# Neural-Network-Based Quark-Gluon Plasma Trigger for the CBM Experiment at FAIR
## 2 Materials and Methods
### Neural Networks  
Feedforward Neural Networks or MultiLayer Perceptrons (MLPs) [11] are among the architectures used for classification in this study. Using MLP enables the construction of models that are easy to implement as well as to train. MLPs are very popular models for supervised learning and are commonly used for classification and regression tasks [12]. A supervised learning procedure means that the network builds a model based on labeled data. A MLP comprises three types of layers (input, hidden, and output) each with several nonlinear computational units (also called neurons). The information flows from the input layer to the output layer through the hidden layer(s) [13]. Typically neurons from one layer are all connected to neurons in the adjacent fully-connected layers as shown in Figure 3. The connection strengths are represented by weights in the computational process. The weights can be thought of as the parameters of the function the neural network is trying to approximate. The number of neurons in the input layer depends on the number of predictor variables in the examples of the dataset, whereas the number of neurons in the output layer is the same as the number of target or true value variables in the examples of the dataset. It can also be the number of variables required to produce the output for the required task. These multi-layer connections along with the activation function enable such networks to approximate a large class of functions with a high degree based on the number of hidden units [14]. The primary operation in MLPs can be represented as:  
\[\mathbf{a}_{n}=\mathbf{W}_{n}\cdot\mathbf{h}_{n-1}+\mathbf{b}_{n}\]  
Figure 2: Average input distribution from simulated collisions. The panels in anti-clockwise order from the top left show the distribution by particle type, by the absolute value of momentum \(|p|\), by inclination angle \(\theta\) and by azimuthal angle \(\varphi\). \[\mathbf{h}_{n}=F_{A}(\mathbf{a}_{n})\]  
where neurons in the \(n\)-th hidden layer are constructed from the neurons in the \((n-1)\)-th, with \(0\)-th layer being the input layer and the final layer being the output layer. Since every neuron in one layer is used to create a single neuron of the next layer, the corresponding weight matrix \(\mathbf{W}_{n}\) would be \(n_{l-1}\times n_{l}\) where \(n_{l-1}\) and \(n_{l}\) are the number of neurons in the \((n-1)\)-th layer and \(n\)-th layer, respectively, see Figure 4. Here, \(\mathbf{b}_{n}\) is the bias parameter for the \(n\)-th layer, which helps in learning an overall shift for the output and would have the same size as the number of neurons in that layer. \(F_{A}\) is the activation function that usually serves the purpose of introducing non-linearity to the network and increasing its representative capacity. \(\mathbf{h}_{n}\) and \(\mathbf{h}_{n-1}\) are neurons in the \(n\)-th and \((n-1)\)-th layers, respectively. The number of fully-connected hidden layers or network depth can be increased in an attempt to capture the optimal representational capacity of the network for this specific type of input and task that should be performed. A comparison of the performance of MLP models with different network depths for their architectures has been carried out by [15]. Figure 4: Each neuron in a fully-connected layer in a MLP is constructed from all the neurons of the previous layer. Each of the neurons in \(l\)-th layer, shown in purple, has connections or has input from every neuron in the \((l-1)\)-th layer, shown in green. Figure 3: Structure of the fully-connected neural network used for QGP detection. The blue color is the forward propagation of information, and the red color is the backpropagation of information. The other type of network used is the Convolutional Neural Network (CNN) [16]. As compared to the MLPs explained above, these types of networks are more capable of capturing position-dependent features of the data. CNNs are commonly used for grid-like data in multi-dimensional space.