## 6 Results and Discussion
### Model Implementation and Training Protocols
#### 6.2.5 Comparative Overview  
Among the four architectures, the RBF network provided the best balance of accuracy (highest R\({}^{2}\), lowest RMSE/MAE) by automatically determining its topology and leveraging localized activations. The Bayesian model offered robust generalization via adaptive regularization without a separate validation set. SCG and LM delivered faster convergence and built-in regularization, but with slightly reduced predictive performance. This systematic implementation and evaluation framework ensures a transparent comparison of nonlinear modeling strategies for geomechanical data. Therefore, the RBF neural network delivers the best predictive accuracy about UCS. However, we must monitor validation performance because RBFs can overfit if the spread parameter is not well tuned. Nevertheless, the Bayesian regularized network is stronger for noisier data, trading off a bit of accuracy for added robustness as demonstrated in Table 4.  
Figure 17 presents boxplots of the mean squared error (MSE) distributions over 30 independent runs for each algorithm, showing the minimum, first quartile (Q1), median, third quartile (Q3), maximum, and outliers for both the training and testing sets. Table 5 complements these results by reporting each method's median MSE on the training and test data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **R\({}^{2}\)** &
\begin{tabular}{c} **RMSE** \\ **(MPa)** \\ \end{tabular} & **MAE (MPa)** \\ \hline RBF neural network & 0.972 & 1.313 & 1.029 \\ Bayesian regularized neural network & 0.967 & 1.413 & 1.164 \\ Scaled conjugate gradient neural network & 0.964 & 1.49 & 1.264 \\ Levenberg–Marquardt neural network & 0.951 & 1.737 & 1.453 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Aggregate performance metrics over 30 runs for all neural network models. Figure 17: Comparison of Mean Squared Error on training and testing sets across neural network algorithms. Figure 17 demonstrates that the radial basis function (RBF) network performs best, exhibiting the lowest median MSE and the tightest interquartile range on training and testing data. The Bayesian regularized (BR) network ranks second, with a slightly higher median error and moderate variability. The scaled conjugate gradient (SCG) network occupies third place, showing increased median MSE and greater spread, indicating sensitivity to data splits. The Levenberg-Marquardt (LM) network performs least favorably, with the highest median error and the broadest distribution. Crucially, the close correspondence between training and testing MSE across all four models suggests minimal overfitting. To evaluate whether the observed differences in test-set MSE among the four neural network algorithms were statistically significant, we applied the Friedman test with Benjamini-Hochberg correction [52] to all pairwise comparisons (Table 6). The null hypothesis for each comparison assumes equal median MSE; at \(\alpha=0.01\), significant results are marked with an asterisk. The RBF network differed significantly from both the Bayesian regularized model (adjusted \(p=3.7\times 10^{-5}\)) and the SCG model (adjusted \(p=1.0\times 10^{-5}\)). Similarly, BR (adjusted \(p=2.0\times 10^{-4}\)) and SCG (adjusted \(p=1.6\times 10^{-4}\)) outperformed the Levenberg-Marquardt network. In contrast, no significant differences were found between BR and SCG (adjusted \(p=0.600\)) or between RBF and LM (adjusted \(p=0.157\)), indicating comparable median performance for those pairs. To further quantify predictive accuracy, we calculated the coefficient of determination (R\({}^{2}\)) from a linear regression of observed versus predicted values for each algorithm. Figure 18 presents boxplots of the R\({}^{2}\) distributions over 30 independent runs, showing the minimum, first quartile, median, third quartile, maximum, and outliers for both training and testing sets. Table 7 complements this by reporting each method's median R\({}^{2}\) on the training and test data.