## 2 Theory and Calculation
### ANN Structure
#### 2.3.1 ANN-Model's Equation  
Referring to the computational process performed by neurons (Section 2.1) and to the architecture of multiple-layer ANNs, the vector equation of a one-hidden layer model (Equation (7)), two-hidden layer model (Equation (8)), and three-hidden layer model (Equation (9)) are reported in the following:  
\[y=\mathrm{f_{2}}(\textbf{LW2},\textbf{1}\textbf{-}\textbf{f_{1}}(\textbf{IWI },\textbf{1}\textbf{-}\textbf{p}+\textbf{b1})+b2), \tag{7}\]  
Figure 2: Representation of the k-fold cross-validation process. \[y=\mathrm{f_{3}}(\mathbf{LW3},\mathbf{2\cdot f_{2}}(\mathbf{LW2},\mathbf{1\cdot f_{1}}(\mathbf{ IW1},\mathbf{1\cdot p}+\mathbf{b1})+\mathbf{b2})+b3), \tag{8}\]  
\[y=\mathrm{f_{4}}(\mathbf{LW4},\mathbf{3\cdot f_{3}}(\mathbf{LW3},\mathbf{2\cdot f_{2}}(\mathbf{LW2}, \mathbf{1\cdot f_{1}}(\mathbf{IN1},\mathbf{1\cdot p}+\mathbf{b1})+\mathbf{b2})+\mathbf{b3})+b4), \tag{9}\]  
where \(\mathbf{p}\) is the R length input vector and \(y\) is the network scalar output associated with the considered mechanical parameter. All ANN models were elaborated on the basis of four numerical input variables, namely the bitumen content (% by weight of mix), filler to bitumen ratio (%), air voids (%), and voids in the mineral aggregates (%), and two categorical input variables, namely the bitumen type and aggregate type. Such input data have been considered as effective in properly representing the composition of HMA mixes, which were designed and produced with different aggregate (limestone or diabase) and bitumen (standard 50/70 or modified 25-55/75) types. Predictive models characterized by such types of input data allow modifications of the mixture composition to be numerically simulated, obtaining an estimation of the mechanical parameter under analysis without the necessity of performing further experimental tests, thus representing an extremely useful design tool during the optimization of the material. The parameters' meaning of Equation (8) is explained in the following, while a graphical representation is reported in Figure 3. Vector and scalar variables are indicated in bold and italics type, respectively. Each layer is characterized by a connections' weight matrix, \(\mathbf{W}\), adds a vector, \(\mathbf{b}\), of neurons' bias, and returns an output vector, \(\mathbf{a}\). To distinguish one layer from another, the number of the layer is added to the variable of interest; furthermore, a distinction between the weight matrix that is connected to inputs and ones that are connected between layers is made by associating letters, \(\mathbf{I}\) and \(\mathbf{L}\), to \(\mathbf{W}\), respectively. A pair of indexes is also assigned to each weight matrix to identify the destination (first index) and the origin (second index) of represented connections. Thus, in a three-layer feedforward network (Figure 3), the S\({}_{1}\) neurons in the first hidden layer add an S\({}_{1}\times 1\) bias vector, \(\mathbf{b1}\), to the dot product of \(\mathbf{IW1},\mathbf{1}\) (i.e., the S\({}_{1}\times\)R weight matrix of connections between R inputs and S\({}_{1}\) neurons) and the input vector, \(\mathbf{p}\). The sum of the bias, \(\mathbf{b1}\), and the product, \(\mathbf{IW1},\mathbf{1\cdot p}\), is the so-called net input, \(\mathbf{n1}\), with the dimension, S\({}_{1}\times 1\). This sum is processed by the transfer function, \(\mathbf{f_{1}}\), to get the neurons' S\({}_{1}\times 1\) output vector, \(\mathbf{a1}\), of the first hidden layer, which is also the input vector of the second hidden layer. This layer, made of S\({}_{2}\) neurons, elaborates the incoming information by the S\({}_{2}\times 1\) bias vector, \(\mathbf{b2}\), the S\({}_{2}\times\)S\({}_{1}\) layer weight matrix, \(\mathbf{LW2},\mathbf{1}\) (having a source 1, neurons' outputs of the first layer, and a destination 2, neurons of the second layer), and the transfer function, \(\mathbf{f_{2}}\), to produce the S\({}_{2}\)x1 output vector, \(\mathbf{a2}\).