## 3 Results
### Artificial Neural Network (ANN) Performance  
Equations (6)-(11) provide definitions for the various parameters that can be employed to evaluate an ANN's performance. These parameters include _mean absolute error_ (MAE), _mean squared error_ (MSE), _root mean squared error_ (RMSE), _mean squared logarithmic error_ (MSLE), _root mean squared logarithmic error_ (RMSLE) and _coefficient of determination_ (\(R^{2}\)). \[\mathrm{MAE}=\frac{\sum_{N}\bigl{(}X_{\mathrm{m}}-X_{\mathrm{p}}\bigr{)}}{N} \tag{6}\]  
\[\mathrm{MSE}=\frac{\sum_{N}\bigl{(}X_{\mathrm{m}}-X_{\mathrm{p}}\bigr{)}^{2}} {N} \tag{7}\]  
\[\mathrm{RMSE}=\sqrt{\frac{\sum_{N}\bigl{(}X_{\mathrm{m}}-X_{\mathrm{p}}\bigr{)} ^{2}}{N}} \tag{8}\]  
\[\mathrm{MSLE}=\frac{\sum_{N}\bigl{(}\log_{10}\bigl{[}X_{\mathrm{m}}+1\bigr{]} -\log_{10}\bigl{[}X_{\mathrm{p}}+1\bigr{]}\bigr{)}^{2}}{N} \tag{9}\]  
\[\mathrm{RMSLE}=\sqrt{\frac{\sum_{N}\bigl{(}\log_{10}\bigl{[}X_{\mathrm{m}}+1 \bigr{]}-\log_{10}\bigl{[}X_{\mathrm{p}}+1\bigr{]}\bigr{)}^{2}}{N}} \tag{10}\]  
\[R^{2}=\left[\frac{\sum_{N}\bigl{(}X_{\mathrm{m}}-\overline{X_{\mathrm{m}}} \bigr{)}\bigl{(}X_{\mathrm{p}}-\overline{X_{\mathrm{p}}}\bigr{)}}{\sum_{N} \bigl{(}X_{\mathrm{m}}-\overline{X_{\mathrm{m}}}\bigr{)}^{2}\sum_{N}\bigl{(}X _{\mathrm{p}}-\overline{X_{\mathrm{p}}}\bigr{)}^{2}}\right]^{2} \tag{11}\]  
where \(N\) = number of datasets; \(X_{\mathrm{m}}\) and \(X_{\mathrm{p}}\) = actual and predicted values, respectively; and \(\overline{X_{\mathrm{m}}}\) and \(\overline{X_{\mathrm{p}}}\) = average of the actual and predicted values, respectively. **Note:** Ideally, the model should have \(R^{2}\) = 1 and MAE, MSE, RMSE, MSLE and RMSLE values of 0. To implement data-driven models, two types of datasets are necessary, namely training and testing datasets. In this study, 80% of the main database was assigned to training and the remaining 20% to testing. Tables 4 and 5 present descriptive statistics for the training and testing datasets, respectively. As is evident from these tables, the statistical descriptors for these two datasets are quite close, which can improve the accuracy of AI-based models. In addition, Table 6 displays the linear correlation matrix for both input and output parameters. Based on the findings, it can be concluded that there is no significant linear correlation observed among any of the parameters. Figure 15: Effects of number of loading cycles, along with (**a**) vertical stress and (**b**) CSR, on the damping ratio. Several parameters, including the number of hidden layers and neurons, can impact the accuracy of ANN models. To obtain the optimal and most accurate ANN model, multiple models were generated and assessed to determine the best one. The performance of the selected model was evaluated by comparing the predicted damping ratio values against those of the testing dataset, as illustrated in Figure 16. The results demonstrate the high accuracy of the ANN model in predicting the damping ratio of sand. To elaborate further on the results of the ANN model, it should be noted that the model had one hidden layer, and the number of neurons in the input layer was five, corresponding \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Variable** & **Number of Cycles** & \(S\) & \(R\) & \(\rho\) & **Vertical Stress** & **CSR** & \(D\) \\ \hline Number of Cycles & 1 & 0.705 & 0.715 & 0.727 & 0.000 & 0.000 & \(-0.256\) \\ Sphericity, \(S\) & 0.705 & — & 0.705 & 0.780 & 0.193 & 0.227 & \(-0.044\) \\ Roundness, \(R\) & 0.715 & 0.705 & — & 0.771 & 0.193 & 0.206 & \(-0.040\) \\ Regularity, \(\rho\) & 0.727 & 0.780 & 0.771 & — & 0.198 & 0.223 & \(-0.043\) \\ Vertical Stress & 0.000 & 0.193 & 0.193 & 0.198 & — & 0.324 & 0.798 \\ CSR & 0.000 & 0.227 & 0.206 & 0.223 & 0.324 & — & 0.277 \\ Damping Ratio, \(D\) & \(-0.256\) & \(-0.044\) & \(-0.040\) & \(-0.043\) & 0.798 & 0.277 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Correlation matrix for the input and output variables.