Hyperparameters have a significant impact on the predictive performance of ML models. Optimizing the hyperparameters of ML models can enhance their predictive performance (Chen and Seo, 2023). BO is an efficient global optimization method suitable for scenarios where the objective function is non-analytical, computationally expensive, or noisy (Zhang et al., 2020). It can address the limitations of grid search, which has a large computational burden, and random search, which is prone to getting stuck in local optima. BO constructs probabilistic models, such as Gaussian processes, to predict the behavior of the objective function and uses acquisition functions, such as expected improvement and upper confidence bound, to balance exploration and exploitation (Frazier, 2018). By doing so, BO can intelligently select evaluation points and iteratively approach the optimal solution. Its core idea is to update the posterior distribution using prior knowledge, thereby guiding the optimization process.

Mathematical models play an indispensable role in understanding, simulating, and predicting complex environmental phenomena and systems. In various application domains, the evaluation of ML prediction results is an essential component (Feng et al., 2022; Zhou X. et al., 2023; Liu et al., 2024). To accurately assess the predictive performance of ML models, it is advisable to consider commonly used statistical metrics such as coefficient of determination ( R 2 ), mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE) (Li et al., 2023; An et al., 2024a). The calculations for these metrics are as Equations 8–11. R 2 = 1 − ∑ i = 1 n y ^ i − y i 2 ∑ i = 1 n y ¯ − y i 2 (8) M A E = 1 n ∑ i = 1 n y ^ i − y i (9) M A P E = 1 n ∑ i = 1 n y i − y ^ i y i × 100 % (10) R M S E = 1 n ∑ i = 1 n y ^ i − y i 2 (11)

Owing to the black-box nature of ML models, comprehending their output results presents a significant challenge (Ribeiro et al., 2016). In order to enhance the transparency of ML models, interpretability methods for these models have garnered substantial attention from researchers (Li et al., 2019; Love et al., 2023; Longo et al., 2024). The SHapley Additive exPlanations method is an additive explanation approach that interprets model output by calculating the contribution of each input variable to the prediction result, known as the SHAP value (Lundberg and Lee, 2017). SHAP method analyzes the impact of each feature to explain the model’s prediction and sums the SHAP values of all features to obtain the final prediction result (Lundberg et al., 2019; Qu and Zhang, 2025). The calculation of SHAP values is given by Equation 12. y i − y base = f x i 1 + f x i 2 + ⋯ + f x i k (12) where f x i k represents the contribution of the k-th feature x i k in the i-th sample to the model prediction y i ; y base is the baseline value. When f x i k > 0 , the feature has a positive impact on the model prediction; conversely, when f x i k < 0 , the feature has a negative impact.

For a prediction problem involving three parameter variables, the SHAP method is illustrated in Figure 5. Here, f x i 1 indicates the change in the current prediction value caused by input variable 1, which is a positive impact leading to an increase in the prediction value. Similarly, the addition of input variable two increases the current prediction value, while the addition of input variable 3 decreases the current prediction value by f x i 3 . SHAP method can effectively explains the model’s prediction results, enhancing the understanding of how the model operates (Fu et al., 2022; Wang et al., 2023).

The TBM tunnel crown convergence dataset was standardized and randomly divided into training and test sets in an 8:2 ratio for training and testing of the 6 ML models. After data standardization and train-test splitting, the training set and test set are divided into input data and output data, respectively. The input data and output data are input into the ML models using a function ‘.fit ()’, which is used to train the models. Similarly, the input data of the test set are input into the well-trained models to yield the output using the function ‘.predict ()’. The model development is conducted using the scikit-learn, xgboost and lightgbm packages with Python.

In this study, to achieve precise prediction of CC in TBM tunnels, six classical ML models—SVR, DT, RF, LightGBM, XGBoost, and KNN—were meticulously constructed and optimized using BO. The hyperparameters of each model were finely tuned via BO to enhance their predictive capabilities for CC. Negative Mean Squared Error (NMSE) was selected as the objective function for BO to minimize prediction error. To prevent overfitting, five-fold cross-validation was employed during model training. Each model underwent 100 optimization iterations, as shown in Figure 6. The results showed that the NMSE of all six models gradually increased during the BO process and eventually converged, demonstrating the effectiveness of BO in improving the models’ performance on the training set.

The optimal hyperparameters and optimization results for the 6 ML models are presented in Supplementary Tables A1-A6 in the supplementary materials. Using the optimal hyperparameters from Supplementary Tables A1-A6, the best-performing SVR, DT, RF, LightGBM, XGBoost, and KNN models were constructed for CC prediction of TBM tunnel. The prediction performance of the six models on the training set is shown in Table 2. From Table 2, it can be seen that the XGBoost model achieved the best prediction performance on the training set, with RMSE, MAE, MAPE, and R ² values of 0.7506, 0.5158, 0.0537, and 0.9744, respectively. The LightGBM model ranked second in prediction performance on the training set, with RMSE, MAE, MAPE, and R ² values of 0.8170, 0.4773, 0.0537, and 0.9673, respectively.

To further underscore the superiority of the BO, Grey Wolf Optimization (GWO) (Mirjalili et al., 2014), Beluga Whale Optimization (BWO) (Zhong et al., 2022), and Genetic Algorithm (GA) (Albus et al., 2024) are used to optimize the hyperparameters of the LightGBM model to provide comparison. The optimization results are illustrated in Figure 7 and in Table 3. It can be observed from Table 3 that the optimal NMSEs of the are −0.0028 mm, which is smaller than the NMSE of BO. Moreover, the GWO, BWO, and GA take much more time than that of BO. The optimization comparison of LightGBM indicates the superior tuning efficiency of BO and better tuning performance.

The prediction performance of the 6 ML models on the test set is shown in Table 4. From Table 4, it can be observed that the LightGBM model outperformed the other models in terms of prediction accuracy on the test set. The LightGBM model achieved the lowest RMSE (0.9211 mm), MAE (0.6027 mm), and MAPE (0.0644), as well as the highest R ² (0.9636). This indicates that among the 6 ML models evaluated, LightGBM is the best-performing model for predicting TBM tunnel crown convergence. The XGBoost model, which performed best on the training set (as shown in Table 2), also demonstrated satisfactory performance on the test set, with RMSE, MAE, MAPE, and R ² values of 1.2195 mm, 0.7760 mm, 0.0802, and 0.9227, respectively. Although XGBoost’s performance on the test set was slightly inferior to LightGBM, it still ranked second, highlighting its good generalization ability on unseen data. The RF model also showed competitive performance, with RMSE, MAE, MAPE, and R ² values of 1.2979 mm, 0.8028 mm, 0.0839, and 0.9211, respectively. The RF model’s prediction performance on the test set was comparable to that of XGBoost, indicating that ensemble methods are generally suitable for this prediction task.

Compared to ensemble methods, the DT model has a simpler structure but still provides reasonable prediction results, with RMSE, MAE, MAPE, and R ² values of 1.5250 mm, 0.9001 mm, 0.0940, and 0.8910, respectively. However, its performance was significantly lower than that of LightGBM, XGBoost, and RF models, suggesting that more complex models may be needed to capture the nonlinear relationships in the data. The SVR and KNN models had relatively higher prediction errors compared to the other models. The SVR model achieved RMSE, MAE, MAPE, and R ² values of 1.5971 mm, 1.2990 mm, 0.1682, and 0.8805, respectively, while the KNN model achieved values of 1.6904 mm, 1.1675 mm, 0.1388, and 0.8661, respectively. These results indicate that SVR and KNN are less effective than ensemble methods for the specific task of predicting TBM tunnel crown convergence, likely due to their limitations in handling complex nonlinear interactions between input features. For instance, the SVR model’s training set performance is inferior to that of ensemble learning models like LightGBM, and it shows no significant improvement on the test set. This indicates that the SVR model may struggle to capture the complex nonlinear characteristics in TBM tunnel crown convergence data.

As indicated in Tables 2,4, the XGBoost, RF, DT, and KNN models exhibit markedly better predictive performance on the training set compared to the test set. This suggests potential overfitting during their training process. Conversely, the LightGBM model demonstrates comparable predictive performance on both the training and test sets, implying its strong generalization ability and lower risk of overfitting. This might be attributed to LightGBM’s unique algorithmic optimization strategies, such as gradient boosting and the histogram algorithm, which effectively reduce overfitting.

Figure 8 shows the prediction results of the six models on the test set, which depicts the actual versus predicted crown convergence values for each model, clearly comparing their prediction capabilities. Referring to Figure 8, the LightGBM model’s predicted values were closest to the actual values, with the smallest deviations, further confirming its superior performance. In contrast, the SVR and KNN models showed larger deviations from the actual values, especially for larger crown convergence values, indicating their limitations in accurately predicting extreme cases. Overall, the prediction performance of the 6 ML models for TBM tunnel crown convergence can be ranked as follows: LightGBM > XGBoost > RF > DT > SVR > KNN. This outcome offers a key reference for choosing a TBM tunnel crown convergence prediction model. It indicates that in similar complex engineering tasks, opting for advanced ensemble learning models like LightGBM is likely to yield superior predictive results.

Table 5 shows the improvement in prediction performance of the LightGBM model compared to the other five models. From Table 5, it can be observed that the LightGBM model significantly outperformed the other 5 ML models in terms of prediction accuracy on the test set. Specifically, the LightGBM model achieved average improvements of 36.80%, 36.46%, 37.92%, and 7.58% in RMSE, MAE, MAPE, and R ², respectively, compared to the XGBoost, RF, DT, SVR, and KNN models.

The results of this study have important implications for engineering practice, particularly in the context of TBM tunnel construction. Accurate prediction of crown convergence is crucial for ensuring the safety and stability of tunnels during and after construction. The LightGBM model, with its high prediction accuracy, can serve as a powerful tool for engineers to predict and mitigate potential risks associated with tunnel deformation. By integrating this model into real-time monitoring systems, engineers can make more informed decisions regarding support measures and construction schedules, leading to safer and more efficient tunnel construction processes.

Although the developed ML models demonstrated satisfactory prediction performance in the task of predicting TBM tunnel crown convergence, particularly the LightGBM model, which achieved a low MAPE of 6.44% on the test set, the “black-box” nature of ML models makes their internal decision-making mechanisms difficult to understand (Adadi and Berrada, 2018). To explore the importance of input features and improve the interpretability of the ML models, the SHAP method was used to analyze the LightGBM model in depth, with the results shown in Figure 9. According to Figure 9, the average SHAP values of the six input variables were ranked as follows: T (0.1366) > RG (0.0871) > DR (0.0528) > SA (0.0200) > SCS (0.0093) > RQD (0.0047). This ranking of average SHAP values reveals the importance of each input feature to the LightGBM model’s predictions.

The time factor (T) had the highest average SHAP value (0.1366), indicating that it has the most significant impact on CC prediction. This result aligns with engineering practice, as tunnel deformation typically accumulates over time, especially after the completion of initial support, where the creep effect of surrounding rock significantly affects tunnel stability. Therefore, the time factor is a key variable in predicting CC. The RG had an average SHAP value of 0.0871, suggesting that it also has a significant impact on CC prediction. Rock grade reflects the strength and stability of the surrounding rock; higher grades indicate better stability and smaller tunnel deformation. The DR had an average SHAP value of 0.0528, demonstrating that it also has some influence on crown settlement prediction. However, compared to the T and RG, the influence of the depth ratio is relatively small. The SA had an average SHAP value of 0.0200, indicating that the stiffness of the initial support also contributes to crown settlement prediction. The steel arch height reflects the stiffness of the initial support; higher stiffness leads to better support effects and smaller tunnel deformation. However, compared to the T and RG, the influence of initial support is relatively small. The SCS and RQD had average SHAP values of 0.0093 and 0.0047, respectively, suggesting that they have a relatively small impact on CC prediction. This may be because these features are correlated with other features (e.g., RG), resulting in their independent contributions being relatively small.

Figure 10 presents the dependence between the four most influential features. The relationship between T and RG indicates that time has less impact on CC at lower RG (better rock quality), but a more pronounced impact at higher RG (poorer rock quality). This aligns with engineering reality, as poor rock quality implies lower stability, leading to more significant deformation over time due to enhanced creep effects. RG and DR are interdependent, with DR exerting a greater influence on CC at higher RG. When rock quality is poor, the tunnel’s stability mainly relies on geometric and burial factors, so changes in DR directly affect the stress environment and deformation characteristics, amplifying its impact on CC. DR and SA also show dependence, where steel arch height has a more noticeable effect on CC at larger DR. Deeply buried tunnels face higher geo-stress and deformation risks, and increasing steel arch height can better resist rock pressure and reduce deformation. In contrast, at smaller DR, steel arch height has a relatively minor impact on CC, as shallow tunnels have a milder stress environment, and lower steel arch height may already suffice for support.

Through SHAP analysis, not only can the contribution of each input feature to the model’s predictions be quantified, but the interactions between features and their specific impacts on CC can also be understood. This analysis not only improves model interpretability but also provides valuable insights for engineering practice. For example, engineers can focus on the factors that have the greatest impact on crown settlement (e.g., time factor and rock grade) based on the SHAP analysis results, enabling them to take more targeted support measures to ensure tunnel safety and stability.

Previous studies have constructed reliable ML models for predicting tunnel convergence using different tunnel datasets. Table 6 summarizes the prediction performance of ML models in tunnel convergence prediction tasks from related studies. According to the statistical results in Table 6, the LightGBM model developed in this study achieved an R ² of 0.9636 on the test set, which is higher than the R ² values of most ML models in the related studies. This indicates that the LightGBM model developed in this study has satisfactory prediction performance comparable to that of ML models from previous studies, providing an effective tool for the safe and efficient construction of TBM tunnels.

Despite the accurate prediction of crown convergence in the TBM water conveyance tunnel, this study has some limitations that need to be addressed. Firstly, the generalization of the dataset needs improvement as some features have overly limited values, such as SCS having only 4 values and SA having only 3 values. Secondly, the support form only considered the steel arch height as an input feature, neglecting the stiffness provided by other support measures such as rock bolts, steel mesh, and concrete. Thirdly, the dataset in this study was constructed from a single tunnel, lacking regional and engineering diversity.

The TBM water conveyance tunnel in Xinjiang, China, traverses strata primarily composed of tuffaceous sandstone, tuff, calcareous sandstone, tuff breccia, breccia lava, andesite porphyry, basalt porphyry, dacite porphyry, feldspar sandstone, crystal tuff sandstone, and carbonaceous sandstone from the Devonian and Carboniferous systems; siliceous slate and schistose tuff from the Silurian system; and conglomerate, sandstone, argillaceous sandstone, and carbonaceous mudstone from the Permian and Triassic systems. The tunnel also passes through Hercynian intrusions such as biotite granite, granodiorite, migmatitic granite, and potassic granite. Most of the tunnel sections are located in fresh rock masses with poorly developed fractures, and the surrounding rock is generally blocky and thick-layered, with relatively intact rock masses. The groundwater along the tunnel is mainly bedrock fissure water, with low flow rates. The tunnel crosses five major fault zones, with fracture zones generally ranging from 90 to 100 m wide and a maximum fracture zone width of 800 m.

The total length of the tunnel is 283.41 km, with a burial depth ranging from 150 to 774 m and an average burial depth of 428 m. The maximum burial depth is 774 m. The designed flow rate is 40.0 m³/s. The elevation of the tunnel inlet is 638.23 m, and the elevation of the outlet is 528.494 m. The TBM excavation length is 227.08 km, and the drill-and-blast method is used for 56.33 km. The drill-and-blast method uses a horseshoe-shaped cross-section with an excavation diameter ranging from 6.56 to 8.96 m, while the TBM uses a circular cross-section with an excavation diameter of 7.00 m. Secondary lining is mainly applied in sections with poor rock integrity (Class III) and in Class IV and V rock sections, using C35 reinforced concrete lining. In sections with better rock integrity (Class II and III), shotcrete and rock bolting are used as the primary support.

Six typical sections (K63 + 510, K63 + 530, K63 + 550, K63 + 630, K63 + 650, and K63 + 670) were selected from a TBM water conveyance tunnel in Xinjiang, China and used to validate the LightGBM model’s prediction performance. The geological and design information of the tunnel sections are displayed in Table 7. To enhance the understanding of engineers in relevant fields, the implementing steps of the application in real-world TBM tunnels are illustrated in Figure 11. To make it more convenient for the workers on site to use, a graphical user interface is designed to implement these models, as shown in Figure 12. In this user interface, the users can choose different well-trained models to predict the CC of the tunnel sections with the input features of the sections. The prediction results can be shown as a curve plot to enhance user’s understanding.

Table 8 shows the prediction performance metrics of the LightGBM model for these six sections. According to Table 8, the LightGBM model achieved high prediction accuracy in the K63 + 510, K63 + 530, K63 + 650 and K63 + 670 sections, with R ² values of 0.9951, 0.9941, 0.9822 and 0.9882, respectively. In the K63 + 550 and K63 + 630 sections, the prediction accuracy was slightly lower, with R ² values of 0.8093 and 0.9275, respectively.

Overall, the LightGBM model achieved an average RMSE of 0.9097 mm, an average MAE of 0.7347 mm, an average MAPE of 5.95%, and an average R ² of 0.9494 across the six sections. This indicates that the LightGBM model has high prediction accuracy and generalization ability in actual TBM tunnel engineering. Additionally, Figure 13 shows the prediction curves of the LightGBM model for the six sections compared to the actual observed values. It can be observed that the model’s prediction curves closely match the actual observed values in most sections, particularly in the K63 + 510, K63 + 530, K63 + 650, and K63 + 710 sections, where the prediction curves almost completely overlap with the actual observed values. In the K63 + 550 and K63 + 630 sections, although the prediction errors increased slightly, the model still captured the trend of crown convergence well, with small deviations between the prediction curves and the actual observed values.

The high prediction accuracy and generalization ability of the LightGBM model in actual TBM tunnel engineering provide strong support for tunnel support design and construction decision-making. By integrating this model into real-time monitoring systems, engineers can predict crown convergence in real-time and adjust support measures and construction schedules accordingly, ensuring the safety and stability of the tunnel. For instance, in the K63 + 550 and K63 + 630 sections, despite a slight increase in prediction errors, the model still captures the crown convergence trend effectively, offering a reliable basis for risk warning and decision support during construction.