# Modeling Liquid Thermal Conductivity of Low-GWP Refrigerants Using Neural Networks
## 3 Artificial Neural Network  
Neural networks, also known as artificial neural networks (ANNs) or simulated neural networks (SNNs), are the core element of deep learning algorithms [50]. Their name and structure are inspired by the human brain, emulating how biological neurons send signals to each other [51]. ANNs are composed of layers of nodes comprising an input layer, one or more hidden layers, and an output layer [52]. Each node, or artificial neuron, is connected to other nodes and has associated weights and thresholds. A neural network can be thought of as a series of connections where information flows from input neurons to output neurons. A group of different neurons that have the same depth within the network is called a layer. Each node has as inputs real numbers that are modified by the weights of each connection; therefore, each neuron that has \(N\) inputs is represented in this way:  
\[z=\sum_{i=1}^{k}w_{i}\cdot x_{i}+b_{i} \tag{5}\]  
where \(w_{i}\) are the weights of each \(x_{i}\) that is n-th input and \(b_{i}\) is the bias. In fact, once a neuron receives inputs from all the other connected neurons, a constant value, known as bias, must be added to the calculation involving the mentioned neuron. The bias is also a factor that is modified during training, together with the weights of the inputs. A neural network can accurately model a problem, such as predicting the thermal conductivity, because of the adjustment of weights and bias for each neuron until the network approaches an acceptable solution. The final output is obtained by processing the neural network using an activation function, which is used to map the inputs to the output. The activation function (\(h\)) helps an ANN to learn complex relationships and patterns in the data and is defined as:  
\[h=f(z)=f\Big{(}\sum_{i=1}^{k}w_{i}\cdot x_{i}+b_{i}\Big{)} \tag{6}\]  
The choice of the activation function is one of the most critical aspects of network training. There are several types of activation functions including:  
* The threshold function has a simple mechanism; the function returns 1 when the weighted sum of the input signals is greater than or equal to zero and 0 in the remaining cases. It is useful if a binary output signal is needed. * The rectified linear unit is a widely used activator function. It returns 0 if the weighted sum of the input signals is less than or equal to zero, and \(\sum_{i=1}^{N}w_{i}x_{i}\) in the other cases. * The hyperbolic tangent is useful for non-linear problems and has the following form: \[\text{Tanh}(x)=\frac{e^{2x}-1}{e^{2x}+1}\] (7) In this function, the codomain ranges from \(-1\) to \(+1\). * The sigmoid is very similar to the hyperbolic tangent, but it differs in the simple fact that its codomain ranges from 0 to +1: \[\sum(x)=\frac{1}{1+e^{-x}}\] (8) A multilevel network has a much higher approximation capability than a single-level network; with enough hidden units, a multilevel network can represent any continuous function over the domain of the input variables. By stacking several layers, so that the output of one layer is given as the input to another, we obtain the multilayer perceptron (MLP) architecture. An MLP neural network is characterized by several layers of nodes connected as a directed graph between the input and output layers. In general, we canimagine our neural network as a black box that, given certain inputs (\(x\)), produces outputs (\(y\)), which are estimations of the data used for its training:  
\[y=g(x,\theta) \tag{9}\]  
where \(\theta\) is the set of values provided by the network. The network uses specific training data to train itself and to attempt to minimize errors between these data and the output data. The model's error estimation is completed using the so-called loss functions that estimate the error committed by the network.