## 4 Results and Discussion
### Machine Learning Results Discussion  
As far as data modeling is concerned, the implementation of the LOOCV resampling method into the SNNs training and testing processes resulted in a smooth and accurate regression of the few available observations. In particular, the grid search allowed the optimal model structure to be identified (Figure 9): the SNN, characterized by \(N=21\) neurons in its hidden layer and \(FUN=\) TanH activation function, showed a mean squared error \(MSE_{CV}\) equal to 0.0032 and a correlation coefficient \(R_{CV}\) of 0.9988. This network has a total number of parameters \(\delta\) equal to 238, of which 96.4% (=\(ENP\)) contribute to effectively reduce the loss function. Such results led to the highest \(L_{CV}\) value of 0.9609. In the following, this model is referred to as TanH-SNN. The TanH activation unit, which gives the non-linearity required by the problem to the proposed model, has shown greater effectiveness compared to the more recently introduced ELU function (widely used in the computer vision field) for the greater number of parameters (i.e., \(ENP\) equal to 0.9641 for TanH, 0.9319 for ELU) that actually contribute to defining the predictive performance of the SNN model, although the higher number of neurons that comprise the TanH-SNN hidden layer increases the calculation time compared to the ELU-SNN (i.e., \(N\) is 21 for TanH, 16 for LogS, as shown in Figure 9). No relevant differences were found between the best-performing models with TanH and LogS activation function (Figure 9). All grid search results (for each number of neurons and activation function considered) are provided in the Supplementary Materials (Table S2). To verify that the TanH-SNN had returned a smooth regression of the data (without excessive oscillations, not physically acceptable), the seven target variables were predicted, separately for the two filler materials, in increments of 0.05% bitumen content from 4.5 to 6.0% and of 0.10% filler content from 4.0 to 8.0%. In this way, the regression surface of the TanH-SNN model for each target variable was plotted in a space whose axes represent the bitumen content (\(x\)-axis), the percentage of filler by volume of the mix (\(y\)-axis), and the desired target variable (\(z\)-axis). In addition, a contour map was included in the plane defined by the bitumen and filler contents to outline the course of the regression surface. In Figure 10, these surfaces are shown for the Marshall stability parameter before (MS) and after aging (MSA). Regression surfaces of the remaining variables are included in the Supplementary Materials (Figures S1 and S2). The azimuth and elevation angles of the line of sight were changed between the surface graphs of the different target variables for a better understanding of the trends. The blue spheres in Figure 10 represent the experimentally observed data. The prediction surfaces in Figure 10 show an almost perfect adaptation of the model to the experimental data and demonstrate that TanH-SNN has correctly identified the mechanical (and volumetric) response of the mixes: in fact, all the findings reported in Section 4.1 can be verified anew by analyzing these surfaces. As a result, it can be concluded that the modeling procedure followed had a positive outcome on the fitting of the experimental data presented. Figure 10: Model prediction surface over feature variability ranges for MS and MSA parameters. Figure 9: Results of the grid search approach for each considered activation function. The neural model developed can enhance the asphalt mixture optimization by means of the numerical prediction of the mix response for different compositions without the necessity to perform any other laboratory tests.