## 2 Methodology
### Data Acquisition  
A dataset was compiled from input parameters for the Mohr-Coulomb and the Hoek-Brown criteria.  
The datasets for the Mohr-Coulomb criteria were from reference [8]. A total of 168 independent input and output parameters and their range of variation are presented in Table 1. Although the datasets are those adopted from reference [8], the presented ANN models have the following superiority: When a new dataset is available, ANN helps to have a prediction based on the relationships they have found between previously used input and output parameters.  
It should be noted that, in this table, the unit of \(U_{i}\) is mm, and the remaining parameters follow the units presented in the nomenclature.  
Similar to the Mohr-Coulomb criteria, the dataset for the Hoek-Brown criteria was collected from the cases studied in references [5; 6; 9; 36] and [14; 37]. In total, 547 datasets were applied. Table 2 shows all the studied input and output parameters together with their range of variation. Differently from Table 1, in Table 2, the unit of \(U_{i}\) is m.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Range of Variation** & **Standard Deviation** & **Coefficient of Variation (\%)** \\ \hline \(GSI^{peak}\) & 21.4–64.9 & 17.48 & 34.45 \\ \(GSI^{res}\) & 15.1–33 & 6.91 & 26.20 \\ \(\sigma_{ci}\) & 23–162 & 60.09 & 60.49 \\ \(\gamma\) & 26–26.7 & 0.33 & 1.27 \\ \(E\) & 1.1–24 & 10.57 & 92.47 \\ \(\theta\) & 0.25–0.3 & 0.02 & 8.27 \\ \(c^{peak}\) & 0.34–3.7 & 1.55 & 83.51 \\ \(\varphi^{peak}\) & 24.81–57.8 & 14.26 & 33.24 \\ \(c^{res}\) & 0.27–0.96 & 0.31 & 51.67 \\ \(\varphi^{res}\) & 15.69–51 & 15.42 & 42.51 \\ \(m_{i}\) & 10–20 & 4.47 & 27.61 \\ \(\psi\) & 0–14 & 6.14 & 90.30 \\ \(\sigma_{0}\) & 10.4–26 & 7.27 & 41.89 \\ \(\eta^{*}\) & 0.0465–0.1394 & 0.037 & 43.43 \\ \(U_{i}\) & 8.86–1263.69 & 307.86 & 147.24 \\ \(P_{i}\) & 0–2 & 0.55 & 71.55 \\ \hline \hline \end{tabular} \(GSI^{peak}\): peak geological strength index, \(GSI^{res}\): residual geological strength index, \(\sigma_{ci}\): uni-axial compressive strength, \(\gamma\): unit weight, \(E\): Young’s modulus, \(\theta\): Poisson’s ratio, \(c^{peak}\) : peak cohesion, \(\varphi^{peak}\) : peak friction angle, \(c^{res}\) : residual cohesion, \(\varphi^{res}\) : residual friction angle, \(m_{i}\): mi constant, \(\psi\) : dilation angle, \(\sigma_{0}\) : in-situ rock stress, \(\eta^{*}\) : critical softening parameter, \(U_{i}\) : radial displacement, \(P_{i}\) : support pressure.  
\end{table}
Table 1: Input and output parameters and their range of variation used in the prediction of the \(P_{i}\) (Mohr–Coulomb case).  
The multi-layer perceptron (MLP) ANN is still used by researchers in engineering [38]. The transfer function applied in this study is the TANSIG transfer function, according to the following formula:  
\[TANSIG(e_{x})=\frac{2}{1+e^{(-2e_{x})}}-1, \tag{1}\]  
where \(e_{x}\) is the weighted sum of the inputs for a neuron [39].  
The radial basis function (RBF) ANNs are also popular in engineering. In RBF-ANN, by increasing the Euclidean distance between the input vector and its center, the output attained using the activation function will tend to zero. The width of RBF controls the decreasing rate of the output. Equation (2) shows the Gaussian basis function:  
\[y_{j}=\exp\left(\frac{-\|\overrightarrow{x}-\overrightarrow{\mu}_{j}\|^{2}}{ \sigma_{j}^{2}}\right), \tag{2}\]