## 3 Results and Discussion
### Prediction Model of Thermal Conductivity of Fine Sandy Soil in the Near-Phase-Transition Zone  
To improve the model's prediction accuracy, multivariate regression fitting formulas for the thermal conductivity in the drastic and stable phase transition zones were established. The MR prediction formulas are as follows:  
\[\left\{\begin{array}{l}\lambda_{T}=-0.004506|T_{e}|^{-0.94496}+0.27677_{1} \lambda_{u}+0.73366\lambda_{f}\left(T_{f}>T\geq-3.5\,^{\circ}\text{C}\right)\\ \lambda_{T}=0.05-0.008T_{e}+0.753\lambda_{u}+0.239\lambda_{f}\ \ \ \ \left(T<-3.5\,^{\circ}\text{C}\right) \end{array}\right., \tag{12}\]  
Figure 9 exhibits the prediction results of the MR model in different temperature ranges. It shows that the predictions of the MR model are in good agreement with the experimental values, with most of the sampling points distributed within \(\pm 10\%\) error, which proves the model's effectiveness and value for engineering application. However, it is worth noting that the prediction results in the drastic phase transition zone still have a certain degree of error; some predicted values deviate from the experimental values by more than 50%. This is attributed to the highly nonlinear characteristics of thermal conductivity in this temperature interval.  
A comparison of the values predicted by the RBFNN model with the experimental results is given in Figure 10. It can be seen that the prediction accuracy of the RBFNN model is better than that of the MR model. All prediction points are distributed within the \(\pm 10\%\) error lines. Moreover, the RBFNN model has a better performance in the drastic phase transition temperature zone. The largest prediction error is only 8.69%, which further demonstrates the superiority of this machine learning method over traditional formula fitting methods.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Sample No.**} & **Initial Water Content/\%** & **Natural Density /g cm\({}^{-3}\)** & \multirow{2}{*}{\(a_{0}\)} & \multirow{2}{*}{\(b_{0}\)} & \multirow{2}{*}{\(R^{2}\)} & **Average Slope** & \multirow{2}{*}{\(T_{\text{limit}}\)/\({}^{\circ}\)C} \\  & & & & & & & \\ \hline
1 & 15.5 & 1.95 & 5.99 & 0.61 & 0.986 & 0.548 & \(-\)3.24 \\
2 & 9.2 & 2.  
Table 7 shows a comparison of the mean absolute error (_MAE_), root mean square error (_RMSE_), determination coefficient (\(R^{2}\)), and proportion of error within \(\pm 10\%\) (\(P_{\pm 10\%}\)) of different thermal conductivity prediction models. It can be seen that both prediction models have a high accuracy, with all the \(P_{\pm 10\%}\) values above 90%. The \(R^{2}\) of the RBFNN model in drastic and stable phase transition zones is 0.991 and 0.986, respectively, while the \(R^{2}\) of the MR model is 0.82 and 0.97, respectively. In addition, the _MAE_ and _RSME_ of the RBFNN model in drastic and stable phase transition zones are 0.011, 0.021, 0.022, and 0.038, respectively, while those of the MR model are much higher, reaching 0.079, 0.166, 0.037, and 0.049, respectively. It can be inferred that the prediction effect of the RBFNN model is better than that of the MR model, proving that the RBFNN model can more effectively capture and replicate the relationship between conductivity and predictors.  
Figure 10: Prediction results of the RBFNN model: (**a**) drastic phase transition zone; (**b**) stable phase transition zone.  
Figure 9: Prediction results of the MR model: (**a**) drastic phase transition zone; (**b**) stable phase transition zone.