## 2 Theory and Calculation
### Artificial Neural Networks (ANN)  
An ANN is a machine-learning method used for evaluating and processing of information. The general structure of an ANN consists of an input layer, one or more hidden layers and an output layer.  
The formation of an ANN model undergoes three phases: the network modeling phase, network training phase and performance evaluation phase.  
To define the network architecture, a range of parameters are used, such as:  
* number of input parameters;
* number of layers and neurons in them;
* number of output data;
* selection of activation functions of hidden and output neurons;
* type of training function, whether the network is forward- or backward-oriented.  
In this paper, the available data set is divided into training data, validation data and testing data. In the training phase, the model is trained with the learning algorithm Levenberg-Marquardt (LM), a second-order numerical optimization technique which combines the advantages of the Gauss-Newton and the steepest descent algorithms. It is very effective when training networks that have up to several hundred units of weight [30].  
Supervised learning involves two basic steps, a forward and a backward one. Neural network training incorporates the following steps for the fine-tuning of weights and biases, in order to obtain the most accurate solutions at the output [26; 31]:  
Step 1: Model initialization: initializes weights and biases;  
Step 2: Forward propagation: using the given input \(x_{i}\), weights \(w_{i}\) and biases \(b\), the linear combination of inputs and weights (\(z_{i}\)) is calculated for each layer. After that, the output values (\(a_{i}\)) are calculated by applying the activation function to (\(z_{i}\)), represented in the Equation (1):  
\[a_{i}=f(z_{i})=f\Big{(}\sum_{j=1}^{R}x_{j}w_{i,j}+b\Big{)} \tag{1}\]  
In the current study, the logistic sigmoid (LogS) was used for the hidden activation function, while the output layer was associated with the linear activation function (pure-lin) [32]. Activation functions and expressions are shown in Figure 1.  
Step 3: Compute the loss function, based on which the difference between the obtained prediction and the given target values is reached. The calculation is based on the mean square error, represented in Equation (2):  
\[MSE=\frac{1}{n}\sum_{1}^{n}(y_{i}-a_{i})^{2} \tag{2}\]  
Step 4: Backpropagation is conducted, in which the gradients of the loss function are calculated. Using these gradients, the parameters are adjusted from the last layer to the first one.  
Step 5: Iterations are repeated until it is observed that the loss function has been minimized, without overfitting the training data.  
The described process of the feed-forward neural network and backpropagation is schematically shown in Figure 2.  
In this paper, the input neurons include the measured deflections and surface temperature of the asphalt layer. The best ANN architecture for the E\({}_{\text{AC}}\) prediction comprises eight input neurons, one hidden layer with eleven neurons and one output neuron (Figure 3).  
The learning rate and adaptation parameter Mu affect the convergence speed of the backpropagation algorithm. A learning rate of 0.001 and an adaptation parameter Mu of 0.001 were chosen. The epoch (iteration number) was limited to 1000 during training. The database for ANN modeling is divided in the following way: 85% for the training set, 10% for the validation set and 5% for the testing set.  
Figure 1: Activation functions.  
Figure 2: Representation of a two-layer feed-forward neural network and backpropagation.