance of Airfoils with Different Shapes, Angles of Attack, and Reynolds Numbers with the DNN  
The DNN was trained using dataset 2. For training, the batch size was 32 and the number of epochs was 1000. The ADAM optimizer was used to train the DNN model by setting \(\beta\)1 = 0.9, \(\beta\)2 = 0.999, and \(\varepsilon\) = 10\({}^{-8}\). The initial learning rate was set to 0.001. The test dataset had a mean square error (MSE) of 0.0087, mean absolute error (MAE) of 0.04, and  
Figure 7: Prediction results of pix2 pix for the flow fields of three airfoil shapes at constant angles of attack of 0\({}^{\circ}\), 5\({}^{\circ}\), and 15\({}^{\circ}\) and Reynolds numbers of 3.0 \(\times\) 10\({}^{6}\), 1.5 \(\times\) 10\({}^{6}\), and 1.5 \(\times\) 10\({}^{6}\), respectively: (a) DU 00-W2-401, (b) DU 97-W-250, and (c) DU 97-W-250.  
mean absolute percentage error (MAPE) of 9.45%. Figure 8 shows that the values predicted by DNN were positively correlated with the true values.