## 3 Results
It is essential to point out that the ANN is working properly; thus, the proposed methodology is correct. However, for application on the specific parts of the jet engine, the CFD model must be optimized, not only from the aerodynamic and numerical simulation point of view. Furthermore, the preliminary results obtained by the ANN can serve for the specification of the parts of the model that must be optimized. Based on the presented results, it is possible to decide if the mesh size is suitable for the integration of the CFD model into the ANN calculations. Figure 7: Statistical data for the training process of the ANN for temperature field prediction at specific nodes. The nodes in Figure 8 corresponds to the coordinates specified, where these specific points are directly shown on the surface of the turbine blade of the iSTC-21v jet engine. Figure 8: Predicted temperatures in specific nodes representing the chosen positions on the surface of the turbine blade. Figure 9: Computed absolute errors of the temperatures determined for each node for the specified operational regimes. Qualitative coefficients of the performed analyses in the form of the estimated errors are summarized in Table 3. It is essential to emphasize that the trained neural network can now be used for other regimes to predict temperatures. In Table 3, mean absolute error (MAE) [47] was calculated based on the predicted temperatures according to Equation (3):  
\[MAE=\ \frac{1}{n}\sum_{n}^{1}\lvert y_{i}-\hat{y}_{i}\rvert \tag{3}\]  
The mean absolute percentage error (MAPE) was calculated according to the relationship:  
\[MAPE=\ \frac{100}{n}\sum_{n}^{1}\frac{\lvert y_{i}-\hat{y}_{i}\rvert}{y_{i}} \tag{4}\]  
The computed error for each point is shown in Figure 9, where the development of the error for the chosen nodes can be clearly seen. The maximal absolute error reached the value of 44.7 \({}^{\circ}\)C. For the comparison, in Figure 9 also the visualization of the absolute percentage error with the mean value of 0.5% can be seen. Estimated errors in Table 3 were computed in the Matlab software. The maximum difference is the maximal difference between the temperature predicted by the CFD methodology and the temperature estimated using the methodology based on the ANN. In Figure 9, each line corresponds to the errors calculated for the particular regimes. There are 10 lines, each for one regime, and dots correspond to the specific node on the surface of the turbine blade of the iSTC-21v experimental jet engine. As it was already mentioned, ANN is trained on 10 operation regimes; furthermore, trained ANN is used for predicting temperatures for more regimes. Thus, in Figure 10, the overview of predicted temperatures for 1800 boundary conditions (engine operation conditions) for the chosen 24 nodes representing the points at the turbine blade of the iSTC-21v jet engine can be seen. It is necessary to emphasize that only data from operation regimes 400 to 1400 are relevant due to the training process of ANN. A created ANN is used for the prediction of other regimes apart from ten trained regimes, and the results are in Figure 10; in the figure, it can be seen that data from 0 to 400 and from 1400 to 1800 regimes cannot be included for the result evaluation. The reason for the exclusion of these regimes is that during the training process, there were not included data for lower regimes, so, for example, there are not computed temperatures for the ambient conditions, etc. It is essential to emphasize that the main purpose of the proposed article is to demonstrate methodology and other regimes that can be included in the additional research; nevertheless, the results have shown a high level of accuracy and proved the fact that it is possible to predict temperatures using ANN. \begin{table}
\begin{tabular}{c c c} \hline \hline
**MAE (\({}^{\circ}\)C)** & **MAPE (\%)** & **Max. Difference (\({}^{\circ}\)C)** \\ \hline
5.59 & 0.5 & 44.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimated errors of the predicted temperatures.