## 3 Test Results and Discussion  
Out of the 1484 samples, 90% (1336 samples) were randomly selected for training, while 10% (148 samples) were reserved for validation without being used in the training process. Figure 4a shows the performance of the neural network during the training and validation phases. The best performance, measured by the mean squared error (MSE), was achieved after 1000 iterations (epochs) in the training phase, with an error of 0.04236. In the validation phase, the best result was obtained at the 245 th iteration, with an error of 0.10629. At that point, the MSE for training was 0.06364. The complete training of 1000 iterations took 11 s, as detailed in Table 1. Table 1 also presents the correlation values (R\({}^{2}\)) for the artificial neural network (ANN) phases and for the polynomial regressions. The R\({}^{2}\) value achieved by the ANN for 90% of the samples during training was 0.99998, and for the validation phase, with 10% of the samples, it was 0.99997, outperforming the polynomial regression techniques of degrees 1, 2, and 3, respectively. In Table 1, the MAE value of 0.21532 indicates that the model is, on average, off by about 0.215 units. Meanwhile, the MAPE of 1.8356% suggests that the average percentage error relative to the actual values is very low, which is a strong indication of accuracy. Figure 4b displays the error histogram, representing the difference between the desired output and the obtained output (\(\mid\text{Y}_{\text{des}}-\text{Y}_{\text{ob}}\mid\)) in relation to zero error. It is evident that the error remained close to zero for the analyzed data. Figure 4: ANN performance: (a) training and validation performance (MSE), (b) error histogram (targets Y\({}_{\text{des}}-\) outputs Y\({}_{\text{ob}}\)) for ANN with 20 intervals for the 1484 output samples. Figure 5 presents the correlations for the samples in the model validation phase, that is, 148 samples. Figure 5a shows the correlation between the desired and obtained outputs via the ANN. Figure 5b-d illustrate the correlations between the desired and obtained outputs using polynomial regressions of degrees 1, 2, and 3, respectively. Among the regressions, the best result was obtained with degree 1 regression with an R\({}^{2}\) of 0.99823, although it was still inferior to the result obtained with the ANN (R\({}^{2}\) = 0.99997). In addition to illustrating the direct comparison between the methods, Figure 5 highlights the ANN's superior capacity to match the desired outputs during the validation phase. The nearly perfect correlation observed in Figure 5a emphasizes the ANN's ability to capture the nonlinear complexities of the data, resulting in highly accurate predictions. By contrast, the polynomial regression models, particularly those of higher degrees, show more \begin{table}
\begin{tabular}{l c c} \hline \hline
**ANN** & **Specified Values** & **Achieved Values** \\ \hline Iterations & 1000 & * 1000 \\ Time (s) & 60 & 11 \\ Best Validation Performance (Iteration) & 1000 & 245 \\ Performance (MSE) Training (ANN) & 0.001 & 0.06364 \\ Performance (MSE) Validation (ANN) & 0.001 & 0.10629 \\ Performance (MAE) Validation & 0.001 & 0.21532 \\ Performance (MAPE) Validation & 0.00\% & 1.8356\% \\ Correlation (R\({}^{2}\)) Training (ANN) & 1.0 & 0.99998 \\ Correlation (R\({}^{2}\)) Validation (ANN) & 1.0 & 0.99997 \\ First-degree regression (R\({}^{2}\)) & 1.0 & 0.99823 \\ Second-degree regression (R\({}^{2}\)) & 1.0 & 0.99717 \\ Third-degree regression (R\({}^{2}\)) & 1.0 & 0.99684 \\ \hline \hline \end{tabular}  
* Achieved criterion.  
\end{table} 
Table 1: Values specified and achieved in training and validation of ANN compared to desired output Y\({}_{\text{des}}\). Figure 5: Correlation between Y\({}_{\text{des}}\) and Y\({}_{\text{ob}}\) outputs in validation phase (148 samples), (**a**) obtained via ANN; (**b**) first-degree regression; (**c**) second-degree regression; (**d**) third-degree regression. deviation in their predictions, as seen in Figure 5c,d.