## 4 Machine Learning Methodology
### Evaluation Metrics
However, compared with BPNN, the noticeable differences in MAE, RMSE, and U\({}_{95}\) suggest that tree-based models may still face limitations in regression tasks when capturing subtle variations in continuous permeability coefficients, likely due to an insufficient level of smoothness [47]. Figure 7: Predictive performance of machine learning models in the testing dataset (the 4 th training). (**a**) BPNN; (**b**) GPR; (**c**) RF. For the case derived from [48], the relative error \(\delta\) of the empirical model ranges from 2.0-88.3%, whereas that of the BPNN model ranges from 1.5-19.4%, demonstrating the great stability. For the case derived from [49], prediction accuracy analysis revealed that the empirical model achieved a coefficient of determination of \(R^{2}\) = 0.9031, whereas the BPNN model attained \(R^{2}\) = 0.9947, marking a 10.13% improvement. This enhancement indicates that the BPNN model, through its nonlinear mapping mechanism, more precisely captures the complex relationship between gradation parameters and permeability coefficients, with its fitted curve exhibiting a significantly better alignment with the experimental data than traditional empirical formulas. The data-driven approach demonstrated consistently high precision on both the training and validation sets, overcoming the inherent biases imposed by the model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Serial Number**} & \multicolumn{1}{c}{**DJ-1**} & \multicolumn{1}{c}{**DJ-2**} & \multicolumn{1}{c}{**DJ-3**} & \multicolumn{1}{c}{**DJ-4**} & \multicolumn{1}{c}{**DJ-5**} \\ \hline Measured value/\(10^{-5}\) m\(\cdot\)s\({}^{-1}\) & 12.3237 & 11.4610 & 10.6601 & 10.1054 & 9.11984 \\ Predicted & Empirical model & 1.43591 & 18.2085 & 14.0025 & 9.90161 & 7.89281 \\ value/\(10^{-5}\) m\(\cdot\)s\({}^{-1}\) & BPNN model & 11.5674 & 13.6789 & 12.4435 & 10.7384 & 8.9854 \\ Relative error \(\delta\)/\% & Empirical model & 88.3 & 58.8 & 31.3 & 2.0 & 12.4 \\ BPNN model & BPNN model & 6.1 & 19.4 & 16.7 & 6.3 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the prediction results for CGS permeability coefficients between empirical model and a BPNN model [48]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Serial Number**}} & \multicolumn{1}{c}{**Input 1**} & \multicolumn{1}{c}{**Input 2**} & \multicolumn{1}{c}{**Input 3**} & \multicolumn{1}{c}{**Input 4**} \\  & \(\alpha\) & \(\beta\) & \(d_{\text{max}}\) & \(m\) \\ \hline JP-1 & 0.51592 & 0.33487 & 45 & 0.154 \\ JP-2 & 0.74735 & 0.25519 & 45 & 0.154 \\ JP-3 & 0.88186 & 0.22519 & 45 & 0.154 \\ JP-4 & 0.93975 & 0.26377 & 45 & 0.180 \\ JP-5 & 1.47297 & 0.18312 & 45 & 0.180 \\ JP-6 & 2.11912 & 0.13452 & 45 & 0.180 \\ JP-7 & 1.95894 & 0.18182 & 45 & 0.201 \\ JP-8 & 2.76691 & 0.13193 & 45 & 0.201 \\ JP-9 & 4.99095 & 0.07468 & 45 & 0.201 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of negative exponential grading equation and porosity derived from [49]. Figure 9: Comparison of the prediction results for CGS permeability coefficients between empirical model and a BPNN model [49]. (**a**) Empirical model; (**b**) BPNN. by theoretical assumptions in empirical models and providing a more reliable predictive tool for evaluating the permeability of coarse-grained soils.