## 4 Artificial Neural Networks  
The artificial neural network is a non-linear mapping system consisting of large numbers of computational units with connections in a parallel structure and analogous to simplifying biological neurons in a brain [19; 20]. As a flexible and powerful tool for approximation, an ANN commonly consists of three parts: an input layer, a hidden layer(s), and an output layer. The basic principles of ANNs are shown in Figure 5. An ANN uses the training data to learn the relation between input and output. The system receives inputs; then, each neuron receives weighted results. After summing up all the weighted results \begin{table}
\begin{tabular}{c c c} \hline \hline  & **LT Circuit** & **MT Circuit** \\ \hline Compressor & Bitzer 4G-20.2Y semi-hermetic reciprocating & Bitzer 8 GC-60.2Y semi-hermetic reciprocating \\ Evaporator & Alfa Laval PHE & Alfa Laval AC-120-150-EQ PHE \\ Condenser & Alfa Laval CB-52-50H PHE & Alfa Laval AC-120-110-EQ PHE \\ Subcooler & Alfa Laval Brazed PHE & Alfa Laval Brazed PHE \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Components of the supermarket refrigeration system. and passing through the activation function, the predictive output is delivered [21]. The sum of the weighted inputs is calculated as presented in Equation (1):  
\[n=\left(\sum_{i=1}^{p}w_{i}x_{i}\right)+b \tag{1}\]  
where \(P\), \(w_{i}\), and \(b\) are the number of elements, the interconnection weight of the input vector \(x_{i}\) in the preceding layer, and the bias term of the neuron, respectively. The output is then calculated using the sum of the weighted inputs with a bias through an activation function. In this case, the output is obtained as presented in Equation (2). \[f(n)=f\left[\left(\sum_{i=1}^{p}w_{i}x_{i}\right)+b\right] \tag{2}\]  
where \(f\) is an activation function. To train an ANN system, many input-target output pairs are used. The adjustment of the values of connections between the elements is based on comparing the output and the target. This process is repeated until reaching the point that the network output and the target are matched using the optimal set of weights. The error of the modelling is produced by comparing the output of the model with the desired response. The network used in this study has four inputs, including the middle evaporating and condensing temperatures and the superheating and subcooling temperatures. In the hidden layer, the number of neurons was controlled and adjusted to avoid overfitting. In the output layer, there is only one output. Therefore, three ANNs were designed and trained to predict the values for the output parameters, including the COP, the cooling capacity, and the compressor discharge temperature. The field measurement data were employed to train the model and test the predicted results (75% for training and 25% for testing). For computational programming of the ANN, Matlab software was employed. The optimal number of hidden layers and neurons for each layer were selected by trial and error. Therefore, no rule was established for this purpose. The neural network performance  
Figure 5: Schematic diagram of ANNs (adopted from [19]). was controlled using the mean square error (MSE), which is defined as presented in Equation (3). \[MSE=\frac{\sum(y_{sim}-y_{obs})^{2}}{N} \tag{3}\]  
where \(y_{sim}\) and \(y_{obs}\) are the predicted output and the observed output, and \(N\) is the number of data points. To solve non-linear least squares problems and supervise the learning way called backpropagation, the Levenberg-Marquardt algorithm (LMA) was used [22; 23]. This algorithm is known as damped least-squares (DLS) method. The LMA, as a commonly used algorithm to solve generic curve-fitting problems, computes derivations according to the training data. To avoid overfitting, the best results according to the validation data found during the previous iterations were remembered by the algorithm. Although the standard backpropagation is known as a slow algorithm, the LMA has been proved to have the fastest convergence on networks.