## 3 Methodology  
### Artificial Neural Networks  
ANNs are nonlinear mathematical models that aim to simulate the human brain processing schemes [56; 57; 58; 59]. In a first approximation, an ANN is the result of the weighted and biased connections of logistic regression units, i.e., the artificial neurons. In a multilayer feedforward network, such neurons are organized into hidden layers between an input layer, consisting of source nodes (the components of the input vector), and an output layer composed of one or more computational neurons that compute the response of the network. Each neuron is characterized by an activation function that limits the range of its response to a continuous value between 0 and 1. The input, i.e., the feature vector or the hidden layer output, is weighted, meaning that a certain weight is associated to each input signal (the components of the input feature vector or the hidden neurons' individual responses). These network parameters are learned through a supervised training process so that the network itself (at this point interpretable as a nonlinear parametric function) can perform a specific task. In regression problems, the network's task is to predict the experimental target associated with a given feature vector, i.e., to represent the implicit relationship between the input and the ground-truth output. In formal terms, given a data set of input-output vector pairs:  
\[D=\Big{\{}\Big{(}\mathbf{x}^{(d)}\,,\mathbf{y}^{(d)}\Big{)}\Big{\}}^{|D|}_{d=1} \tag{1}\]  
\[\text{where}\ \mathbf{x}^{(d)}=\Big{[}\mathbf{x}^{(d)}{}_{0},\ldots,\mathbf{x}^{(d)}{}_{F-1} \Big{]}\in R^{F}\ \text{and}\ y^{(d)}=\Big{[}\mathbf{y}^{(d)}{}_{0},\ldots,\ y^{(d)}{}_{T}\Big{]} \ \in R^{T} \tag{2}\]  
and the input layer of an ANN is referred as being the input feature vector \(\mathbf{x}^{(d)}\). The first hidden layer's output is the vector:  
\[\mathbf{h}^{(1)}=\Big{[}h^{(1)}_{0},\ldots,h^{(1)}_{N-1}\Big{]} \tag{3}\]  
where each item \(h^{(1)}_{j}\) is obtained as:  
\[h^{(1)}_{j}=\phi\Bigg{(}\sum_{i=0}^{F-1}w^{(1)}_{ij}.x_{i}+b^{(1)}_{j}\Bigg{)}. \tag{4}\]  
In this expression, \(w^{(1)}_{ij}\) are the weights of the connections between the neurons \(\mathbf{x}^{(d)}{}_{i}\) and the neurons \(h^{(1)}_{j}\), \(b^{(1)}_{j}\) are the biases and \(\phi\) is the activation function. The subsequent layer, \(\mathbf{h}^{(2)}\), is computed in a similar way, by considering the items of the vector \(\mathbf{h}^{(1)}\) as input layer, with corresponding weights \(w^{(2)}_{ij}\) and biases \(b^{(2)}_{j}\):  
\[h^{(2)}_{j}=\phi\Bigg{(}\sum_{i=0}^{N-1}w^{(2)}_{ij}.h^{(1)}_{i}+b^{(2)}_{j} \Bigg{)}. \tag{5}\]  
This process is repeated to compute the activations of each layer \(\mathbf{h}^{(l)}\), \(l\in\{1,2,\ldots,L\}\) and the output \(\mathbf{y}^{(d)}=\phi^{(out)}\Big{(}\mathbf{h}^{(L)}\Big{)}\). All the learnable parameters are considered to form the set:  
\[W=\Big{\{}w^{(l)}_{ij}\,,b^{(l)}_{j}\ \Big{|}\ l\ \in\{1,2,\ldots,L\}\Big{\}}. \tag{6}\]  
In the proposed ANN: the number of source nodes is equal to the number of input features (F = 3), namely, the bitumen content (BC), the filler-to-bitumen ratio (FtB), along with a categorical variable that distinguishes the bituminous mixture in terms of bitumen type, maximum nominal grain size, aggregate type, and production site (values: 1 for M1,2 for M2, 3 for M3 and so on). These features are fundamental parameters of the pavement engineering, related to the composition of the bituminous mixtures. Each hidden layer is equipped with N neurons and passed to \(\phi(\cdot)\) being either an ELU, TanH or ReLU function (Figure 5). The output layer consists of \(T=4\) neurons (corresponding to MS, MF, ITSM, and AV) and the identity activation function is considered \(\phi^{(out)}(.)\).