## 2 Methodology
### Machine Learning (ML)
#### 2.3.5 Artificial Neural Networks  
The back-propagation neural network (BPNN) is a type of commonly known artificial neural network (ANN). In order for the fundamental BPNN design to function properly, there must be input, hidden, and output layers. The data obtained from outside sources are transferred to the input layer, where they are then processed by the hidden levels before being transmitted to a receptor located outside the system. The incoming signals' accuracy is improved with the use of weight factors \(W_{ij}\), which are networks of weight. The aggregate of the altered signals is then modified through the application of an exchange function that has a sigmoidal form [55; 56; 57]. The outcome of a regression ANN is a single node, denoted by the letter (y), which is determined by the activation function f and is derived from any node (j) that has been active in the past using the following Equation (2):  
\[y=\mathrm{f}\left(\sum_{i=1}^{m}\left(W_{ij}x_{i}+b_{j}\right)\right) \tag{2}\]  
where \(x_{i}\) represents the input to the layers, \(b_{j}\) represents the bias term that is given to the node, and \(W_{ij}\) is the weight of the path that can be taken between any two nodes (i and j) in the network layer. The neural network that was used for the AMCs can be seen with its input and output settings in Figure 1. Equation (3) was used to determine the amount of error that exists between the experimental data and the expected values for AMC responses. \[\mathrm{Difference\ error\ \%}=\frac{\mathrm{Target-Output}}{\mathrm{Target}} \times 100 \tag{3}\]  
Figure 1: AMC’s artificial neural network. The graph of the correlation between the input factors and the output variables is shown in Figure 2. The correlation coefficients of the input data matrix are displayed for each of the several sets of variables. Every off-diagonal subplot is a scatter plot with two variables and a least-squares reference line. The slope of this line is equal to the correlation coefficient that was provided. This number is derived from an estimate of the Pearson coefficient, which is a metric that determines the degree to which two sets of data are linearly correlated with one another. Assuming that the two different sets of data are related in a linear fashion, the Pearson correlation coefficient, denoted by the symbol \(\rho(x,y)\), for this graph can be determined as follows:  
\[\rho(x,y)=\frac{1}{N-1}\sum_{i=1}^{N}\left(\frac{x_{i}-\mu_{x}}{\sigma_{x}} \right)\,\left(\frac{y_{i}-\mu_{y}}{\sigma_{y}}\right) \tag{4}\]  
where \(\mu_{x}\), \(\mu_{y}\) and \(\sigma_{x}\), \(\sigma_{y}\) are the mean and standard deviation of the variables \(x\) and \(y\), respectively. The value that represents the correlation coefficient perpetually falls between \(-1\) and \(1\). The chart has a diagonal symmetry, and the values that are positive for the two variables indicate an upward trend, while the values that are negative signal a descending inclination. A strong correlation is often indicated by coefficients whose values are close to or equal to \(1\), whereas a weak correlation is typically indicated by coefficients whose values are close to or equal to \(0\). The correlation study can be used to obtain an initial idea of the weights assigned to each input feature, as well as the significance of those features in the process of creating the prediction model. Bar charts display the data distribution for each variable in the chart as opposed to showing the correlation coefficient between each variable, which is always \(1\). The correlation analysis in Figure 2 shows that the wt. % of Al\({}_{2}\)O\({}_{3}\) had a positive correlation with the hardness, whereas the H/D was negatively correlated with the RD%, with a value of \(0.41\), and positively correlated with the fracture strain, with a value of \(0.32\). On the other hand, the pressure shows a tremendously high positive correlation with the compression strength, yield strength, and hardness, with values of \(0.94\), \(0.91\), and \(0.9\), respectively.