## 4 Results and Discussion  
Figure 8 shows the k-fold CV score of neural models that were designed in the 500 iterations of the BO algorithm. The [PERSON]'s R coefficient averaged over the 5 validation folds tended to become uniform in value among the different combinations tested by the BO optimizer at least during the first 350 iterations (Figure 8) but showed more marked variations afterwards. The algorithm has detected a region of the search space where the validation error was not significantly changing and, having assessed an over-exploration of a specific zone, it has decided to focus its search on an unexplored area that might have high-performing solutions (however, identifying worse combinations). This result highlights the regulation between exploitation and exploration performed by the UCB algorithm. Despite the large size of the search space, the optimal model (i.e., showing the minimum validation loss MSE\({}_{\text{CV}}\) = 0.249 and then the maximum value of the Pearson coefficient R\({}_{\text{CV}}\) = 0.868) was found at iteration 54 (Figure 8). The hyperparameters discovered by the BO algorithm defined an ANN with \(L\) = 5 layers, \(N\) = 37 neurons, and hyperbolic tangent activation function (_tanh_), that was trained for \(E\) = 3552 iterations, with a learning rate of \(\alpha\) = 0.01 and weight decay \(\beta\) = 1:\(10^{-6}\). Table 5 shows the validation MSE (second column) and the R-score of the optimal model for each of the 5 folds (last column). In addition, the final average results for each mechanical characteristic and volumetric property are  
Figure 8: Average R-score on the 5 validation folds for the 500 algorithm iterations. reported in the last row of Table 5. Figure 9 shows the relation between network output and experimental target for each fold. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Fold**} & \multirow{2}{*}{**MSEmean**} & \multicolumn{4}{c}{**R—Pearson Correlation Coefficient**} & \multirow{2}{*}{**Rmean**} \\ \cline{3-3} \cline{5-6}  & & **ITSM** & **MS** & **MF** & **AV** & \\ \hline
0 & 0.219 & 0.837 & 0.866 & 0.842 & 0.964 & 0.877 \\
1 & 0.203 & 0.963 & 0.835 & 0.725 & 0.973 & 0.874 \\
2 & 0.254 & 0.872 & 0.836 & 0.799 & 0.917 & 0.856 \\
3 & 0.223 & 0.918 & 0.826 & 0.917 & 0.956 & 0.905 \\
4 & 0.346 & 0.841 & 0.731 & 0.834 & 0.912 & 0.829 \\ \hline \(\text{CV}_{\text{result}}\) & 0.249 & 0.886 & 0.819 & 0.824 & 0.944 & 0.868 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Number and codes of Marshall specimens. Figure 9: Regression analysis on the 5 folds. Although there is a high variability in the data set, which can be explained by considering the different properties of the mixtures analysed, the optimal BO model returns very successful results (Figure 9). The 5-folds averaged Pearson coefficient for each mechanical and physical parameter is always greater than 0.819 (last row of Table 5). Fluctuations in values of the MSE and R parameters (second and seventh column in Table 5) can be attributed to the different distribution of training and test data that characterize each fold. This result shows how a fixed split of the data set can lead to an incorrect assessment of the prediction error, which can be worse (R4 = 0.829) or better (R3 = 0.906) than the most likely situation represented by the k-fold CV (RCV = 0.868). Random and grid searches based on the prediction error by a fixed split training test may find solutions that are not optimal [49], due to fluctuations resulting from considering one partition rather than another. Figure 10 shows the comparison between experimental targets and predicted outputs, for the four parameters analyzed, as regards fold 4. Values calculated by the ANN model characterized by the highest prediction error (MSEmean = 0.346, Table 5) are very close in value to the experimental data, whatever variable is considered.