## 3 Methodology
### Leave-One-Out Cross-Validation  
The standard practice of splitting the available dataset into 2 random subsets for training and testing (also called hold-out method) may result in biased performance evaluations due to the different distribution of data within such splits (the test error rate can be highly variable depending on precisely which observations are included in each set), along with the risk of missing some relevant trends in training data. These effects are particularly marked when the dataset is relatively small [58]. Conversely, the leave-one-out cross-validation (LOOCV) method [59] repeatedly performs a partition of the experimental dataset into a training set containing all but 1 observation and a test set that contains only that observation (Figure 4). Therefore, if the available set of observations consists of \(n\) data, the ANN model is trained on \(n-1\) observations and the fitted model is used to predict the response for the excluded observation. Such procedure is repeated for \(n\) splits: the first training set contains all but observation 1, the second training set contains all but observation 2, and so on. Each fitted model produces a squared error \(MSE_{i}\) computed as the squared difference between the prediction \(\hat{\mathbf{y}}_{i}\) and the experimental value \(\mathbf{y}_{i}\) of the excluded observation: \(MSE_{i}=\|\hat{\mathbf{y}}_{i}-\mathbf{y}_{i}\|_{2}^{2}\). The test error of the ANN model is then estimated by averaging the \(n\) resulting \(MSE\)'s:  
\[MSE_{CV}=\frac{1}{n}\sum_{i=1}^{n}MSE_{i} \tag{5}\]  
Thereby, the LOOCV allows the model to be trained on an almost entire dataset, resulting in the absence of a designated test set, and provides an unbiased estimate of the prediction error rate [60]. Furthermore, by removing the randomness in the hold-out splits, LOOCV always returns the same results. Nevertheless, the model presented in this study aims to describe the physical properties and mechanical response of 2 groups of mixtures characterized by different filler materials. For this reason, a cross-validation was implemented by excluding 1 observation  
Figure 4: Schematic representation of the leave-one-out cross-validation procedure. for each type of filler used, defining 12 stratified couples of observations. In other words, the model was fitted on 22 observations and tested on an independent data couple, repeating the procedure 12 times. In this way it was possible to reduce the computational cost of fitting 24 models (\(n=24\) is the size of available dataset) and to decrease the variance of the cross-validation procedure because the overlap between the 12 training sets is lower, and consequently, the model outputs are somewhat less correlated with each other. Among the possible combinations of \(N\) and \(FUN\), a grid search approach was performed to select the combination that maximized the difference between the effective number of parameters (i.e., \(\gamma=\{0,\cdots,\delta\}\) is a measure of how many weights and biases in the neural network are effectively used in reducing the loss function [55] divided by the total number of network weights and biases), \(ENP\), and the averaged test error, \(MSE_{CV}\), of the twelve fitted models:  
\[ \begin{split} L_{CV}(\delta_{i},\alpha_{i},\hat{y}(\mathbf{W}_{i}), y,W_{i})&=ENP-MSE_{CV}=\\ &=\sum_{i=1}^{12}\frac{\delta_{i}-2\alpha_{i}\text{tr}\big{(} \
abla^{2}F(\mathbf{W}_{i})\big{)}^{-1}}{\delta_{i}}-\sum_{i=1}^{12}\lVert\hat{y}_ {i}(\mathbf{W}_{i})-\mathbf{y}_{i}\rVert_{2}^{2}\end{split}  \tag{6}\]  
where \(W_{i}\) is the matrix of weights and biases of the i-th SNN model trained in the leave-one-out cross-validation procedure and \(\delta_{i}\) is the total number of parameters in the same \(i\)-th SNN. The trace of Hessian matrix can be computed by making use of the Jacobian matrix of the training set errors and the regularization parameters \(\alpha_{i}\) and \(\beta_{i}\), as described in [PERSON] et al. [55].