## 4 Machine Learning Methodology
### Evaluation Metrics  
In this investigation, a multidimensional evaluation framework was established using the coefficient of determination (\(R^{2}\)), root mean square error (RMSE), mean absolute error (MAE), and 95% uncertainty interval width (U\({}_{95}\)) to systematically assess the model's fitting capability and predictive accuracy. \(R^{2}\) quantifies the extent to which the model explains the variability of the target variable, thereby statistically reflecting the congruence between the model and the data distribution; its range of [0, 1] provides a standardized basis for comparing models across different scales. RMSE amplifies larger deviations through the squaring operation, thus accentuating the model's sensitivity to outliers, while its dimensional consistency with the original data intuitively conveys the physical significance of the prediction error. In contrast, MAE, through linear computation, evenly weighs the error contributions of individual samples, offering a more robust accuracy assessment in the presence of outliers. Together, these metrics complement one another: \(R^{2}\) emphasizes the overall ability of the model to capture data trends, RMSE underscores the impact of significant deviations, MAE reflects the stability of the model's predictions, and U\({}_{95}\) measures the prediction stability and reliability of different models or the same model under different input conditions. The combined analysis not only reveals the model's theoretical explanatory power but also enables a multidimensional diagnosis of its error characteristics in practical applications, thereby providing directional guidance for further model optimization. The details for evaluation metrics are demonstrated in Table 3.  
Figure 6: Variation of fitness values for each model with the number of iterations. The three models exhibited significant differences in predicting the permeability coefficient of coarse-grained soils, as evidenced by the cross-validation results (see Figure 8). The BPNN demonstrated the best overall performance, achieving an average (\(R^{2}\)) as high as 0.99 and mean absolute error (MAE), root mean square error (RMSE), and 95% uncertainty interval width (U\({}_{95}\)) values of only 1.5, 2.9, and 0.4, respectively. These metrics indicate that BPNN possesses a high degree of fit to the data characteristics and exhibits extremely low prediction bias, thereby capturing the complex nonlinear relationships between the permeability coefficient and the input parameters effectively. Its superior performance may be attributed to its deep feature extraction capabilities, which characterize high-order interactions among critical factors such as particle gradation and pore structure effectively. In contrast, GPR exhibited relatively lower predictive accuracy, with an (\(R^{2}\)) of 0.92, MAE of 3.4, RMSE of 4.5, and U\({}_{95}\) of 1.2. This reduced performance may be associated with the selection of its kernel function and suboptimal parameter tuning. The noise sensitivity inherent in GPR becomes problematic when processing coarse-grained soil permeability data, particularly when experimental measurements include errors or the sample distribution is uneven. Moreover, potential threshold effects in permeability prediction, such as those arising from a critical void ratio, may not be well accommodated by the default Gaussian assumptions in GPR, thereby diminishing its ability to predict extreme values accurately. The performance of the RF model fell between the two, with an (\(R^{2}\)) of 0.97, MAE of 2.1, RMSE of 3.7, and U\({}_{95}\) of 0.9. RF maintained high accuracy and exhibited robust performance, owing to its ensemble of decision trees. This behavior can be attributed to the inherent compatibility between the piecewise linear relationships of the permeability coefficient with the feature variables and the decision tree's splitting mechanism.