## 3 Results and Discussion
### Flow around the SD7003 Airfoil
#### 3.2.2 Local Model Training  
In this section, we present the results obtained using the FFNN for the SD7003 test case. As mentioned earlier, we have ground truth data for two angles of attack for this test case. In Section 3.2.1, we demonstrated the superiority of using local coefficients instead of global ones by showing the different results of field inversion using different goal functions. However, to maintain coherence with the NACA0021 test case for which only global coefficients are available, we train the neural network using only global coefficients. Specifically, we use the results obtained by using the CL error as the objective function. These results are used to train both the FFNN and U-Net models. The details of the FFNN were already explained in Section 2.3. The loss function is defined as the mean squared error between the simulated correction field and the one obtained from the field inversion process. In Figure 12, we show the trend of the train and validation loss. As explained, we use an early stopping method to automatically stop the training in order to avoid overfitting. We used the R-squared (\(R^{2}\)) coefficient to determine the goodness of the neural network training. The \(R^{2}\) value measures the proportion of the variance in the dependent variable that is predictable from the independent variables. It is calculated as:  
\[R^{2}=1-\frac{SS_{res}}{SS_{tot}} \tag{39}\]  
where \(SS_{res}\) is the sum of squares of residuals defined as:  
\[SS_{res}=\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2} \tag{40}\]  
and \(SS_{tot}\) is the total sum of squares defined as:  
\[SS_{tot}=\sum_{i=1}^{n}(y_{i}-\overline{y_{i}})^{2} \tag{41}\]  
In these equations, \(n\), \(y_{i}\), \(y_{i}\) and \(\overline{y_{i}}\) are, respectively, the total number of data points, the ground truth value, the predicted value and the mean of the ground truth value. The \(R^{2}\) values range from 0 to 1, where 1 indicates perfect prediction and 0 indicates that the model  
Figure 11: Comparison of local coefficients for different goal function definitions for SD7003 at \(\alpha=8^{\circ}\). (**a**) Pressure coefficient. (**b**) Skin friction coefficient on the suction side. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **CL** & **CD** \\ \hline LES [54] & 0.92 & 0.043 \\ SA & 0.895 & 0.0435 \\ SA (FI\(-\)CL) & 0.917 & 0.0436 \\ SA (FI\(-\)CD) & 0.894 & 0.044 \\ SA (FI\(-\)CP) & 0.939 & 0.0372 \\ SA (FI\(-\)CF) & 0.935 & 0.0411 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of lift and drag coefficients for SD7003 airfoil at \(\alpha=8^{\circ}\). does not explain any of the variability. Thus, higher \(R^{2}\) values indicate better fit and more accurate predictions by the model. The value obtained in this training was \(R^{2}=0.862\). Figure 13 shows the results of using the FFNN implemented in the CFD solver. The simulation starts from the converged solution of the original SA model. To better understand the results, Table 4 presents the results in terms of CL and CD. It is evident that for \(\alpha=8^{\circ}\), we had a good match in terms of CL with the optimized result of the field inversion. This is not the case for \(\alpha=4^{\circ}\), where it is clear that the trained model did not learn the correct relation to predict the correct value of CL, which remains the same as the original SA model. By examining these results, it becomes apparent that while the FFNN showed promising results for certain angles of attack, it did not consistently achieve the desired accuracy across different conditions. This suggests that the model may require further tuning or a different approach to improve its generalization and prediction capabilities. We left an angle of attack \(\alpha=6^{\circ}\) out of the training set and did not perform field inversion for this angle. Instead, we used it as a test to understand how well the trained model can predict an angle not included in the training set. Figure 14 shows the results in terms of velocity and turbulent viscosity fields.