# Aerodynamic Analyses of Airfoils Using Machine Learning as an Alternative to RANS Simulation
## 4 Results and Discussions  
The BPNN was trained in MATLAB using a diverse number of nodes or neurons in the hidden layer to obtain the best root mean squared error (RMSE) value. The results that were obtained with the varying numbers of neurons in the hidden layer are summarized in Table 2 with their respective RMSE values. It can be seen that the best RMSE value was attained when the number of neurons in the hidden layer was 10, with RMSE value of \(3.57\times 10^{-7}\) for the validation case with 28 epochs. The correlation values (\(R\)) were very close to 1 for all of the training, validation and testing cases, which was an indication of the high accuracy of the predicted values. It is also evident from the table that the RMSE was sensitive to the number of neurons in the hidden layer, although the correlation value (\(R\)) did not show any visible variations with changes in the number of neurons in the hidden layer. Figure 5 illustrates the association between the number of neurons in hidden layer and the RMSE value. It is evident that the minimum value of \(3.57\times 10^{-7}\) for RMSE was achieved with 10 neurons in the hidden layer. The RMSE value was sensitive to the number of neurons; however, it did not show a continuous trend or an established relationship. Initially, when the number of neurons increased from 10 to 40, the value of RMSE also increased, after which it decreased as the number of neurons increased to 50. The RMSE again increased when the number of neurons further increased to 60; however, on increasing the number of neurons further, the RMSE started to decrease until the number of neurons reached 80 in the hidden layer, after which the RMSE started to increase again until the end of the experiment. The number of epochs vs. RMSE plot is presented in Figure 6. The network was trained with different numbers of neurons in the hidden layer. In each case, the training was continued until the best RMSE value was achieved, irrespective of the number of epochs that it took to achieve the best RMSE value. In each of the training sessions, the RMSE varied with the number of epochs. For the best case, the number of epochs was 28 with an RMSE value of \(3.57\times 10^{-7}\). Depending on the number of neurons in the hidden layer, the number of epochs varied to achieve the desired output. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**S No. **} & **No. of** & **RMSE** & **No. of** & **Corr Coeff** & **Corr Coeff for** & **Corr Coeff** \\  & **Neurons** & \(\times 10^{-7}\)** & **Epochs** & **for Training** & **Validation** & **for Testing** & **Overall** \\ \hline
1 & 10 & 3.57 & 28 & 0.9996 & 0.99994 & 0.99994 & 0.99971 \\
2 & 20 & 5.61 & 34 & 0.99969 & 0.9999 & 0.99983 & 0.99974 \\
3 & 30 & 16.2 & 11 & 0.99  
A performance plot of the RMSE value and the number of epochs vs. the network response for the training, testing and validation datasets is depicted in Figure 7. The figure shows that the RMSE value was relatively high at the start of the training and gradually reduced as the training progressed. After 28 epochs, the RMSE value became constant at 3.57 \(\times\) 10\({}^{-7}\) and did not fall any further. This performance plot of RMSE vs. the number of epochs represents for the best RMSE case only. It can be seen that the RMSE values for the training data are relatively large compared to the testing and validation data, which was attributed to the limited number of training data points (440 cases) that was used in the present work [36]. Figure 5: Number of neurons in the hidden layer vs. RMSE. Figure 6: Number of epochs vs. RMSE. A regression plot for the training, testing and validation subsets (for the best RMSE case) is depicted in Figure 8. It can be observed that the values of R were very close to 1 in all instances, which indicated a very high correlation between the predicted and actual target values. This means that a high accuracy of predicted values was achieved using this trained model. The training of the BPNN model was performed using historical data.