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<p>Scouring is a natural phenomenon in water bodies that exposes an inevitable risk to hydraulic structures. This study proposes a stacking method for estimating equilibrium scour depth around bridge piers for enhancing sustainable and safety design. It consists of 16 machine learning (ML) models, including advanced ensemble methods, with the top-performing ML model as the meta-learner. The comparative analysis also entails 9 commonly used empirical equations over a dataset consisting of 859 field samples. Moreover, reliability assessments and interpretability techniques were conducted to provide deeper insights into the key factors affecting scour depth. Among individual ML models, ensemble boosting techniques, like CatBoost Regressor (CBR), Gradient Boosting Regressor, and Histogram Gradient Boosting Regressor, performed as the most robust ones, yielding a ranking index higher than 0.91. Furthermore, Gaussian Process Regression and K-Nearest Neighbors (KNN) exhibited similarly high accuracy, coupled with the highest reliability percentages, with training reliabilities both higher than 96% and testing reliabilities of 62.02% and 55.81%, respectively. The results revealed that ML models consistently outperformed empirical methods in predictive performance. Given that CatBoost Regressor outperformed other individual ML models, it was chosen as the foundation for the stacking approach. The results of the stacked CBR model demonstrated superior performance, achieving the best overall results in both training and testing phases. Additionally, the SHapley Additive exPlanations analysis identified the shape correction coefficient and the upstream flow depth divided by the width of the pier as the most effective key factors influencing scour depth, providing actionable insights for infrastructure design. Furthermore, it indicated that KNN and CBR predictions are the most influential components contributing to the accuracy of the staking predictions. The findings suggest that ML-based stacking models, particularly the ensemble boosting methods, offer a reliable and versatile approach for estimating scour depth, thereby contributing significantly to practical applications by offering a reliable methodology to enhance the design, cost-efficiency and safety of bridge structures.</p> Graphical Abstract <p>This study proposes a novel stacking machine learning (ML) approach to estimate the equilibrium scour depth around bridge piers using. The graphical abstract offers a visual roadmap of the research workflow, starting with the use of 859 field-based samples collected from various researches existing in the literature, comprising seven non-dimensional hydraulic and geometric input parameters. Sixteen distinct ML models were individually developed. They are Artificial Neural Network, Support Vector Regression, Gaussian Process Regression, Multiple Linear Regression, K-Nearest Neighbors, Bayesian Ridge, Huber Regressor, Stochastic Gradient Descent Regressor, Decision Tree Regressor, Random Forest Regressor, AdaBoost Regressor, Gradient Boosting Regressor, Light Gradient Boosting Regressor, Histogram Gradient Boosting Regressor, CatBoost Regressor (CBR), and eXtreme Gradient Boosting Regressor. All standalone ML models were evaluated using normalized performance metrics—including Root Mean Square Error, Mean Absolute Error, Coefficient of Determination, and Nash–Sutcliffe Efficiency—and subsequently ranked using a comprehensive Ranking Index (RI). The graphical abstract also presents comparative analysis results, where the stacked CBR model outperformed all individual ML models and nine traditional empirical equations (RI = 0.972). Based on the comparative analysis, CatBoost Regressor achieved the highest overall performance among all individual models and was therefore selected as the meta-learner in the stacked ML model. Reliability analysis further confirmed its strong generalization capability, with 57.75% of predictions falling within acceptable error thresholds on the test dataset. The findings offer critical insights for hydraulic engineers and researchers by presenting a high-performing, interpretable, and generalizable ML-based framework for predicting scour depth, thereby supporting safer and more cost-efficient bridge design practices.</p>

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Enhancing Prediction of Equilibrium Scour Depth Around Bridge Piers Using Staking Machine Learning Models

  • 期刊论文

  • 发表日期：

    2025年8月5日

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  • Reza Piraei,
  • Majid Niazkar,
  • Alessio Cislaghi,
  • Seied Hossein Afzali,
  • Ali Mohammadi

摘要

Scouring is a natural phenomenon in water bodies that exposes an inevitable risk to hydraulic structures. This study proposes a stacking method for estimating equilibrium scour depth around bridge piers for enhancing sustainable and safety design. It consists of 16 machine learning (ML) models, including advanced ensemble methods, with the top-performing ML model as the meta-learner. The comparative analysis also entails 9 commonly used empirical equations over a dataset consisting of 859 field samples. Moreover, reliability assessments and interpretability techniques were conducted to provide deeper insights into the key factors affecting scour depth. Among individual ML models, ensemble boosting techniques, like CatBoost Regressor (CBR), Gradient Boosting Regressor, and Histogram Gradient Boosting Regressor, performed as the most robust ones, yielding a ranking index higher than 0.91. Furthermore, Gaussian Process Regression and K-Nearest Neighbors (KNN) exhibited similarly high accuracy, coupled with the highest reliability percentages, with training reliabilities both higher than 96% and testing reliabilities of 62.02% and 55.81%, respectively. The results revealed that ML models consistently outperformed empirical methods in predictive performance. Given that CatBoost Regressor outperformed other individual ML models, it was chosen as the foundation for the stacking approach. The results of the stacked CBR model demonstrated superior performance, achieving the best overall results in both training and testing phases. Additionally, the SHapley Additive exPlanations analysis identified the shape correction coefficient and the upstream flow depth divided by the width of the pier as the most effective key factors influencing scour depth, providing actionable insights for infrastructure design. Furthermore, it indicated that KNN and CBR predictions are the most influential components contributing to the accuracy of the staking predictions. The findings suggest that ML-based stacking models, particularly the ensemble boosting methods, offer a reliable and versatile approach for estimating scour depth, thereby contributing significantly to practical applications by offering a reliable methodology to enhance the design, cost-efficiency and safety of bridge structures.

Graphical Abstract

This study proposes a novel stacking machine learning (ML) approach to estimate the equilibrium scour depth around bridge piers using. The graphical abstract offers a visual roadmap of the research workflow, starting with the use of 859 field-based samples collected from various researches existing in the literature, comprising seven non-dimensional hydraulic and geometric input parameters. Sixteen distinct ML models were individually developed. They are Artificial Neural Network, Support Vector Regression, Gaussian Process Regression, Multiple Linear Regression, K-Nearest Neighbors, Bayesian Ridge, Huber Regressor, Stochastic Gradient Descent Regressor, Decision Tree Regressor, Random Forest Regressor, AdaBoost Regressor, Gradient Boosting Regressor, Light Gradient Boosting Regressor, Histogram Gradient Boosting Regressor, CatBoost Regressor (CBR), and eXtreme Gradient Boosting Regressor. All standalone ML models were evaluated using normalized performance metrics—including Root Mean Square Error, Mean Absolute Error, Coefficient of Determination, and Nash–Sutcliffe Efficiency—and subsequently ranked using a comprehensive Ranking Index (RI). The graphical abstract also presents comparative analysis results, where the stacked CBR model outperformed all individual ML models and nine traditional empirical equations (RI = 0.972). Based on the comparative analysis, CatBoost Regressor achieved the highest overall performance among all individual models and was therefore selected as the meta-learner in the stacked ML model. Reliability analysis further confirmed its strong generalization capability, with 57.75% of predictions falling within acceptable error thresholds on the test dataset. The findings offer critical insights for hydraulic engineers and researchers by presenting a high-performing, interpretable, and generalizable ML-based framework for predicting scour depth, thereby supporting safer and more cost-efficient bridge design practices.