Developing new machine-learning intelligent models to predict the excavation-tunnel displacements

Definition of the artificial neural network (ANN)-base model

Artificial neural networks (ANNs) were chosen as the core algorithm for developing the intelligent prediction model due to their exceptional capacity for modeling complex, nonlinear relationships through biologically inspired computation. In addition, nonlinear activation functions used in excavation-tunnel response of this study, allow the modeling of diverse system behaviors, while distributed representations provide inherent noise tolerance-crucial for real-world datasets with inherent variability. The architectural framework addressed in this context entails the creation of a neural network model with seven neurons in the hidden layer. This framework, illustrated in Fig. 11, has been iterated four times, with each iteration designating one of the displacement parameters as the target parameter. Subsequently, two neural network models were developed to estimate the \({\delta }_{hrm}\) and \({\delta }_{vm}\) values while two additional models were crafted to ascertain the values of the \({\delta }_{htm}\) and \({\delta }_{vtm}\).

In the training phase for each of the four neural network models in this study, 70% of the all dataset was employed. To evaluate accuracy and test the models, the remaining 30% of the data was utilized. Following the determination of the final models, they were employed as the foundational models to establish the mathematical framework and formulate the equations for estimating displacement values. These aspects are presented and elucidated in the subsequent sections.

Computational framework for determination of displacement values caused by an excavation

In the earlier section, the proposed ANN demonstrated satisfactory error rates. However, the complexity of the ANN structure makes it challenging to ascertain the target accurately. To address this issue, this section focuses on deriving an equation from the previously obtained ANN. The methodology begins by assigning a fixed value to each input variable, termed the reference value. This constant can be the mean or another statistical parameter. Notably, the chosen value for each variable should be non-zero, as the technique involves subtracting values from the reference numbers. In the current study, we determined the average value for the each four input variable parameters based on the designated database (360 datasets) for this purpose. Subsequently, a sensitivity analysis was conducted for each input variable parameter. This involved altering one input variable (e.g., 100 times) while maintaining fixed reference values for the other variables. By performing this analysis, the sensitivity of the output to each input could be gauged.

The intricacies of calculating the target from the complex ANN structure prompted the extraction of an equation from the network. The methodology involved selecting reference values for input variables, emphasizing the need for non-zero values due to the subtraction-based technique. To extend these principles to the surface settlement scenario, a parallel approach is adopted. The reference values, derived from statistical parameters such as the median or mean, are applied to the input variables in the sensitivity analysis. The subsequent analysis reveals the sensitivity of the output to variations in each input variable. Drawing inspiration from the chart variable identified in the horizontal context, an equivalent chart variable is pinpointed for the horizontal displacement of the retaining wall and the surface settlement determination. The selection process hinges on assessing the maximum relative importance, computed from the values of weight of the trained neural network. The method introduced by Milne⁵¹, aids in calculating the relative importance of each input variable on the output for the \({\delta }_{vm}\) values (see Figures A1 and A2 in the Supplemental Materials part of the paper).

With the crucial chart variable identified, a new database is established to exert to the ANN for determining the \({\delta }_{vm}\) values. Similar to the \({\delta }_{hrm}\) values scenario, the database creation involves systematically varying the values of input variables. Figures A3 and A4 in the Supplemental Materials part of the paper shows input variables. The resulting variation graphs, representing the relationship between input changes and the \({\delta }_{vm}\) values, are critical for further analysis.

To avoid redundancy, the methodologies employed for the \({\delta }_{hrm}\) values determination are referenced here without delving into repetitive details. The principles of sensitivity analysis, identification of the chart variable, and database creation remain consistent. Figures A5 and A6 in the Supplemental Materials illustrated the average graphs for \({\delta }_{hrm}\) and \({\delta }_{vm}\) values, respectively.

Based on Fig. A1, conducting statistical regression allows the derivation of Eq. (5), which depicts the relationship between variable \({X}_{1}\) and the target parameter (in this instance, \({\delta }_{hrm}\)). Using graphs shows in Fig. A5 and regression analysis on their outcomes, Eqs. (6), (7), (8) can be established. Ultimately, the \({\delta }_{hrm}\) value can be determined using Eq. (9).

Upon establishing the relationship between the chart parameter (Eq. (5)) and the average curves (Eqs. (6), (7), (8)), the \({\delta }_{hrm}\) value could be calculated with simplicity using Eq. (9).

Likewise, one can derive relationships for determining the \({\delta }_{vm}\) value by employing Eqs. (10), (11), (12), (13) and (14).

Computational framework for determination of the tunnel displacement values

Building upon the insights gained in the preceding section, which focused on the determination of displacements cussed by an excavation, this section delves into the intricacies of ascertaining tunnel displacements including \({\delta }_{htm}\) and \({\delta }_{vtm}\) values. The groundwork laid in section “Computational framework for determination of displacement values caused by an excavation” involving the application of ANN and sensitivity analyses serves as a valuable foundation for addressing the vertical aspect. Results of these sensitivity analyses for the tunnel problem illustrated in Figs. A7 and A8 of the Supplemental Materials part.

The objective of the sensitivity analysis is to identify a primary variable, termed the chart variable, significantly influencing the network formulation. The selection of this variable holds substantial importance as it directly contributes to the final equation. The approach is based on determining the maximum relative importance that is determined according to the values of weight of the trained neural network. Milne’s method⁵¹ was employed to predict the relative importance of each input variable on the output, with X₁ identified as the most crucial parameter, possessing the highest value of relative importance (28.565%). Following this, a new database was created for applying to the ANN, determining the output value for each input vector. The database comprised 50 datasets for each network implementation, allowing for a comprehensive representation of variable changes. The creation process involved incrementing the values of each variable in 50 steps, generating 10 sets of 50 databases for each variable. The variation graphs, representing the relationship between input changes and output, were constructed for each input (see Figs. A9 and A10 in the Supplemental Materials).

To utilize the obtained change graphs, the average of these graphs was determined, and subsequently, the best-fitted curve for each variable was identified through regression analysis as presented in Figs. A11 and A12 of the Supplemental Materials. The relationships for each mean curve were expressed in Eqs. (16), (17) and (18) for the \({\delta }_{htm}\) value problem.

Upon establishing the relationship between the chart parameter (Eq. (15)) and the average curves (Eqs. (16), (17) and (18)), the \({\delta }_{htm}\) value could be calculated with simplicity using Eq. (19).

Utilizing a comparable methodology, one can derive relationships pertaining to the determination of the \({\delta }_{vtm}\) value. These relationships are discernible in Eqs. (20), (21), (22) and (24).

Assessment of the accuracy of the proposed model for the excavation problem

A comprehensive comparison of two models, namely the ANN and a Formula-based approach, in predicting the \({\delta }_{hrm}\) values provides in Table 5. The evaluation metrics encompass Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) for both training and test datasets, as well as their cumulative values. Focusing on the \({\delta }_{hrm}\) and \({\delta }_{htm}\) values details, the MAE and RMSE metrics are presented for the training data, test data, and the overall dataset. The ANN model emerges as the superior performer, showcasing lower MAE and RMSE values across all three datasets, indicating its effectiveness in accurately predicting the \({\delta }_{hrm}\) and \({\delta }_{htm}\) values. In contrast, the Formula-based model, while delivering reasonable predictions, exhibits higher error metrics, suggesting a less precise fit to the data (see Fig. 12).

It is essential to consider the practical implications of the models. Despite the higher error metrics, the Formula-based technique may be more practical and applicable due to its user-friendly and straightforward nature. This aspect should be taken into account when deciding between the ANN and Formula models for predicting the \({\delta }_{hrm}\) value scenarios. The accompanying Fig. 12 provides visual insights into the predictive performance of both models.

Examining Table 6, which centers on the \({\delta }_{vm}\) values, a consistent trend is observed. The ANN model consistently outperforms the Formula model across metrics such as MAE and RMSE in both training and test datasets, as well as the combined dataset. The lower MAE and RMSE values for the ANN model indicate its proficiency in capturing the \({\delta }_{vm}\) patterns. Conversely, the Formula-based model displays higher errors, as illustrated in Fig. 13, indicating potential limitations in accurately predicting the \({\delta }_{vm}\) values. While the ANN model excels in precision, it is crucial to consider practicality and applicability. Despite its higher error metrics, the Formula-based technique may be more practical and applicable due to its user-friendly and straightforward nature. This consideration becomes significant when deciding between the ANN and Formula models for predicting the surface settlement scenarios. Figure 13 displays a visual representation of the predictive performance, aiding in a comprehensive evaluation of both models.

While the ANN model demonstrates superior predictive accuracy in both the \({\delta }_{hrm}\) and \({\delta }_{vm}\) values, it’s essential to consider practical applicability. The Formula-based model, despite exhibiting higher error metrics, might be more advantageous in certain scenarios due to its user-friendly nature and simplicity. The Formula model offers a more straightforward approach that is easier to understand and implement, making it potentially more practical for users who prioritize simplicity and ease of application over the absolute precision of predictions. The choice between the ANN and Formula models should therefore be guided by a balance between predictive performance and the practical considerations of usability and simplicity in real-world applications.

Assessment of the accuracy of the proposed model for the tunnel problem

A thorough assessment of the performance of two models, the ANN and a Formula-based model, in predicting the \({\delta }_{htm}\) values presents in Table 7. The evaluation metrics encompass MAE and RMSE for both training and test datasets, as well as their cumulative values. The ANN model consistently outperforms the Formula-based model across all scenarios, demonstrating lower MAE and RMSE values. While this highlights the ANN model’s adeptness in fitting the training data and its capacity to generalize to new, unseen data, it’s crucial to consider practical applicability. The Formula-based model, despite yielding higher error metrics, may be deemed more practical and applicable due to its user-friendly nature and simplicity. This aspect should be carefully weighed when deciding between the ANN and Formula models for predicting the \({\delta }_{htm}\) value scenarios. For a detailed visual representation, regression plots for the horizontal model can be referenced in Fig. 14.

Transitioning to Table 8, the examination now focuses on the \({\delta }_{vtm}\) values. Consistent with the observations in the \({\delta }_{htm}\) values, the ANN model showcases superior performance across all metrics compared to the Formula-based model. The ANN model’s commendably low MAE and RMSE values on the training data underscore its robust fit. However, a slight uptick in errors on the test data suggests challenges in generalization. Despite this, the ANN model maintains competitive accuracy levels when considering the entire dataset. In contrast, the Formula-based model consistently reveals higher errors, accentuating the overall superiority of the ANN model in predicting both \({\delta }_{htm}\) and \({\delta }_{vtm}\) values.

It’s important to note that while the ANN model excels in precision, the practical applicability of the models should be carefully considered. The Formula-based technique, despite yielding higher errors, may be deemed more practical and applicable due to its user-friendly nature and simplicity. This aspect should be taken into account when deciding between the ANN and Formula models for predicting vertical displacements in tunnel scenarios. Refer to Fig. 15 for detailed regression plots illustrating the \({\delta }_{vtm}\) values.

While the results from both tables highlight the robust predictive performance of the ANN model, it’s essential to consider practicality and user-friendliness in real-world applications. Despite the ANN model’s superior accuracy, the Formula-based model may offer advantages in terms of simplicity and ease of implementation. The Formula model’s user-friendly structure and framework make it a potentially more practical choice for scenarios where interpretability and straightforward application are critical. Engineers and practitioners often value models that are not only accurate but also accessible and easy to integrate into existing systems. Therefore, the Formula-based model’s user-friendly nature could make it an effective solution in certain practical contexts, balancing performance with ease of use for more straightforward implementation in engineering and construction applications.