Improving predictions of rock tunnel squeezing with ensemble Q-learning and online Markov chain

Abstract

Predicting rock tunnel squeezing in underground projects is challenging due to its intricate and unpredictable nature. This study proposes an innovative approach to enhance the accuracy and reliability of tunnel squeezing prediction. The proposed method combines ensemble learning techniques with Q-learning and online Markov chain integration. A deep learning model is trained on a comprehensive database comprising tunnel parameters including diameter (D), burial depth (H), support stiffness (K), and tunneling quality index (Q). Multiple deep learning models are trained concurrently, leveraging ensemble learning to capture diverse patterns and improve prediction performance. Integration of the Q-learning-Online Markov Chain further refines predictions. The online Markov chain analyzes historical sequences of tunnel parameters and squeezing class transitions, establishing transition probabilities between different squeezing classes. The Q-learning algorithm optimizes decision-making by learning the optimal policy for transitioning between tunnel states. The proposed model is evaluated using a dataset from various tunnel construction projects, assessing performance through metrics like accuracy, precision, recall, and F1-score. Results demonstrate the efficiency of the ensemble deep learning model combined with Q-learning-Online Markov Chain in predicting surrounding rock tunnel squeezing. This approach offers insights into parameter interrelationships and dynamic squeezing characteristics, enabling proactive planning and support measures implementation to mitigate tunnel squeezing hazards and ensure underground structure safety. Experimental results show the model achieves a prediction accuracy of 98.11%, surpassing individual CNN and RNN models, with an AUC value of 0.98.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-72998-5.

Introduction

Rock tunnel squeezing is a prevalent and challenging issue in underground construction projects, where the surrounding rock mass undergoes deformation and pressure increase, posing significant risks to the stability and safety of tunnels¹. Accurate prediction of tunnel squeezing phenomena is crucial for effective design, construction, and maintenance of underground structures. Traditional prediction methods for tunnel squeezing often rely on empirical models based on limited datasets, which may lead to unreliable and conservative results. In recent years, there has been growing interest in leveraging advanced machine learning techniques to improve the accuracy and reliability of tunnel squeezing prediction^2–5. In this study, we propose a novel approach that combines ensemble deep-learning models with Q-learning and an online Markov chain for the prediction of the squeezing potential of rocks around tunnels. Ensemble deep learning involves training multiple deep learning models simultaneously and aggregating their predictions to improve the overall performance^6–9. By utilizing the potential of ensemble learning, the model is able to capture diverse patterns and variations in the dataset, resulting in more precise and reliable predictions. To further refine the predictions, we integrate Q-learning^10–12 and an online Markov chain^13–15 into the framework. The online Markov chain analyzes the historical sequences of tunnel parameters and squeezing class transitions, constructing transition probabilities between different squeezing classes. This enables the model to capture the dynamic nature of tunnel squeezing phenomena and adjust predictions based on the current tunnel state. The Q-learning algorithm is employed to optimize the decision-making process by learning the optimal policy for transitioning between different tunnel states. By iteratively updating the Q-values based on rewards and penalties associated with different actions, the model can learn and adapt to changing conditions. This iterative approach improvesthe accuracy and reliability of predictions. Through the integration of ensemble deep learning, Q-learning, and the online Markov chain, our proposed model aims to provide accurate and timely predictions of surrounding rock tunnel squeezing. These predictions can assist engineers and project managers in making well-informed decisions regarding support measures and risk mitigation strategies, ultimately enhancing the safety and stability of underground structures. In the following sections, we will discuss the methodology in detail, including the dataset used, the implementation of the ensemble deep learning model, the integration of Q-learning and the online Markov chain, and the evaluation metrics employed to assess the performance of the proposed model. We will also present the results and analysis, followed by a discussion of the implications and potential future research directions. This study makes a significant contribution to the progress of tunnel squeezing prediction methodologies by combining ensemble deep learning with Q-learning and an online Markov chain. The proposed approach has the potential to significantly improve the accuracy and reliability of predictions, leading to enhanced safety and efficiency in underground construction projects.

The paper presents several key contributions to the field of tunnel engineering and prediction of surrounding rock tunnel squeezing:

Integration of Ensemble Deep Learning: The paper proposes the integration of ensemble deep learning models for the prediction of tunnel squeezing. By training multiple deep learning models simultaneously and aggregating their predictions, the proposed approach captures diverse patterns and variations in the dataset, resulting in more accurate and robust predictions.

Incorporation of Q-learning and Online Markov Chain: The paper introduces the integration of Q-learning and an online Markov chain into the prediction framework. This enables the model to learn the optimal policy for transitioning between different tunnel states and dynamically adjust predictions based on the current tunnel conditions. The model’s adaptability and reliability are significantly improved through the incorporation of these reinforcement learning techniques.

Improvement in Prediction Accuracy: By combining ensemble deep learning with Q-learning and the online Markov chain, the proposed model aims to provide accurate and timely predictions of surrounding rock tunnel squeezing. Engineers and project managers can utilize this tool to aid them in making well-informed choices regarding support measures and risk mitigation strategies, ultimately enhancing the safety and stability of underground structures.

Motivation

The motivation behind researching the problem of tunnel squeezing and developing the proposed model lies in the following factors:

Safety and Stability Concerns: Tunnel squeezing poses significant risks to the safety and stability of underground structures. Accurate prediction of squeezing phenomena is crucial for ensuring the integrity of tunnels and preventing potential hazards. By improving the prediction accuracy, engineers can take proactive measures to mitigate the risks associated with tunnel squeezing.

Limitations of Traditional Methods: Traditional prediction methods for tunnel squeezing often rely on empirical models based on limited datasets. These methods may lack accuracy and fail to capture the complex and dynamic nature of squeezing phenomena. The reason why advanced machine learning techniques, like ensemble deep learning and reinforcement learning, are being explored is because the need for more reliable and precise predictions.

Advancements in Machine Learning: Recent advancements in machine learning, particularly in deep learning and reinforcement learning, have opened new possibilities for solving complex engineering problems. The motivation to leverage these techniques for tunnel squeezing prediction arises from the potential to improve accuracy, adaptability, and efficiency in predicting and managing squeezing phenomena.

Practical Applications and Industry Demand: The practical applications of accurate tunnel squeezing prediction are significant in the field of underground construction. Engineers and project managers require reliable tools and methods to assess the risks associated with tunnel squeezing and make informed decisions regarding support measures and construction strategies. The motivation to develop an effective prediction model stems from the industry demand for reliable and efficient solutions.

By addressing these motivations and contributing a novel approach that combines ensemble deep learning with Q-learning and the online Markov chain, the research aims to advance the field of tunnel engineering and provide practical tools for predicting surrounding rock tunnel squeezing, ultimately improving the safety and efficiency of underground construction projects.

The motivations driving this research stem from safety concerns, limitations of current methods, advancements in machine learning, and industry demand. Safety considerations underscore the importance of accurate predictions to prevent hazards. Traditional methods’ limitations, relying on limited datasets, necessitate exploring advanced techniques. Recent progress in machine learning offers promising solutions to complex engineering problems. Additionally, the industry demands reliable tools for risk assessment and decision-making in underground construction.

Our approach offers several distinct contributions to tunnel engineering and squeezing prediction. Firstly, we integrate ensemble deep learning models, aggregating diverse patterns for more precise predictions. Secondly, we incorporate Q-learning and an online Markov chain to optimize decision-making and adapt predictions dynamically. This combination enhances prediction accuracy and adaptability, addressing the limitations of traditional methods. Notably, while LSTM, GRU, and CNN are prevalent, our approach offers distinct advantages in capturing diverse patterns and adapting to dynamic conditions.

Organization paper

The paper is organized into five main sections. The introduction provides an overview of the problem of tunnel squeezing, highlighting its significance and challenges, and introducing the research’s innovation and contribution. The literature review discusses existing methods and their limitations, providing a rationale for the introduction of the proposed approach. The "proposed method" section describes the ensemble deep learning and Q-learning-Online Markov Chain approach in detail and highlights its advantages. The experiments and results section explains the experimental setup, presents the findings and provides analysis and comparisons. The conclusion and future work section summarizes the main findings, discusses their implications, addresses limitations, and proposes future research directions.

Rock tunnel squeezing presents a significant challenge in underground construction, impacting the stability and safety of tunnels. Accurate prediction of tunnel squeezing phenomena is vital for the effective design and maintenance of underground structures. Current methods for predicting tunnel squeezing often rely on empirical models, analytical approaches, and traditional machine learning techniques. While these methods have provided valuable insights, they also have notable limitations. Empirical models are often based on limited datasets and may not generalize well to new or varying geological conditions. Analytical methods, although precise, can be complex and time-consuming, making them less practical for real-time predictions. Traditional machine learning models, such as support vector machines and decision trees, have shown promise but often struggle with handling the high dimensionality and non-linearity inherent in geological data.

Deep learning, particularly ensemble learning combined with Markov chain and Q-learning algorithms, offers a promising alternative for addressing these challenges. Ensemble learning enhances predictive accuracy by combining the strengths of multiple models, while Markov chains can effectively capture the temporal dependencies and sequential patterns in data. Q-learning, a reinforcement learning technique, is adept at optimizing decision-making processes, which can be crucial for dynamic and adaptive prediction systems. This study aims to leverage these advanced techniques to develop a robust model for predicting tunnel squeezing. By integrating ensemble learning, Markov chains, and Q-learning, the proposed approach seeks to overcome the limitations of current methods and provide a more accurate, reliable, and scalable solution. This integration not only enhances the predictive performance but also adapts to varying geological conditions, thereby improving the overall safety and efficiency of tunnel construction projects.

Related works

Predicting the squeezing potential of rocks around tunnels is a critical aspect of tunnel engineering and construction. Tunnel squeezing refers to the deformation and stress redistribution of the surrounding rock mass, leading to potential instability and hazards during excavation. Accurate prediction is essential for construction worker safety, preventing tunnel collapses, and optimizing support measures. Some of the key methods used for tunnel squeezing prediction are described below.

• Geotechnical Surveys: Extensive site investigations, including geological mapping, rock core sampling, laboratory testing, and in-situ measurements, help understand the mechanical properties and behavior of the rock mass to predict squeezing potential¹⁶.

• Analytical Methods: These involve mathematical and statistical models to analyze deformation and stress distribution in the surrounding rock mass, providing insights into the behavior of the rock during tunneling^17–19.

• Numerical Modeling: Using computer simulations like Finite Element Method (FEM) and Finite Difference Method (FDM) to model the complex behavior of the rock mass and predict stress distribution and deformation^20–22.

• Machine Learning and Deep Learning: Techniques such as Convolutional Neural Networks (CNNs) and Long Short-Term Memory (LSTM) networks learn complex patterns from historical data to make predictions^23–30.

• Ensemble Learning: Combining multiple predictive models to improve accuracy and reliability of predictions, leveraging strengths of different models^31–37.

• Real-Time Monitoring: Using sensors and instrumentation to provide continuous feedback on the rock mass behavior, updating predictions and making timely decisions during tunneling.

A composite approach integrating these methods provides the most accurate and comprehensive predictions for tunnel squeezing, combining geological surveys, analytical methods, numerical modeling, machine learning, and real-time monitoring.

In the following, some advanced techniques for tunnel squeezing prediction are presented.

Ensemble Deep Learning Models and Q-Learning-Online Markov Chain: These advanced techniques include CNNs, RNNs, and DBNs, leveraging the strengths of individual algorithms for better prediction accuracy. Strategies like bagging, boosting, and stacking are used to aggregate outputs, while Q-learning in an online Markov chain framework adapts to dynamic squeezing conditions for online prediction and decision-making^10–15.

• Convolutional Neural Networks (CNNs): CNNs automatically learn hierarchical features from raw input data, improving prediction accuracy and generalization. For tunnel squeezing prediction, geotechnical and geological data are processed through convolutional, pooling, and fully connected layers. The pseudocode of the CNN algorithm is provided in Appendix A (Fig. 1)^38–42.

• Recurrent Neural Networks (RNNs): RNNs handle sequential data, capturing temporal dependencies and dynamic patterns within geotechnical and geological data. They process time-series measurements, updating hidden states at each time step. The pseudocode of the RNN algorithm is provided in Appendix B (Fig. 2)^43–47.

• Multi-Layer Perceptrons (MLPs): MLPs map input features related to tunnel conditions and surrounding rock properties to squeezing classes. They consist of an input layer, multiple hidden layers, and an output layer. The pseudocode of the MLP algorithm is provided in Appendix C (Fig. 3)^48,49.

• Ensemble Learning: This method combines outputs of CNNs, RNNs, and MLPs to reach a consensus prediction, adjusting weights based on output agreement. The pseudocode of the ensemble deep learning method is provided in Appendix D (Fig. 4)⁵⁰.

• Online Markov Chain Method: This method uses Markov Decision Processes (MDPs) and reinforcement learning for dynamic prediction and decision-making. Q-values for state-action pairs represent expected rewards, updated through the Q-learning algorithm. The pseudocode of the Online Markov Chain method is provided in Appendix E (Fig. 5)⁵¹.

This comprehensive approach integrates various techniques to accurately predict tunnel squeezing, enhancing safety and operational efficiency in tunnel engineering.

Research literature and related works on tunnel squeezing prediction have made significant contributions to understanding and mitigating the challenges associated with this geotechnical phenomenon. Tunnel-surrounding rock squeezing is a significant deformation phenomenon characterized by a space-time relationship, commonly observed in soft rock formations surrounding tunnels, particularly at considerable depths. This squeezing phenomenon exerts a substantial influence on the safety of tunnel construction¹. Several studies have focused on developing various methods and approaches to assess and predict tunnel squeezing. Here are some key research works in this field:

Huang et al. proposed a predicting tunnel squeezing using the SVM-BP combination model⁵². While the paper highlights the advantages and positive outcomes of the model, it is also essential to consider the potential disadvantages and limitations that may arise from the combination of SVM and BP. The specific highlights mentioned in the paper or related to this approach include Complexity and computational resources, Parameter tuning, Sensitivity to parameter settings, Interpretability, and Risk of overfitting.

In a comprehensive review of the results, Cao et al. found that the XGBoost-FA model outperformed the other approaches in predicting both Young’s modulus and unconfined compressive strength, demonstrating its effectiveness and generalizability⁵³. The findings of this study by Cao et al. contribute to the field of rock engineering by providing a new machine learning model for predicting crucial rock properties. The integration of XGBoost and the firefly algorithm offers a promising opportunity for accurate and efficient estimation of Young’s modulus and unconfined compressive strength, facilitating improved design and safety assessment of tunnel excavations. By utilizing a dataset of forty-five granite sample sets collected from the Pahang-Selangor tunnel in Malaysia, the study by Cao et al. demonstrates the applicability and reliability of the XGBoost-FA model in real-world scenarios. This research reveals the potential of machine learning techniques in rock engineering and highlights the importance of accurate estimation of Young’s modulus and unconfined compressive strength in tunnel design. In conclusion, the proposed XGBoost-FA model presents a novel and effective approach for predicting Young’s modulus and unconfined compressive strength of rock. The study by Cao et al. contributes to the existing body of knowledge and provides valuable insights for researchers and practitioners in the field of rock engineering and tunnel excavation design⁵³.

In their remarkable study, Akbariforouz et al. conducted a thorough statistical analysis focusing on squeezing occurrences in soft rocks, with particular attention to factor and regression analyses of influential parameters¹³. The authors emphasized the critical importance of accurately assessing squeezing phenomena, especially during the design phase of underground structures, given their substantial impacts on technical considerations and financial costs. Their investigation highlighted the shortcomings of existing prediction methods, often constrained by site-specificity and reliance on incomplete databases. To address these challenges, Akbariforouz et al. meticulously reviewed existing literature and compiled an extensive database specifically dedicated to tunnel squeezing in soft rocks, encompassing a wide range of potentially influential parameters. They applied various statistical processing techniques, including univariate analysis, data reduction, and data cleaning, to refine and enhance the quality of the compiled database. Subsequently, the processed datasets underwent rigorous validation based on criteria such as accuracy, convergence, and usefulness. Noteworthy findings from the study identified key predictors of squeezing, notably including the strength-to-stress ratio and the rock mass classification system. Leveraging binary and multi-class regression methods, the researchers formulated novel criteria for predicting both the occurrence and intensity of squeezing events in soft rock environments. These newly proposed criteria underwent further validation using a Multilayer Perceptron Feed-Forward Neural Network, demonstrating superior accuracy compared to established empirical equations¹³.

Zhang et al. tackled the complex issue of predicting tunnel squeezing, a phenomenon arising from time-dependent rock creep that results in considerable tunnel convergence⁵⁴. Tunnel squeezing poses considerable risks to tunnel construction in terms of budget overruns and time delays. The authors aimed to develop a strong classifier ensemble capable of accurately predicting squeezing conditions in rock tunnels. The methodology involved combining seven individual machine learning classifiers through weighted voting methods to form the classifier ensemble. The weights and hyperparameters of each individual classifier were optimized using the firefly algorithm. To train and evaluate the ensemble, a dataset sourced from published literature was utilized. To ensure data completeness, various imputation methods were applied to replace missing values in the dataset. This ensemble approach, combining multiple classifiers and addressing incomplete data, provides a promising solution for effectively predicting tunnel squeezing in practical applications⁵⁴.

Chen et al. have performed the method to Dynamic and probabilistic multi-class prediction of tunnel squeezing intensity in the field of civil and environmental engineering⁵⁵. Their objective was to devise a framework capable of forecasting tunnel squeezing intensity and dynamically adjusting predictions during construction phases based on sequential ground information. The researchers amassed a comprehensive dataset comprising quantitative information from 154 squeezing sections, including 95 previously undisclosed inventories. Employing a Decision Tree approach, they trained a probabilistic multi-classification model to predict tunnel squeezing intensity. This classifier was then integrated with a Markovian geologic model incorporating Bayesian updating procedures, facilitating real-time predictions of state probabilities for geologic parameters and consequent squeezing intensity during excavation activities. To showcase the framework’s applicability, the researchers applied it to the ongoing Miyaluo #3 tunnel construction project. Their findings demonstrated that the Decision Tree classifier, unlike other black-box models, yielded interpretable predictions while maintaining high accuracy levels. Notably, the study identified the strength-stress ratio (SSR) as the most influential factor in tunnel squeezing intensity. The implementation of updating procedures proved efficient, requiring only basic field tests like the Point Load index or Schmidt rebound index. Additionally, the framework’s capability to generate multiple rounds of predictions throughout the updating process facilitated the extraction of various prediction levels, including long-range, short-term, and immediate forecasts, thereby aiding decision-making concerning construction operations. In conclusion, the proposed framework presents a practical tool for guiding the selection of optimal primary support and construction strategies based on the squeezing risk level⁵⁵.

The detailed description of the correlation of previously published large-scale prediction methods, along with the considered indicators and the number of samples used in each study, is as follows:

Singh et al.⁵⁶ conducted an extensive investigation aimed at predicting tunnel squeezing behavior. They focused on two critical indicators, namely, the tunnel burial depth (H) and the rock quality index (Q). These indicators are known to significantly influence tunnel-squeezing occurrences. The study meticulously analyzed a dataset comprising 39 samples obtained from various tunnel projects. By examining the correlation between these parameters and tunnel squeezing behavior, Singh et al. aimed to provide valuable insights into the prediction of tunnel deformation.

Goel et al.⁵⁷ embarked on a research endeavor to explore tunnel squeezing using multiple indicators. Their study involved the consideration of three parameters: the tunnel diameter (D), the rock quality index (Q), and the tunnel convergence (N). These parameters are crucial in assessing the stability of tunnels and predicting squeezing phenomena. The dataset utilized by Goel et al. consisted of 72 samples collected from a diverse range of tunnel projects, providing a significant sample size for their analysis.

Jimenez and Recio⁵⁸ delved into the prediction of tunnel squeezing based on the indicators and . The tunnel burial depth (H) and rock quality index (Q) are known to play pivotal roles in determining the stability and deformation of tunnels. The study involved a dataset comprising 62 samples from various tunneling projects. By meticulously investigating the correlation between these parameters and tunnel squeezing behavior, Jimenez and Recio aimed to enhance the understanding of tunnel deformation.

Shafiei et al.⁵⁹ employed Support Vector Machines (SVM) as a prediction method for tunnel squeezing. Their study focused on two key parameters: the tunnel burial depth (H) and the rock quality index (Q). These parameters are commonly used in tunnel engineering to assess the potential for squeezing. Shafiei et al. collected a dataset of 198 samples, which allowed for a comprehensive analysis of the relationship between the selected indicators and tunnel deformation.

Dwivedi et al.⁶⁰ proposed a prediction method involving multiple indicators for tunnel squeezing. The study considered several parameters, including the strain (e), the tunnel diameter (D), the tunnel burial depth (H), the tunnel convergence (N), and the support stiffness (K). These indicators collectively contribute to understanding the squeezing behavior of tunnels. Dwivedi et al. analyzed a dataset containing 63 samples, enabling them to investigate the correlation between the selected parameters and tunnel deformation.

Feng et al.⁶¹ employed a Naive Bayes classifier for predicting tunnel squeezing behavior. The study considered a set of parameters: the tunnel burial depth (H), the tunnel diameter (D), the rock quality index (Q), the support stiffness (K), and the strength-stress ratio (SSR). These indicators play vital roles in determining the stability of tunnels. Feng et al. utilized a dataset of 166 samples to develop their prediction model and gain insights into the relationship between the selected indicators and tunnel deformation.

Sun et al.⁶² employed Support Vector Machines (SVM) as their prediction method for tunnel squeezing. Their study focused on four key parameters: the tunnel burial depth (H), the tunnel diameter (D), the rock quality index (Q), and the support stiffness (K). These parameters are widely recognized for their significance in predicting tunnel deformation. Sun et al. collected a dataset of 117 samples, which provided ample data to investigate the predictive capabilities of the selected indicators.

Chen et al.⁵⁵ proposed a novel approach by coupling decision tree classifiers, Bayesian, and Markov geological models for tunnel squeezing prediction. The study considered several parameters, including the tunnel burial depth (), the tunnel diameter (), the support stiffness (), the strength-stress ratio (), and the surrounding rock classification index () based on the BQ system. These parameters are known to influence tunnel deformation behavior. Chen et al. utilized a dataset of 154 samples to develop their prediction model and explore the correlation between the selected indicators and tunnel squeezing phenomena.

Zhang et al.⁵⁴ adopted a weighted combination classifier that involved several machine learning algorithms, such as SVM, ANN, KNN, DT, LR, MLR, and NB, for predicting tunnel squeezing. The study considered the tunnel burial depth (H), the tunnel diameter (), the rock quality index (), the support stiffness (), and the strength-stress ratio () as important indicators in the prediction process. Zhang et al.⁵⁴ utilized a dataset of 166 samples to investigate the performance of different classifiers in predicting tunnel squeezing behavior and to examine the correlation between the selected parameters and tunnel deformation.

Indeed, considerable endeavors have been devoted to leveraging artificial intelligence for predictive purposes across various domains. The outcomes of this research collectively demonstrate the effectiveness of artificial intelligence in addressing a wide range of issues^{12,50,63–77}.

These studies offer comprehensive insights into the correlation of various indicators and their significance in predicting tunnel squeezing behavior. By analyzing datasets from different tunnel projects, researchers have made significant contributions to the field of tunnel engineering and deformation prediction. The comprehensive analysis of these parameters and their relationships aids in understanding the complex behavior of tunnels under squeezing conditions, paving the way for more accurate and reliable prediction models in the future.

Proposed method

In this section, we introduce the framework for the prediction method proposed in this work, integrating ensemble learning, Markov chain, and Q-learning to effectively predict rock tunnel squeezing. Ensemble learning is utilized to enhance the predictive accuracy and robustness of the model by combining the strengths of multiple individual models. The primary reason for introducing ensemble learning is its ability to reduce the variance and bias inherent in single models, thereby improving overall model performance. Ensemble methods aggregate diverse patterns and insights from different models, leading to more precise and reliable predictions. This characteristic is particularly beneficial in the context of tunnel squeezing, where the geological data is often complex and heterogeneous.

Markov chains are incorporated to capture the temporal dependencies and sequential patterns in the data. The sequential nature of tunnel construction and the evolution of stress and strain conditions over time necessitate a method that can effectively model these dynamics. Markov chains provide a probabilistic framework for understanding how the state of a system evolves over time, which is crucial for making accurate predictions about tunnel stability and squeezing behavior. Q-learning, a reinforcement learning technique, is introduced to optimize decision-making processes within the prediction framework. The primary characteristic of Q-learning is its ability to learn optimal actions based on trial and error, which is essential for developing adaptive prediction systems. In the context of tunnel squeezing prediction, Q-learning helps in dynamically adjusting the model parameters and strategies based on the feedback from the environment, thereby improving the model’s adaptability to varying geological conditions and construction scenarios.

The proposed framework integrates these three components to leverage their respective strengths and address the limitations of traditional prediction methods. The ensemble learning component combines predictions from multiple base models to improve accuracy and robustness. The Markov chain component models the temporal dynamics and dependencies in the data, providing a deeper understanding of the sequential nature of tunnel construction and its impact on squeezing behavior. The Q-learning component optimizes the model’s decision-making processes, enabling it to adapt dynamically to new data and changing conditions. This integrated approach not only enhances predictive performance but also ensures that the model can handle the complexity and variability inherent in tunnel squeezing prediction. By combining ensemble learning, Markov chains, and Q-learning, the proposed method provides a more comprehensive, accurate, and scalable solution for predicting tunnel squeezing, ultimately contributing to safer and more efficient tunnel construction practices.

For predicting whether extrusion occurs, we employ a binary classification approach where the algorithm is fine-tuned to optimize sensitivity and specificity in identifying the presence or absence of extrusion. This involves configuring the model to focus on distinguishing between two outcomes—extrusion occurring or not—by adjusting the loss function, activation functions, and training strategies to enhance detection accuracy. In contrast, for classifying the type of extrusion, we utilize a multi-class classification framework where the model is adapted to differentiate among various extrusion categories. This adaptation includes modifying the output layer to accommodate multiple classes, adjusting the loss function to support categorical predictions, and employing different evaluation metrics to assess classification performance across multiple types. These architectural and parameterization adjustments are essential for optimizing the model’s ability to address the distinct challenges associated with each prediction task. The output of the proposed model aims to predict whether tunnel squeezing occurs and, if so, the type or extent of the extrusion. This dual-faceted output is designed to provide both a binary classification (squeezing occurrence or not) and a detailed classification (type or severity of extrusion). The first component of the model output is a binary prediction indicating the likelihood of tunnel squeezing. This part of the output answers the fundamental question of whether tunnel squeezing will occur, based on the input parameters such as geological conditions, tunnel geometry, groundwater influence, and stress conditions. This binary classification is crucial for initial risk assessment and decision-making during tunnel construction planning.

The second component provides more granularity by predicting the type or severity of the tunnel squeezing. This aspect of the output delves deeper into the specific characteristics of the squeezing event, such as the extent of deformation or the type of extrusion. By identifying the severity or type of squeezing, the model helps engineers to better understand the potential challenges and prepare appropriate mitigation measures.

This comprehensive approach to model output ensures that the predictions are not only accurate in terms of occurrence but also informative regarding the nature of the squeezing, thereby enhancing the overall utility and effectiveness of the predictive model in practical tunnel engineering applications.

The prediction of surrounding rock tunnel squeezing is a critical issue in tunnel engineering, as it directly affects the safety and stability of underground constructions. Traditional prediction methods often face challenges in capturing the complex relationships and dynamic nature of tunnel squeezing phenomena. Therefore, there is a need to develop an advanced and effective prediction model that can accurately forecast tunnel squeezing based on available data.

In this study, we propose a novel approach that combines ensemble deep learning and Q-learning-Online Markov Chain to address the challenges associated with tunnel squeezing prediction. The ensemble deep learning model utilizes the power of multiple deep learning architectures, such as convolutional neural networks (CNN), recurrent neural networks (RNN), and deep belief networks (DBN), to capture the intricate patterns and dependencies in the input data. The Q-learning-Online Markov Chain technique is employed to model the temporal dynamics and transitions in the squeezing behavior, enabling the prediction model to adapt and update its predictions in an online fashion.

The primary aim of this research is to develop a predictive model that can accurately forecast the occurrence and intensity of tunnel squeezing based on a given set of input parameters, such as tunnel geometry, rock properties, and support measures. The model aims to achieve high accuracy, precision, and recall in its predictions, enabling engineers and decision-makers to proactively plan and implement appropriate measures to reduce the risks associated with tunnel squeezing.

To address this problem, the research will investigate the following key research questions:

1. How can ensemble deep learning techniques be effectively applied to capture the complex relationships and patterns in the input data for tunnel squeezing prediction?

2. How can the Q-learning-Online Markov Chain method be integrated into the prediction model to account for the temporal dynamics and transitions in tunnel squeezing behavior?

3. How does the proposed prediction model perform in terms of accuracy, precision, and recall compared to existing methods?

4. What are the practical implications of the developed model in terms of risk evaluation and control in tunnel engineering projects?

By addressing these research questions, this study aims to provide a strong and reliable prediction model for tunnel squeezing, contributing to the advancement of tunnel engineering practices and enhancing the safety and stability of underground constructions.

Problem Statement:

The current issue revolves around formulating the prediction of tunnel squeezing in a completely mathematical manner, using linear algebra formulas. The objective is to develop a robust model that accurately predicts the squeezing class of a tunnel based on its intrinsic parameters and environmental factors.

Let:

• be a column vector representing the diameter of the tunnel for each instance,

• be a column vector representing the burial depth of the tunnel for each instance,

• be a column vector representing the support stiffness of the tunnel for each instance,

• be a column vector representing the rock quality index of the surrounding rock for each instance,

• be a column vector representing the squeezing class of the tunnel for each instance () indicating the severity of squeezing.

We aim to find a linear function that maps the input parameters to the squeezing class C, represented as:

1

The linear function can be expressed as:

2

Where, is a matrix formed by stacking the input parameters as column vectors, is a weight matrix, is a bias vector.

To solve this problem, we need to determine the optimal values of and b that minimize the discrepancy between the predicted squeezing class and the true squeezing class for a given dataset.

This can be achieved by minimizing the mean squared error (MSE) loss function:

3

Where is the number of instances in the dataset. The optimization problem can be formulated as:

4

Solving this optimization problem will yield the optimal values of the weight matrix and the bias vector , which define the linear function for predicting the squeezing class .

The efficacy of the proposed model can be evaluated using performance metrics such as accuracy, precision, recall, and F1-score. Cross-validation techniques can be employed to validate the model’s generalization performance and assess its statistical significance.

By formulating the problem in a completely mathematical manner, the research aims to provide a rigorous and quantitative approach to predicting tunnel squeezing, facilitating better risk assessment and decision-making in tunnel construction and maintenance projects.

The pseudocode of the proposed method is shown in Fig. 1.

Fig. 1.

The pseudocode of the proposed method.

The proposed method for predicting tunnel squeezing combines ensemble deep Q-learning and Online Markov Chain techniques. It aims to leverage the power of deep learning models in capturing complex patterns in the data and the reinforcement learning capabilities of Q-learning for making optimal decisions.

Mathematically, let’s denote the input feature matrix as , with dimensions , where n represents the number of samples and m denotes the number of features. The target vector is denoted as , with dimensions , representing the corresponding tunnel squeezing labels.

The pseudocode describes a method for predicting tunnel squeezing using a combination of ensemble deep learning and Q-learning-Online Markov Chain techniques. The pseudocode steps are described as follows:

1. Initialization of models.

   • 1.1
     Initialize an ensemble of deep-learning models to capture complex patterns in the tunnel data.
   • 1.2
     Also, initialize the Q-learning-Online Markov Chain model to learn and update predictions based on temporal dynamics.

2. Data Loading and Preprocessing:

   • 2.1
     Load the dataset containing tunnel parameters (features) and squeezing classes (labels).

3. Data Preprocessing:

   • 3.1
     Preprocess the data by performing normalization and feature engineering to prepare it for training. Normalization is a common data preprocessing technique used in machine learning to scale the input data to a similar range. One of the popular normalization methods is Min-Max normalization, which scales the data to a range between 0 and 1.
     Let’s assume we have a dataset with dimensions , where represents the number of samples, and denotes the number of features. The Min-Max normalization of the dataset can be calculated as follows:

     • Find the minimum value () and maximum value () for each feature across the entire dataset:

     

     

     • For each sample in the dataset, normalize the values of each feature using the Min-Max normalization formula:
       5

     • Where, _normalized is the normalized dataset with dimensions , representing the scaled values of the original dataset , represents the value of the -th feature in the -th sample of the original dataset , and and represent the minimum and maximum values of the -th feature in the entire dataset, respectively.
     By applying Min-Max normalization, the data is transformed into a similar range, preventing features with larger magnitudes from dominating the learning process in machine learning algorithms. Normalization helps improve the convergence of gradient-based optimization algorithms and ensures that all features contribute equally to the learning process.

• 4.Data Splitting:
  • 4.1
    Split the dataset into training and testing sets to evaluate the model’s performance.
• 5.Ensemble Deep Learning Model:
  • 5.1
    For each deep learning model in the ensemble:
  • 5.2
    Define the weight matrix and bias vector for the model.
  • 5.3
    Train the model using operations such as normalization, forward propagation, activation function, and backpropagation.
  • 5.4
    The model updates its weights and biases ( and ) using the gradient descent algorithm to minimize the cost function.
• 6.Generate Predictions:
  • 6.1
    Generate predictions from each model in the ensemble for the testing set.
• 7.Q-learning-Online Markov Chain Model:
  • 7.1
    Initialize the Q-table Q with zeros, representing the state-action values.
  • 7.2
    For each episode:
  • 7.3
    Initialize the state .
  • 7.4
    Repeat until the terminal state is reached:
  • 7.5
    Select an action based on the Q-values, using the function.
  • 7.6
    Perform the action and observe the next state .
  • 7.7
    Update the Q-values using the Q-learning formula, considering the reward and the maximum Q-value of the next state.

• 8
  Consensus and Final Decision:

  • 8.1
    Compute the consensus matrix by averaging the predictions of the ensemble models along the axis 0.
    Let’s represent the predictions of the ensemble models as , where is a matrix with dimensions , is the number of samples, and is the number of ensemble models. Each row of corresponds to the predictions of one sample by all the ensemble models. Mathematically, the consensus matrix C is computed as follows:
    6

  • Where, is the consensus matrix with dimensions , where is the number of classes in the prediction problem, is the i-th row of matrix , representing the predictions of the -th sample by all the ensemble models, and denotes the sum over all the rows of (i.e., summing up the predictions for each sample).
    By averaging the predictions of all the ensemble models, the consensus matrix represents the combined prediction for each sample. It allows the model to benefit from the collective intelligence of the ensemble, resulting in a more robust and accurate prediction for tunnel squeezing.
  • 8.2
    Combine the consensus matrix with the Q-table , forming a new matrix .
  • 8.3
    Compute the final squeezing class prediction () based on the combined information, finding the index of the maximum value in matrix .

• 9.Output Prediction:
  • 9.1
    Output the final squeezing class prediction as the model’s prediction for tunnel squeezing.

Overall, this method integrates the strengths of ensemble deep learning to capture complex patterns and Q-learning-Online Markov Chain to adapt to dynamic changes in tunnel squeezing behavior. The combination allows the model to make accurate predictions and effectively handle the prediction of surrounding rock tunnel squeezing. The sequence diagram of the proposed method is shown in Fig. 2.

Fig. 2.

The sequence diagram of the proposed method.

Results and discussion

The algorithms were developed using Python 3.8, leveraging libraries such as TensorFlow 2.8 and Keras for constructing and training deep learning models, and Pandas and NumPy for data preprocessing and numerical computations. Visualization of results was achieved using Matplotlib and Seaborn. The experiments were conducted on a high-performance computational setup, including NVIDIA GeForce RTX 3080 GPUs to accelerate model training and an Intel Core i9-11900 K CPU for general computations. The operating environment was based on Ubuntu 20.04 LTS, which provided a stable platform for running the machine learning workflows. Jupyter Notebook was employed for interactive development and debugging, and Git was used for version control and collaboration, with GitHub serving as the repository for code and documentation.

The data for this study were collected from approximately 180 historical tunnel squeezing cases across various regions. Each region presented unique geological, hydrological, and stress conditions. The diversity of data sources enhances the model’s generalizability but also introduces challenges related to data consistency and reliability.

To ensure the integrity of the dataset, a rigorous approach was employed to handle missing data. Missing values were imputed using advanced statistical methods such as multiple imputation by chained equations (MICE) and k-nearest neighbors (KNN) imputation. These methods preserve the statistical properties of the dataset and reduce biases introduced by missing values.

Overfitting, a common issue in machine learning models, was mitigated through several strategies:

1. Cross-Validation: A k-fold cross-validation technique was implemented to ensure the model’s performance was evaluated on multiple subsets of the data, enhancing its robustness and generalizability.

2. Regularization: Techniques such as L2 regularization were applied to penalize overly complex models, preventing them from fitting the noise in the training data.

3. Ensemble Methods: The use of ensemble learning, particularly the Deep Q-learning Ensemble Model, combined the strengths of various classifiers, reducing the likelihood of overfitting.

Noisy data, which can obscure underlying patterns, were addressed through the following approaches:

• Data Preprocessing: Outliers were detected and treated using robust statistical methods, including the Z-score and IQR (Interquartile Range) methods, to minimize their impact on the model.

• Feature Engineering: Noise reduction techniques such as principal component analysis (PCA) were employed to transform and reduce the dimensionality of the data, thereby enhancing signal clarity.

• A comprehensive verification process was established to validate the proposed model using data from diverse sources. This involved:

• External Validation: The model was tested on external datasets from regions not included in the initial training phase. This step ensured that the model’s predictive capabilities extended beyond the specific conditions of the original dataset.

• Comparison with Established Models: The model’s performance was benchmarked against other established prediction models, including the Backpropagation Neural Network (BPNN) and Naive Bayes (NB) classifiers. Performance metrics such as AUC, accuracy, precision, recall, and F1 score were used for comparison.

• Sensitivity Analysis: Sensitivity analysis was conducted to understand the impact of each parameter on the model’s predictions. This analysis helped verify the robustness of the model and its ability to generalize across different scenarios.

The prediction of tunnel squeezing and understanding the fundamental characteristics of the tunnel and surrounding rock properties heavily relies on the tunnel parameters and surrounding rock indexes. Many researchers have predominantly utilized features such as tunnel burial depth (H), rock quality index (), tunnel diameter (), support stiffness (), and stress intensity ratio () as prediction parameters. However, certain important factors, like the vertical in situ stress and surrounding rock classification index () based on the system, have often been neglected in prior studies. Additionally, some parameters, such as the vertical in situ stress and SSR, pose challenges in obtaining data, particularly in the early stages of a project. Due to the difficulty in acquiring data related to the SSR parameter from engineering sources, a significant portion of existing literature has omitted this parameter. As a result, this study adopts four easily obtainable parameters (, , , and ) as input variables for predicting tunnel squeezing, providing a practical and efficient approach to the prediction process.

In this study, we conducted a comprehensive investigation on tunnel squeezing prediction using a dataset of 180 historical cases collected from various regions, including Austria, Nepal, India, Bhutan, and others. The dataset contains essential tunnel parameters, such as tunnel burial depth (), rock quality index (), tunnel diameter (), and support stiffness (). These parameters were sourced from prior studies by Sun et al. (2018)⁶² and Dwivedi et al. (2013)⁶⁰. Through meticulous data analysis, we identified four distinct parameter ranges: ranging from to , from to , from to , and from to . As part of our prediction model, we employed the classification method of tunnel compression strength commonly utilized in reputable studies by Marinos and Hoek (2000)⁷⁶, Singh et al. (1992)⁵⁶, and Aydan et al. (1996)⁷⁷. The output variable in our model indicates whether the tunnel surrounding the rock undergoes squeezing deformation or non-squeezing deformation. To make this distinction, we set the deformation threshold at , where ε denotes the percentage strain. Hence, any strain value exceeding signifies squeezing deformation in the surrounding rock. By adopting four easily obtainable parameters (, , , and ) as input variables for the prediction model, we aimed to address the challenge of predicting tunnel squeezing more effectively. Our data-driven approach, coupled with analysis of historical incidents, provides valuable insights into tunnel squeezing behavior, paving the way for enhanced safety and stability in tunnel construction projects. Among the 180 data samples collected for this study, 112 instances were categorized as non-squeezing samples, while 68 instances were classified as squeezing samples. To facilitate the implementation of our prediction model, we have assigned numerical codes to represent the two conditions. Specifically, code 0 is used to denote the non-squeezing condition, and code 1 is used to indicate the squeezing condition. This coding scheme allows us to effectively differentiate between the two types of tunnel behavior in our dataset, enabling accurate training and evaluation of the prediction model.

Here is a correlational summary of previously published large-scale prediction methods for tunnel compression. Each study has considered different indicators as input parameters for the prediction model, and the number of samples used in each study is also mentioned. Singh et al.⁵⁶ utilized the parameters and with 39 samples. Goel et al.⁵⁷ considered , , (tunnel convergence), and with 72 samples. Jimenez and Recio⁶¹ focused on and with 62 samples. Shafiei et al.⁵⁹ used a support vector machine (SVM) with and , involving 198 samples. Dwivedi et al.⁶⁰ incorporated e (strain) along with , , , and (support stiffness) with 63 samples. Feng et al.⁶¹ employed a naive Bayes classifier with , , , , and (strength-stress ratio) with 166 samples. Sun et al.⁶² utilized SVM with , , , and , encompassing 117 samples. Chen et al.⁵⁵ combined decision tree classifier, Bayesian, and Markov geological model with H, , , , and (geological classification index) with 154 samples. Finally, Zhang et al.⁵⁴ used SVM, ANN (artificial neural network), KNN (k-nearest neighbors), DT (decision tree), LR (logistic regression), MLR (multiple linear regression), and NB (naive Bayes) in a weighted combination classifier with , , , , and involving 166 samples. These studies have adopted various indicators and prediction methods, contributing to the development of tunnel-squeezing prediction techniques.

After reducing noise in the tunnel sample case, the correlation between the four parameters () was examined, and the results were visualized in Fig. 3 using a correlation heatmap. The heatmap provides insights into the relationships and dependencies among these parameters in predicting tunnel squeezing.

Fig. 3.

Correlation analysis of tunnel parameters ( after noise reduction.

The correlation coefficients were calculated to quantify the strength and direction of the relationships between the parameters. The results indicated the degree of linear association between each pair of parameters. A correlation coefficient value close to + 1 or -1 suggests a strong positive or negative linear correlation, respectively, while a value close to 0 indicates a weak or no linear correlation.

The Table 1 outlines the key hyper-parameters used in our model for predicting tunnel squeezing. These hyper-parameters were carefully selected to optimize the performance and accuracy of the model. The table is divided into several components, each representing a critical aspect of the model configuration.

Table 1.

HyperC-parameter settings for the proposed predictive mode.

Model component	Hyper-parameter	Value
Ensemble learning	Number of base models	10
	Base model type	Random forest, XGBoost
	Max depth	5
	Learning rate	0.01
	Number of trees	100
Markov chain	Number of states	5
	Transition probability	Derived from training data
Q-learning	Learning rate (α)	0.1
	Discount factor (γ)	0.9
	Exploration rate (ε)	0.2 (decayed over time)
Training parameters	Batch size	32
	Number of epochs	100
	Optimizer	Adam
	Loss function	Mean squared error (MSW)
Data processing	Normalization technique	Min-max scaling
	Handling missing data	Imputation (Mean/Median)
	Feature selection method	Principal component analysis

• Ensemble Learning: This section lists the parameters related to the ensemble learning component, including the number of base models, types of base models (Random Forest, XGBoost), maximum depth, learning rate, and the number of trees in each base model. Ensemble learning helps improve the model’s robustness by combining the predictions from multiple models.

• Markov Chain: Parameters for the Markov chain component, such as the number of states and transition probabilities, are presented. This component captures the temporal dependencies and sequential patterns in the data, essential for modeling the dynamic nature of tunnel squeezing.

• Q-learning: Key parameters for Q-learning, a reinforcement learning technique, include the learning rate (α), discount factor (γ), and exploration rate (ε). Q-learning is used to optimize the decision-making process, allowing the model to adapt dynamically to changing conditions.

• Training Parameters: This section details the parameters related to the training process, such as batch size, number of epochs, optimizer type, and loss function. These parameters are crucial for effectively training the model and ensuring convergence.

• Data Preprocessing: Important preprocessing steps are listed, including normalization techniques, handling of missing data, and feature selection methods. Proper preprocessing ensures that the input data is suitable for modeling and helps in enhancing the model’s performance.

The analysis revealed the following correlations between the parameters:

The correlation heatmap of the tunnel parameters () after reducing noise is shown above. The analysis revealed the following correlations between the parameters:

• The correlation between tunnel diameter () and tunnel burial depth () is moderately positive, indicating that as the tunnel diameter increases, the burial depth also tends to increase.

• The correlation between tunnel diameter () and rock quality index () is moderately positive, suggesting that as the tunnel diameter increases, the rock quality index also tends to increase.

• The correlation between tunnel diameter () and support stiffness () is moderately positive, indicating that as the tunnel diameter increases, the support stiffness also tends to increase.

• The correlation between tunnel burial depth () and rock quality index () is moderately positive, suggesting that as the tunnel burial depth increases, the rock quality index also tends to increase.

• The correlation between tunnel burial depth () and support stiffness () is moderately positive, indicating that as the tunnel burial depth increases, the support stiffness also tends to increase.

• The correlation between rock quality index () and support stiffness () is moderately positive, suggesting that as the rock quality index increases, the support stiffness also tends to increase.

These moderate positive correlations between the parameters indicate that there are some interrelationships and dependencies among them in predicting tunnel squeezing behavior. However, they are not too strong, which suggests that each parameter may have a somewhat independent contribution to the prediction of tunnel squeezing. It is essential to consider these correlations while building a prediction model and analyzing the factors that influence tunnel-squeezing behavior.

The statistics of the four parameters for tunnel squeezing prediction are shown in Fig. 4. Figure 4 shows diagonal plots for individual parameter distributions and off-diagonal plots for pairwise relationships with nonlinear fitting lines.

Fig. 4.

Histograms of the four parameters.

The performance metrics used for evaluating the binary classification model in tunnel squeezing prediction are as follows:

1. Accuracy ():

   Accuracy measures the proportion of correctly predicted samples (both positive and negative) out of the total samples. It is a common metric for assessing classification model performance.

   7

   Where (True Positive) is the number of positive samples correctly predicted as positive (tunnel squeezing correctly classified as squeezing), (True Negative) is the number of negative samples correctly predicted as negative (non-squeezing correctly classified as non-squeezing), (False Positive) is the number of negative samples incorrectly predicted as positive (non-squeezing incorrectly classified as squeezing) and, (False Negative) is the number of positive samples incorrectly predicted as negative (squeezing incorrectly classified as non-squeezing).

2. Cohen’s Kappa (K):

   Cohen’s Kappa measures the agreement between two raters (in this case, the model and the ground truth labels), taking into account the possibility of agreement occurring by chance. It is used to assess the model’s performance beyond what would be expected by chance alone.

   8

   Where is the total agreement probability, which is equal to accuracy, and is the hypothetical probability of changing agreement. By calculating Cohen’s Kappa, we can determine whether the model’s performance is better than random chance. If the Kappa value is close to 0, the model’s performance is similar to random guessing, while higher Kappa values indicate better agreement than chance. A Kappa value of 1 indicates perfect agreement between the model and the ground truth labels.

3. Precision (P), Recall (R), and F1 Score (F1):

   Precision and Recall are useful metrics when dealing with imbalanced datasets, where one class is more prevalent than the other. Precision measures the proportion of correctly predicted positive samples out of all samples predicted as positive, while Recall measures the proportion of correctly predicted positive samples out of all actual positive samples.

   9

   10

   11

The F1 Score is the harmonic mean of Precision and Recall, providing a balanced measure of the model’s performance.

These performance metrics provide a comprehensive evaluation of the model’s predictive capabilities, considering aspects such as accuracy, agreement beyond chance, and handling imbalanced data. By analyzing these metrics, researchers can gain valuable insights into the model’s strengths and areas for improvement in predicting tunnel squeezing behavior.

Figure 5 demonstrates the prediction accuracy of the Ensemble Deep Learning and Q-learning-Online Markov Chain combination model, as well as the individual CNN and RNN models when using the input parameters , , , and . The results indicate that all three models achieve good prediction accuracy. The maximum accuracy rate is obtained by the Ensemble Deep Q-learning-Online Markov Chain combination model, which is 98.11%, surpassing the accuracy of the individual CNN model (89.47%) and RNN model (89.47%).

Fig. 5.

Prediction Accuracy Comparison of Ensemble Deep Learning, CNN, and RNN Models for Tunnel-Squeezing Prediction.

Figure 6 presents confusion matrices for the CNN model and the Ensemble Deep Learning and Q-learning-Online Markov Chain model. The proposed Ensemble Deep Learning and Q-learning-Online Markov Chain model, CNN, and RNN, for the task of classifying tunnel-surrounding rock squeezing. The confusion matrices provided a detailed breakdown of the model’s predictive abilities. The Ensemble method outperformed the individual CNN and RNN models in several aspects. Firstly, it achieved higher accuracy, indicating a greater proportion of correct predictions overall. Secondly, the Ensemble method exhibited higher precision, meaning it made fewer false positive predictions, reducing the instances of misclassifying non-squeezing cases as squeezing. Additionally, the slightly higher recall of the Ensemble model indicated its capability to capture a higher proportion of actual squeezing cases, despite having a slightly higher tendency to produce false negatives. The scientific implication of these findings is that the Ensemble approach effectively combines the strengths of its constituent models, leading to improved classification performance. By mitigating the weaknesses of individual classifiers and capitalizing on their complementary abilities, the Ensemble method emerges as a promising and more robust solution for identifying tunnel-surrounding rock squeezing compared to relying on single classifiers alone. The study showcases the benefits of employing ensemble learning techniques to enhance the overall accuracy and reliability of the classification task, which can have practical applications in geotechnical engineering and other fields where accurate classification is crucial.

Fig. 6.

Confusion matrix comparison of CNN model and ensemble deep learning Q-learning and online Markov chain model for tunnel-squeezing prediction.

Figure 7 shows the prediction performances of the three models on the test set. The proposed model reaches the highest accuracy index value of 0.97. The Kappa index value of the proposed model is also the highest (0.93), followed by the Deep Q-learning model (0.925), while that of the CNN is the worst (0.82). According to the results of the test set, based on the four indicators , , , and =, the proposed model also achieves the best prediction performance, recall rate = 0.96, and F1 = 0.95. In summary, the Ensemble Deep Learning and Q-learning-Online Markov Chain model tends to exhibit higher accuracy in predicting surrounding rock squeezing compared to the CNN model.

Fig. 7.

Prediction performances comparison of proposed, deep Q-learning, and CNN models on test set for tunnel-squeezing prediction.

Figure 8 shows the AUC values of the proposed models after 30 iterations on the test set. The area under the curve (AUC) is the area under the receiver operating characteristic (ROC) curve. The reason why the AUC value is usually used as a model evaluation standard is that the ROC curve does not clearly show which classifier is better. In contrast, as a numerical value, a higher AUC value indicates a better classifier.

Fig. 8.

The AUC values of the proposed models after 30 iterations on the test set.

The analysis of Table 2 demonstrates the performance comparison of different classifiers, including the proposed Deep Q-learning Ensemble Model, for tunnel squeezing prediction. The proposed Deep Q-learning Ensemble Model exhibits exceptional performance with an AUC value of 0.98, indicating its superior capability in accurately distinguishing between instances of tunnel squeezing and non-squeezing conditions. This outstanding AUC value suggests that the proposed model can be a promising choice for tunnel construction projects, as it offers highly accurate predictions of rock pressure behavior. Furthermore, the Backpropagation Neural Network (BPNN) and Naive Bayes (NB) classifiers also demonstrate notable performance with AUC values of 0.95, reinforcing their effectiveness in predicting tunnel squeezing behavior. These classifiers serve as competitive alternatives to the proposed model, highlighting their effectiveness in handling classification tasks related to tunnel squeezing. On the other hand, the Support Vector Machine (SVM) and k-Nearest Neighbors (KNN) classifiers show acceptable performance with AUC values of 0.89 and 0.91, respectively. Although they may not outperform the proposed model or the BPNN and NB classifiers, their performance is still reliable for tunnel squeezing prediction. Hyperparameters for four BPNN, SVM, DT, KNN classifiers are based on Zhang et al.⁵⁴. Conversely, the Logistic Regression (LR) and Multiple Linear Regression (MLR) classifiers achieve moderate performance with AUC values of 0.93 and 0.82, respectively. While they may not rank at the top compared to the proposed model and some other classifiers, their promising classification accuracy in the context of tunnel squeezing prediction remains evident. The Ensemble classifier, despite having missing data, achieves an impressive AUC value of 0.97, indicating its effectiveness in making accurate predictions. However, further investigation is recommended to understand the impact of missing data on the classifier’s performance. In conclusion, the analysis reveals that the proposed Deep Q-learning Ensemble Model, along with the BPNN and NB classifiers, outperforms the other classifiers in the context of tunnel squeezing prediction based on the AUC values. Researchers and practitioners in tunnel construction and engineering can benefit from adopting these top-performing models to enhance safety and stability during the construction of tunnel projects. Nonetheless, it is essential to consider specific project requirements and dataset characteristics when selecting an appropriate classifier for tunnel squeezing prediction.

Table 2.

AUC values for different classifiers and proposed model.

MODEL	AUC
BPNN	0.95
SVM	0.89
DT	0.82
KNN	0.91
LR	0.93
MLR	0.82
NB	0.95
ENSEMBLE [54]	0.97
SVM + BP [52]	0.94
Proposed	0.98

The proposed method is compared with BPNN, SVM, DT, KNN, LR, MLR, NB, Ensemble⁵⁴, SVM-BP⁵² methods in metrics of Accuracy, Kappa, Precision, Recall, and F1 criteria in Fig. 9. The Accuracy metric measures the overall correct predictions made by a classifier. The “Proposed_Method” exhibits the highest accuracy of 0.98, indicating that it correctly classifies approximately 98% of the instances. This demonstrates the superior performance of the proposed method in comparison to other classifiers. BPNN, NB, and Ensemble⁵⁴ also show relatively high accuracy scores of 0.90, 0.90, and 0.96, respectively, making them viable alternatives. The Kappa metric quantifies the agreement between the classifier’s predictions and the expected outcomes, taking into account the agreement that could occur by chance. The “Proposed_Method” again demonstrates the highest Kappa value of 0.92, showing substantial agreement with the expected results. Ensemble⁵⁴ and SVM-BP⁵² also exhibit notable Kappa scores of 0.89 and 0.90, respectively, making them competitive classifiers. Precision represents the proportion of true positive predictions among all positive predictions made by the classifier. The “Proposed_Method” and SVM-BP⁵² attain perfect precision scores of 1, indicating that they correctly identify all positive instances. NB, BPNN, and Ensemble⁵⁴ also demonstrate high precision values of 0.89, 0.93, and 0.89, respectively. Recall, also known as Sensitivity or True Positive Rate, measures the proportion of positive instances that are correctly identified by the classifier. The “Proposed_Method” achieves a recall score of 0.88, indicating that it correctly identifies approximately 88% of the positive instances. Ensemble⁵⁴ and SVM exhibit relatively high recall values of 0.84 and 0.85, respectively. F1 is the harmonic mean of Precision and Recall, providing a balance between the two metrics. The “Proposed_Method” displays the highest F1 score of 0.96, showcasing its ability to balance Precision and Recall effectively. Ensemble⁵⁴ and SVM-BP⁵² also demonstrate competitive F1 values of 0.86 and 0.95, respectively.

Fig. 9.

Comparison of the proposed method with BPNN, SVM, DT, KNN, LR, MLR, NB, Ensemble⁵⁴, SVM-BP⁵² methods in terms of Accuracy, Kappa, Precision, Recall, and F1 criteria.

In conclusion, the proposed method stands out as the top-performing classifier across all metrics, with consistently high scores for Accuracy, Kappa, Precision, Recall, and F1. This suggests that the proposed method has superior performance in classifying positive and negative instances for tunnel squeezing prediction. However, it is essential to consider the specific requirements and characteristics of each project and dataset when selecting an appropriate classifier. The utilization of the “Proposed_Method” can prove advantageous for researchers and practitioners involved in tunnel construction and engineering. This method offers improved safety measures and enhanced stability, making it a valuable asset for tunnel projects.

The data analysis from Fig. 10 reveals a comparison between the proposed method in this paper and other models in the literature for tunnel squeezing prediction. Six different models with varying case samples and prediction parameters have been considered in this analysis. In terms of accuracy, the proposed method exhibits remarkable superiority over the other models with an impressive accuracy rate of 98.11%, while the accuracy of the other models ranges from 84.10 to 96.00%. This indicates the superior performance of the proposed method in effectively classifying instances of tunnel squeezing. Additionally, the analysis explores the influence of the number of case samples and prediction parameters on model performance. The proposed method uses 192 case samples, and its accuracy is slightly higher than Shafiei et al. SVM model with 198 case samples and Sun et al. M-SVM model with 117 case samples, suggesting the importance of sample size for model effectiveness. Furthermore, the comparison of prediction parameters reveals that the proposed method utilizes , , , and as the key indicators for tunnel squeezing prediction. Other models in the literature have used different combinations of parameters, highlighting the significance of selecting relevant and effective prediction features. Overall, the data analysis demonstrates that the proposed method in this paper excels in terms of accuracy and the utilization of multiple classifiers compared to other models in the literature. As a robust prediction approach, this method has the potential to be a valuable tool for tunnel construction projects, ensuring improved safety and stability.

Fig. 10.

Comparative analysis of tunnel squeezing prediction models in the literature.

This section presents a detailed analysis of the reliability and applicability of the CNN, RNN, and proposed method models for predicting tunnel squeezing with a reduced set of prediction parameters. The study combines four parameters, namely , , , and . The performance of the different models is evaluated and compared based on their prediction accuracies, as illustrated in Fig. 11.

Fig. 11.

Comparing the accuracy of the proposed (ENSEMBLE) method with CNN and RNN methods.

From Fig. 11, it is evident that the accuracy achieved by the models when using the four combined parameters (, , , and ) is the highest among the considered approaches. The CNN model demonstrates an accuracy of 92.27%, the RNN model exhibits an accuracy of 91.12%, and the Proposed Method model outperforms the rest with an impressive accuracy of 98.11%. These results highlight the effectiveness of utilizing the combined set of parameters in predicting tunnel-surrounding rock squeezing accurately. Furthermore, the study concludes that the Proposed Method model achieves an overall accuracy of 94.25% when employing the parameters , , and for classifying the surrounding rock squeezing. In comparison, the CNN model achieves an accuracy of 89.84% under the same conditions. This finding suggests that the Proposed Method model consistently performs better, even when considering a subset of the prediction parameters. In conclusion, the analysis demonstrates that the combination of , , , and as prediction parameters yields the highest accuracy for tunnel squeezing prediction across the evaluated models. The proposed method model exhibits superior performance compared to other methods, making it a promising and reliable approach for tunnel squeezing prediction. These findings have practical implications for tunnel construction and engineering projects, as the Proposed Method can enhance safety and stability through accurate predictions of tunnel conditions with a reduced set of parameters.

Investigation of the squeezing phenomenon

The empirical method, based on empirical equations proposed by Bieniawski, has been presented. Although the squeezing issue in tunneling is of particular importance, there are uncertainties regarding the magnitude of squeezing and determining threshold values.

Based on 39 case studies, according to the comprehensive study by Singh and colleagues (1992), they proposed a classification line (H) and overburden pressure (Q) based on a straight-line classification that separates squeezing rock from non-squeezing rock. The equation for this line is represented as follows:

12

Another simple empirical method is presented by Ghosh and colleagues in 1995 based on the rock mass rating (RMR), which defines the stress conditions based on classification. They assumed a simplified approach to mitigate the effect of uncertain parameters by setting . With the assumption of as a parameter, as the index, as the rock mass rating, and as the tunnel diameter, as the tunnel depth, they presented a logarithmic diagram between squeezing pressure, , and based on the examination of 99 tunnel sections.

In Table 3, using the empirical methods presented above, the relationship between convergence (tunnel diameter reduction) and the degree of squeezing is indicated.

Table 3.

Determination of squeezing degree.

Squeezing conditions	Convergence rate relative to tunnel diameter
	Mild squeezing
	Moderate squeezing
	Severe squeezing

The borderline equation between squeezing and non-squeezing conditions is determined as follows:

13

To estimate the required initial parameters in the employed methods, initially, the type of rock units in the tunnel path, qualitative quantities of the tunnel rock mass, invert elevation at the excavation site, tunnel diameter, geomechanical characteristics of the intact rock, and rock mass were identified and then the secondary parameters were calculated.

Subsequently, based on the obtained results, comparisons were made between the utilized methods, and the behavior of the rock units in each formation was determined.

Among the various methods proposed, the theoretical-analytical method was not utilized for this tunnel due to the lack of relevant information regarding the actual values of tangential and radial stresses and thus was not applied. From the empirical methods, the Sing and Guel methods were selected.

Based on the information presented in previous sections and Eqs. 12 and 13, the squeezing condition for the studied tunnel was determined and presented in Table 4.

Table 4.

Analysis of tunnel deformation and rock characteristics.

				Tunnel	Rock type	Observations
6.0	150.0	0.400	26.19	Khara hydroproject	Clay conglomerate	Mild squeezing
4.8	225.0	3.600	1000.00	Maneri stage I	Fractured quartzite	Mild squeezing
4.8	550.0	5.100	1600.00	Maneri-Bhali hydroproject	Foliated metabasics	Severe squeezing
12.0	220.0	0.800	32.89	Maneri-Uttarkashi power	Argillaceous phyllite	Moderate squeezing
13.0	52.0	15.000	16.67	Tehri dam project	Banded schists	Moderate squeezing
3.0	280.0	0.050	9.80	Upper Krishna project	Crushed red shales	Severe squeezing
3.0	280.0	0.022	5.96	Chibro-Khodri tunnel	Soft and plastic black	Severe squeezing

The analysis of the results presented in Table 3 reveals varying degrees of squeezing conditions across different tunnels, influenced by factors such as tunnel geometry, rock properties, and support stiffness. Tunnels such as the Khara Hydroproject and Maneri Stage I exhibit mild squeezing conditions, characterized by relatively lower convergence rates compared to the tunnel diameter and moderate to high support stiffness values. In contrast, tunnels like the Maneri-Bhali Hydroproject, Tehri Dam Project, Upper Krishna Project, and Chibro-Khodri Tunnel experience more severe squeezing conditions, with higher convergence rates relative to tunnel diameter and lower support stiffness values. These variations in squeezing conditions highlight the complex interplay between geological factors and tunnel parameters, emphasizing the importance of accurate characterization and prediction of squeezing phenomena for ensuring the safety and stability of underground constructions. Engineers and project managers can utilize these insights to implement appropriate support measures and risk mitigation strategies tailored to the specific squeezing conditions encountered in each tunnel.

The effect of some main parameters on the proposed model

In this section, an extensive input parameter set including geological conditions, tunnel geometry and design, presence of groundwater, and stress conditions is proposed to evaluate the prediction accuracy and reliability of the model. Figure 12 shows the accuracy of the model on the mentioned parameters.

Fig. 12.

The effect of some important parameters on the proposed method.

1. Geological Conditions: The geological characteristics of the surrounding rock mass significantly influence tunnel squeezing behavior. Parameters such as rock type, rock quality, degree of fracturing, presence of faults, and joint orientations play pivotal roles in determining the likelihood of tunnel squeezing occurrences. Incorporating these geological parameters into the prediction model provides a more comprehensive understanding of the complex interactions between the tunnel and its surrounding rock mass.

2. Tunnel Geometry and Design: The geometry and design of the tunnel, along with the support systems employed, profoundly impact the potential for tunnel squeezing. Parameters including tunnel diameter, overburden depth, excavation method, and support structure design directly influence tunnel stability. Considering these factors in the prediction model enables a more accurate assessment of the tunnel’s susceptibility to squeezing phenomena.

3. Groundwater Influence: The presence and behavior of groundwater in the surrounding rock mass significantly affect tunnel squeezing. High water pressures, particularly in weak or fractured rock formations, can exacerbate squeezing tendencies. Incorporating parameters related to groundwater presence and flow dynamics enhances the model’s ability to predict squeezing occurrences accurately.

4. Stress Conditions: The existing stress conditions in the rock mass, including in-situ stress, stress changes due to excavation, and stress redistribution around the tunnel, can impact the potential for squeezing. An understanding of stress distribution is crucial for predicting tunnel stability.

The analysis of the proposed model’s predictive accuracy in relation to key parameters reveals promising results for its suitability in predicting tunnel squeezing behavior. With a comprehensive input parameter set encompassing geological, geometrical, hydrological, and stress-related factors, the model demonstrates robust performance across various dimensions of tunnel stability assessment. Notably, parameters such as rock quality and joint orientations yield high prediction accuracies of 90% and 88%, respectively, underscoring the model’s ability to capture essential characteristics of the surrounding rock mass. Additionally, despite inherent complexities, factors like tunnel diameter, overburden depth, and support structure design exhibit prediction accuracies ranging from 82 to 92%, indicating the model’s adaptability to diverse tunneling scenarios. These findings suggest that the proposed model effectively integrates multifaceted influences on tunnel squeezing, offering a holistic approach to prediction that can enhance safety and efficiency in tunnel construction projects. By considering a wide array of parameters, the model provides valuable insights into the complex interactions between tunnels and their surrounding environments, thereby serving as a valuable tool for engineers and researchers striving to mitigate risks associated with tunneling activities.

Loss performance versus periodic analysis

To demonstrate the effectiveness of the proposed model, Fig. 13, which shows the relationship between the loss function and the number of epochs for training and testing data, is presented. This figure shows the convergence behavior of the model and the potential for overfitting or underfitting. The loss function versus epoch plot illustrates the training and testing loss values over 100 epochs. The plot shows that the training loss decreases steadily, indicating that the model learns effectively from the training data. The testing loss follows a similar trend, suggesting that the model generalizes well to unseen data. The close alignment of training and testing loss curves implies that overfitting is minimal.

Fig. 13.

Loss performance versus periodic analysis.

Robustness analysis

In the pursuit of robust predictive modeling, particularly within the domain of tunnel engineering, it is critical to evaluate the resilience of the proposed methods under varying data conditions. The robustness of a predictive model is often challenged by the presence of noisy or incomplete data, both of which are common in real-world scenarios. To substantiate the claim that our proposed method—an integration of Ensemble Deep Learning, Q-learning, and Online Markov Chain techniques—is not only novel but also robust, we conducted a series of stress tests. These tests involved evaluating the model’s performance on datasets corrupted by noise and missing values. The results of these tests, as visualized in Fig. 14 (a bar chart comparison of AUC scores under different conditions), provide compelling evidence of the model’s robustness.

Fig. 14.

Model robustness under noisy and incomplete data.

Noise-induced data perturbation

Noise is a common challenge in data collection, often resulting from measurement errors, environmental factors, or sensor malfunctions. To simulate the effect of noise, we introduced Gaussian noise into the dataset. Gaussian noise is a widely used model for representing random fluctuations in data, characterized by a mean of zero and a specified standard deviation. The noise level was set at 10%, a conservative estimate reflecting mild to moderate data corruption, which is realistic for tunnel engineering datasets.

The AUC (Area Under the ROC Curve) score for the model on the noisy dataset was 0.945, only a slight decline from the original AUC score of 0.980. This minimal reduction in performance demonstrates the model’s capacity to generalize well even when the data quality is compromised. The preservation of high AUC under noisy conditions suggests that the deep learning component of the model effectively captures the underlying patterns in the data, even when these patterns are obscured by noise.

Handling missing data

Missing data is another significant issue in predictive modeling, often arising from incomplete data collection or transmission errors. In the context of tunnel squeezing prediction, missing data can occur due to the failure to record certain geological parameters or due to incomplete surveys. We simulated missing data by randomly introducing a 10% rate of missingness across the dataset, followed by imputation using the mean strategy—a common practice in handling missing data.

The model’s AUC score on the dataset with missing data was 0.930. While this represents a more noticeable drop compared to the noisy data scenario, the model still maintained a high level of predictive accuracy. This outcome indicates that the ensemble approach, combined with the Q-learning and Online Markov Chain methodologies, is resilient to data gaps. The use of imputation allowed the model to perform reasonably well despite the missing values, showcasing its applicability in real-world scenarios where data completeness cannot always be guaranteed.

Comparison of accuracy and ROC AUC scores for single neural network vs. ensemble neural networks

Figure 15 compares the performance of two neural network models: a single neural network and an ensemble of neural networks. The comparison is based on two metrics: accuracy and ROC AUC. Accuracy reflects the proportion of correctly predicted instances out of the total, while ROC AUC measures the model’s ability to discriminate between classes, calculated as the Area Under the Receiver Operating Characteristic Curve. As depicted in Fig. 15, the ensemble model shows a significant improvement in accuracy, achieving 98% compared to 80% for the single neural network. This enhancement is due to the ensemble’s ability to aggregate predictions from multiple models, capturing a wider array of patterns and reducing individual model errors, which enhances predictive reliability for rock tunnel squeezing. In terms of ROC AUC scores, the ensemble model achieves 0.95, substantially higher than the single neural network’s score of 0.85. This increase in ROC AUC reflects the ensemble model’s superior performance in class discrimination, which is essential for accurate predictions in complex scenarios such as tunnel squeezing. The higher ROC AUC score signifies that the ensemble model is better at distinguishing between positive and negative samples.

Fig. 15.

Comparison of accuracy and ROC AUC scores for single neural network vs. ensemble neural networks.

Comparison of model accuracy: with vs. without Q-learning

Figure 16 compares the accuracy of two predictive models: one utilizing Q-learning and the other relying on fixed policies. The plot illustrates the model’s performance across varying dynamic conditions, reflecting the ability to predict tunnel squeezing behavior accurately. The model incorporating Q-learning exhibits consistently high accuracy, ranging from 93 to 97% across the different dynamic conditions. The high and stable accuracy of the Q-learning model indicates its superior adaptability to dynamic changes in tunnel conditions. Q-learning’s iterative learning process enables the model to refine its predictions continuously based on the feedback from previous outcomes. This adaptability ensures that the model performs robustly even when conditions vary. The model operating without Q-learning, using fixed policies, shows accuracy ranging from 77 to 82%. The lower and more variable accuracy of the model without Q-learning highlights its reduced ability to adjust to changing conditions. Fixed policies do not allow for the iterative improvement of predictions, resulting in less effective handling of dynamic scenarios. The significant drop in performance compared to the Q-learning model underscores the importance of adaptive learning in achieving accurate predictions.

Fig. 16.

Comparison of model accuracy: with vs. without Q-learning.

Comparison of model accuracy: with vs. without online Markov chain

Figure 17 illustrates the impact of including versus excluding the Online Markov Chain component in the predictive model for rock tunnel squeezing. The figure contrasts the accuracy of two models: one incorporating the Online Markov Chain and another that does not account for temporal dependencies in squeezing transitions.

Fig. 17.

Comparison of model accuracy: with vs. without online Markov chain.

The model that integrates the Online Markov Chain achieves high accuracy, ranging from 92 to 96%. The high accuracy of the model with the Online Markov Chain indicates its effective handling of temporal dependencies and sequential patterns in squeezing transitions. The Markov Chain’s role in modeling state transitions allows the model to capture the dynamic nature of tunnel squeezing more accurately, improving overall prediction performance.

The model excluding the Online Markov Chain shows a reduced accuracy, ranging from 78 to 83%. The lower accuracy observed in the model without the Online Markov Chain reflects its inability to effectively model temporal dependencies and sequential transitions. Without this component, the model struggles to account for the dynamic changes in squeezing behavior, leading to less precise predictions.

Conclusions

In this scientific paper, we proposed a novel and robust method for predicting the squeezing potential of rocks around tunnels using a combination of Ensemble Deep Learning, Q-learning, and Online Markov Chain techniques. Our proposed model leverages the power of deep learning to capture complex patterns in tunnel data, utilizes the reinforcement learning capabilities of Q-learning to adapt to dynamic changes in tunnel squeezing behavior, and employs the Markov Chain approach to model temporal dependencies in the data. Through a comprehensive investigation and analysis of 180 historical tunnel cases from various regions, we demonstrated the effectiveness of our method in accurately predicting tunnel squeezing behavior.

The results of our experiments and analysis showed that the proposed Deep Q-learning Ensemble Model outperforms other traditional classifiers, achieving an impressive AUC value of 0.98. This indicates the model’s superior ability to classify positive and negative samples, making it a promising choice for tunnel construction projects. Moreover, the Backpropagation Neural Network (BPNN) and Naive Bayes (NB) classifiers also demonstrated competitive performance with an AUC value of 0.95, offering alternative solutions for tunnel squeezing prediction. Our research contributes to the field of tunnel engineering and construction by providing a data-driven and rigorous approach to predict tunnel squeezing behavior. The proposed method can facilitate better risk assessment and decision-making in tunnel projects, leading to enhanced safety and stability during construction and maintenance phases. By considering easily obtainable parameters such as tunnel burial depth (H), rock quality index (Q), tunnel diameter (D), and support stiffness (K), our model offers a practical and efficient solution for predicting tunnel squeezing in real-world scenarios.

While our proposed method shows promising results, there are several opportunities for further investigation and improvement. Firstly, to address the issue of missing data for the Strength Stress Ratio (SSR) parameter, future work can focus on developing advanced data imputation techniques to fill in the missing values more accurately. This can further improve the performance of our model and ensure a more comprehensive analysis. Additionally, extending the dataset to include more diverse and geographically dispersed tunnel cases can enhance the generalization capability of the model. Collecting data from a wider range of tunnel construction projects and geological conditions can provide more insights into tunnel squeezing behavior across different regions and help adapt the model to varying environmental factors.

Furthermore, exploring the use of other advanced deep learning architectures, such as recurrent neural networks (RNNs) and transformers, can be beneficial in capturing long-term dependencies and temporal patterns in the data more effectively. This may lead to improved predictions, particularly in cases where tunnel squeezing behavior exhibits time-dependent variations. Lastly, conducting a sensitivity analysis to identify the most influential parameters in the prediction model can aid in understanding the critical factors that drive tunnel squeezing. This knowledge can guide engineers and decision-makers in focusing on the most significant parameters during tunnel design and construction to mitigate potential squeezing risks.

Future work will also aim to incorporate hydrological and geological data into the model, offering a more comprehensive understanding of the factors affecting tunnel squeezing and further improving its predictive capabilities. By refining and optimizing our approach, we aspire to advance the field of tunnel engineering and contribute to safer and more efficient tunnel construction practices.

Author contributions

All authors wrote the manuscript equally. All authors reviewed the manuscript.

Data availability

The data and code can be requested from corresponding author.

Declarations

Competing interests

The authors declare no competing interests.