## 4 Results and Discussion
### Artificial Neural Networks Modelling Results
#### 4.1.1 Cross-Validation Results
Instead, regarding the linear positive transfer function, a general trend cannot be observed for the considered indicators, with respect to the number of hidden layers as well as the neurons. Nevertheless, in the case of a single hidden layer, overall, the ANNs performance is poor compared to the cases of two or three hidden layers; it means that, with one hidden layer, the ANNs models show an underfitting problem (i.e., higher \(MSE_{cv}\) and smaller \(\Delta\)). However, the optimal network structure should not be identified by analyzing each indicator individually. Instead, it should be selected by an overall assessment of the four indicators. Indeed, the model with 2 hidden layers and 12 neurons could potentially be considered the best one by focusing the attention only on the R or MSE values (0.891 and 145,174, respectively), but 3 hidden layers and 8 neurons ensure much lower \(\sigma_{cv}\) and \(\Delta\) values (28,282 vs. 39,205 and 34,142 vs. 74,872, respectively), with just a little worsening of the R and MSE results (0.881 and 158,193, respectively). Based on such a consideration, the ITSM's best prediction capacity is achieved by a three hidden layers feedforward network, which makes use of the positive linear transfer function and eight neurons in each hidden layer (Figure 6). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Number of Hidden Layers**} & \multirow{3}{*}{**Test Performance Indicators**} & \multicolumn{6}{c}{**Positive Linear Transfer Function**} \\ \cline{3-8}  & & \multicolumn{3}{c}{**Number of Neurons in Each Hidden Layer**} \\ \cline{3-8}  & & **6** & **8** & **10** & **12** & **14** \\ \hline
1 & \(R_{cv}\) & 0.787 & 0.809 & 0.797 & 0.799 & 0.793 \\  & \(MSE_{cv}\) & 1.071 & 0.964 & 1.034 & 1.028 & 1.062 \\  & \(\sigma_{cv}\) & 0.201 & 0.197 & 0.251 & 0.270 & 0.183 \\  & \(\Delta\)Similar considerations can be outlined for MS models, according to the results reported in Table 5. Therefore, the MS final model is a three hidden layers network characterized by a positive linear transfer function and eight neurons in each hidden layer (Figure 6). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Hidden Layers**} & \multirow{2}{*}{**Test Performance Indicators**} & \multicolumn{6}{c}{**Positive Linear Transfer Function**} \\ \cline{3-7}  & & \multicolumn{3}{c}{**Number of Neurons in Each Hidden Layer**} \\ \cline{3-7}  & & **6** & **8** & **10** & **12** & **14** \\ \hline
1 & \(R_{cv}\) & 0.8  
Table 6 shows that the best results for the MQ parameter are obtained by one hidden layer networks, particularly by means of the positive linear transfer function and 12 neurons (Figure 7). However, less marked differences can be observed between the performance indicators of the MQ models, with respect to the transfer function type, being equal to the number of neurons and hidden layers. In summary, for the MQ parameter, a multiple hidden layers structure is not needed; for the MS parameter, a slight performance improvement can be observed when using three hidden layers, while a substantial predictive capacity improvement can be appreciated by means of a more complex neural network structure for the stiffness modulus. Among the mechanical parameters considered, the stiffness modulus represents the most important one for a rational performance-based characterization of the materials [1; 66; 72; 91]. Therefore, the use of multiple hidden layers architectures and the positive linear transfer function is justified for modeling stiffness laboratory data with regard to the HMAs considered. In the Supplementary Materials, the weights and bias values (Tables S8-S16), as well as normalization factors, are reported for each of the ANN models proposed to give the possibility of other researchers reproducing the ANNs topologies developed in this study.