## Abstract

Most bridge failures occur due to the development of scour holes around the abutment and pier. Therefore, accurate prediction of abutment scour depth is critical for designing and maintaining bridges to ensure their safety and longevity. Traditional methods for predicting abutment scour depth, such as empirical formulas and physical models, have accuracy, applicability, and cost limitations. Machine learning (ML), on the other hand, has the potential to overcome these limitations by leveraging large amounts of data and identifying complex patterns and relationships that are difficult to detect using traditional methods. ML models can be trained on various data sources, including field measurements, laboratory experiments, and numerical simulations, to predict abutment scour depth accurately. Therefore, the present study aims to develop a novel-tuned Custom ensemble ML model for predicting abutment scour depth in clear-water conditions. The proposed Custom ensemble model outperforms the ML models used to predict non-dimensional scour depth at abutments with an accuracy of 95.93%.

## HIGHLIGHTS

Develop a novel machine learning model that understands the physics involved in abutment scour depth and accurately predicts abutment scour depth at abutments.

The proposed Custom ensemble model outperforms the machine learning models used to predict non-dimensional scour depth at abutments with an accuracy of 95.93.

## ABBREVIATIONS

- AdaBoost
Adaptive boosting

- ANFIS
Adaptive-network-based fuzzy inference system

- ANN
Artificial neural network

- BA-Kstar
Bagging-Kstar

- CPU
Central processing unit

- DA-Kstar
Dagging-Kstar

- DT
Decision tree

- GBDT
Gradient-based decision tree

- GEP
Gene-expression programming

- GMDH
Group method of data handling

- GridSearchCV
GridSearch cross-validation

- HWRE
Hydraulic and Water Resource Engineering Laboratory

- LightGBM
Light gradient boosting machine

- MAE
Mean absolute error

- ML
Machine learning

- RC-Kstar
Random Committee-Kstar

- RS-Kstar
Random Subspace-Kstar

- RMSE
Root mean square error

- STD
Standard deviation

- SVM
Support vector machine

- WIHW-Kstar
Weighted Instance Handler Wrapper-Kstar

- XGBoost
Extreme gradient boosting

## INTRODUCTION

Bridges are one of the most significant structures that humankind has constructed for safe transportation purposes. The bridge structure provides safe passage over the river to link two distinct destinations. These structures built on waterways are essential not only to humankind but also for the economic growth of a country. The expense involved in the construction of a bridge is enormous, and the consequences caused by its failure are irreparable (Gazi *et al.* 2019; Kumar & Afzal 2022). Most bridge failures occur mainly due to scour development around the abutment and pier (Afzal *et al.* 2020; Gautam *et al.* 2021). In scour phenomenon, sediment particles are removed from the riverbed around bridge abutments or piers. The scour development weakens the foundation, resulting in the bridge's collapse. Therefore, scouring around bridge abutments has garnered increased attention from researchers during the last several decades.

Various notable studies have been performed on local scour depth around bridge abutments (Laursen & Toch 1956; Wong 1982; Froehlich 1989; Dey & Barbhuiya 2004; Fael *et al.* 2006; Abou-Seida *et al.* 2012). First, Laursen & Toch (1956) conducted a set of experimental studies to examine scour around bridge piers and abutments. However, they could not describe the flow field behavior at the bridge pier and abutment due to the lack of the necessary instrument. Melville (1992) published a laboratory dataset of scour around bridge abutments with varying flow depth, abutment geometry, and alignments. They also investigated the influence of sediment characteristics on scour depth at abutments. Kwan & Melville (1994) performed an experimental investigation on scour at abutments and reported that the flow structures are dominated by a large primary vortex and its associated downflow. They also identified a secondary vortex, with a counter-rotational direction to that of the primary vortex, occurring next to the primary vortex.

Dey & Barbhuiya (2004) conducted experiments for local scour on the vertical wall, 45° wing-wall, and semi-circular abutments and determined equations of maximum equilibrium scour depth under clear-water scour conditions. Fael *et al.* (2006) performed vertical-wall abutment scour under clear-water conditions. Using regression analysis, they gave the morphology of the scour area in terms of volume and plan dimensions. Abou-Seida *et al.* (2012) developed equations to predict equilibrium scour patterns around vertical bridge abutments in cohesive soil. They also presented an equation to predict the development of scour depth with time. Barbhuiya & Mazumder (2014) performed local scour experiments using four uniform cohesionless sediment diameters and five vertical-wall abutments. They proposed an equation to calculate the scour depth values. Recently, Singh *et al.* (2020) studied experimental results of clear-water scour on a sand bed under short contractions. They proposed two analytical equations to calculate time-dependent scour depth and maximum scour at equilibrium conditions. Based on the above-discussed literature, it can be concluded that experimental techniques are expensive and time-consuming. Also, the experimental setup may be less complicated than the actual circumstance, and therefore the generated regressive equation may fail to operate in a real-world context. Several researchers also performed numerical investigations for sediment transport and scour phenomenon (Afzal 2013; Afzal *et al.* 2015, 2020, 2021; Ahmad *et al.* 2015; Gautam *et al.* 2021). However, numerical simulation also requires high cost and time consumption. Thus, machine learning (ML) tools have evolved recently and can be used as an alternative, more reliable, and accurate tool, requiring less money and time consumption.

The ML approaches are user-friendly, precise, and accurately interpret missing data. With the advancement of ML's predictive capabilities, researchers increasingly use ML instead of conventional experimental methods (Dutta *et al.* 2020; Kumar *et al.* 2020, 2022). Muzzammil (2008) predicted scour depth at abutments using an artificial neural network (ANN). He discovered that the ANN approach outperformed traditional empirical equations. He also concluded that predictions based on raw data (dimensional) are superior to those found on non-dimensional characteristics. Further, Muzzammil (2010) extended his work and used an adaptive-network-based fuzzy inference system (ANFIS) for scour depth prediction at abutments. He observed that the ANFIS model performs better than the ANN and conventional regression models. Azamathulla *et al.* (2010) estimated the abutment scour depth using gene-expression programming (GEP). He found that the GEP technique outperforms ANN and other traditional models.

Najafzadeh *et al.* (2013a, 2013b) used the group method of data handling (GMDH) to estimate abutment scour depth in cohesive soils and clear-water and live-bed situations. They found that the GMDH technique outperformed all other traditional models, including the support vector machine (SVM). Azimi *et al.* (2017, 2019) introduced an advanced ANFIS model called the Pareto-evolutionary structure of the ANFIS network, which outperformed the traditional ANFIS model. Parsaie *et al.* (2019) compared SVM, ANN, and ANFIS models and found that SVM had the best performance. Ebtehaj *et al.* (2018) used the extreme learning machine algorithm and showed that it had faster training and better predictive ability than ANN and SVM. Bonakdari *et al.* (2020) also used the extreme learning machine (ELM) method with four input parameters to predict scour depth in clear-water situations. Pandey *et al.* (2020) employed genetic algorithms to predict maximum scour depth and found that it outperformed multiple linear regression. Metaheuristic optimization algorithms such as grasshopper optimization algorithms (Kaveh *et al.* 2021) and firefly algorithms (Kohansarbaz *et al.* 2021) have been integrated with ANN and ANFIS to achieve higher prediction accuracy. Khosravi *et al.* (2021) used the Kstar model with five innovative hybrid algorithms of bagging (BA-Kstar), dagging (DA-Kstar), random committee (RC-Kstar), random subspace (RS-Kstar), and weighted instance handler wrapper to estimate scour depth for clear-water conditions (WIHW-Kstar). They reported that the RC-Kstar model outperformed other models for scour depth prediction around semi-circular and 45° wings. In contrast, the WIHW-Kstar model had the maximum performance in scour depth prediction around vertical abutment shape. Recently, Xu *et al.* (2023) reviewed the use of ML tools in coastal bridge hydrodynamics.

The existing literature highlights the extensive use of ML approaches for predicting abutment scour depth in clear-water conditions. Therefore, this study aims to develop a new Custom ensemble model for the same purpose. The effectiveness of the developed model is evaluated by comparing it with various established models, such as decision tree (DT), AdaBoost, XGBoost, LightGBM, and the Muzzammil (2010) empirical equation. This study presents a new and improved approach to predicting abutment scour depth in clear-water conditions, utilizing a novel-tuned Custom ensemble ML model that has not been developed before.

## MATERIALS AND METHODS

### Dataset collection

The datasets of abutment scour under clear-water conditions were collected from the experimental study of Dey & Barbhuiya (2004). The dataset includes 297 runs conducted at the Hydraulic and Water Resource Engineering Laboratory (HWRE) at the Indian Institute of Technology Kharagpur, India. They used a flume of dimensions 20 m long, 0.9 m wide, and 0.7 m deep to investigate scour at abutments under clear-water conditions. They used three different abutment geometry: vertical wall, 45° wing-wall, and semi-circular wall abutment of various sizes. The abutments were attached to the side wall of the flume and embedded in a sediment bed of 0.3 m deep. They performed these experiments using sand of *d*_{50} ranging from 0.26 to 3.18 mm.

### Dimensional analysis

*et al.*2004). Therefore, the abutment scour can be written as given in the following equation:where

*U*represents the average flow velocity;

*U*

_{c}represents the critical sediment velocity;

*l*represents the transverse abutment length;

*b*represents the stream wise length;

*g*represents the acceleration due to gravity;

*K*

_{s}represents the abutment form factor;

*d*

_{50}represents the median sediment diameter;

*h*represents the approach flow depth;

*ρ*represents the fluid density;

*ρ*

_{s}represents the sediment density; and represents the fluid kinematic viscosity.

*ρ*and

*ρ*

_{s}are constant. Also,

*K*

_{s}is the same for each of the cross sections. Thus, all these parameters can be removed from the function. Most studies in the field of artificial intelligence consider the effective parameters in output parameter estimation to be dimensionless, which leads to good results (Azamathulla

*et al.*2007; Najafzadeh

*et al.*2016). According to the Buckingham

*Π*theorem, the abutment scours depth is normalized. as given in the following equation:where

*F*

_{e}is the excess abutment Froude number calculated from the following equation:where

*U*

_{e}is the excess approaching flow velocity,

*s*is the relative density of sediment particles, and

*g*is the acceleration due to gravity. The term

*b/l*is constant for the dataset and hence has been removed. Therefore, the present study utilizes excess abutment Froude number (

*F*

_{e}), relative flow depth (

*h/l*), relative submergence (

*d*

_{50}

*/h*), and relative sediment size (

*d*

_{50}

*/l*), which are used as the input parameter to estimate non-dimensional scour depth (

*d*

_{s}

*/l*).

### Statistical analysis

Using appropriate input parameters in ML models that match laboratory conditions is essential to achieve optimal results. To develop these techniques for predicting scour at abutment under clear-water conditions, a total of 297 datasets were collected from Dey & Barbhuiya (2004) literature. Further, based on dimensional analysis, the non-dimensional parameters *d*_{s}*/l*, *d*_{50}*/l*, *d*_{50}*/h*, *h/l,* and *F*_{e} datasets from Dey & Barbhuiya (2004) were prepared. The statistical analysis of the dataset collected from Dey & Barbhuiya (2004) shows mean, maximum, minimum, standard deviation, kurtosis, and skewness value shown in Table 1.

Parameter . | Min . | Max . | Mean . | Std. deviation . | Skewness . | Kurtosis . |
---|---|---|---|---|---|---|

d_{s}/l | 0.64615 | 4.35 | 1.80671 | 0.661819 | 0.846851 | 0.5696 |

d_{50}/l | 0.002 | 0.0775 | 0.01361 | 0.013084 | 2.147396 | 5.6943 |

d_{50}/h | 0.00104 | 0.02 | 0.00652 | 0.005211 | 1.115402 | 0.24257 |

h/l | 0.38462 | 6.25 | 2.38072 | 1.406865 | 1.02322 | 0.58423 |

F_{e} | 0.05588 | 0.39442 | 0.13754 | 0.057534 | 1.369378 | 2.11166 |

Parameter . | Min . | Max . | Mean . | Std. deviation . | Skewness . | Kurtosis . |
---|---|---|---|---|---|---|

d_{s}/l | 0.64615 | 4.35 | 1.80671 | 0.661819 | 0.846851 | 0.5696 |

d_{50}/l | 0.002 | 0.0775 | 0.01361 | 0.013084 | 2.147396 | 5.6943 |

d_{50}/h | 0.00104 | 0.02 | 0.00652 | 0.005211 | 1.115402 | 0.24257 |

h/l | 0.38462 | 6.25 | 2.38072 | 1.406865 | 1.02322 | 0.58423 |

F_{e} | 0.05588 | 0.39442 | 0.13754 | 0.057534 | 1.369378 | 2.11166 |

Table 1 presents five parameters such as *d*_{s}*/l, d*_{50}*/l, d*_{50}*/h, h/l,* and *F*_{e}. Each parameter has a minimum, maximum, mean, and standard deviation (Ahani *et al.* 2020a, 2020b; Ahani *et al.* 2022). The skewness and kurtosis values also provide insights into the shape of the distributions. The *d*_{s}*/l*, *h/l*, and *F*_{e} parameters are positively skewed, indicating more values on the lower end of their ranges. The *d*_{50}*/l* and *d*_{50}*/h* parameters have heavily skewed distributions to the right. The *d*_{s}/*l* and *h*/*l* parameters have less kurtosis than the normal distribution. In comparison, *d*_{50}*/l* and *F*_{e} have more kurtosis than the normal distribution. However, the *d*_{50}*/h* parameter has the lowest kurtosis value, meaning fewer outliers than a normal distribution.

### Methodology

*Fe*), relative flow depth (

*h/l*), relative submergence (

*d*

_{50}

*/h*), and relative sediment size (

*d*

_{50}

*/l*). The statistical analysis is implemented to know the properties and trends of these parameters. Further, excess abutment Froude number (

*Fe*), relative flow depth (

*h/l*), relative submergence (

*d*

_{50}

*/h*), and relative sediment size (

*d*

_{50}

*/l*) are used as input parameters to the ML models to predict non-dimensional abutment scour depth (

*d*

_{s}

*/l*). The input parameters are chosen according to the correlation values of each variable. This dataset is divided into training (70%) and testing datasets (30%) for building the model and validation purposes. The ML models use mathematical concepts to find the correlation between dependent and independent variables. This study uses DT, AdaBoost, XGBoost, LightGBM, and a novel Custom ensemble model to predict the abutment scour depth. In addition, Muzzammil's (2010) empirical equations were used for prediction to compare the results of the models mentioned above with the existing regression equations.

### ML models

The present study has used ensemble-based models to predict abutment scour depth. DT, Adaptive boosting regressor (AdaBoost), Extreme Gradient Boosting or (XGBoost), Light gradient boosting machine (LightGBM), and a developed Custom ensemble model are developed to predict abutment scour depth.

#### Decision tree

#### Adaptive boosting regressor

*l*th weak learner or tree , and the pair represents tree predictions in previous and current iterations, respectively, representing the updated or improved coefficient and weak learner.

*L*weak learner.

The AdaBoost technique is prone to overfitting for noiseless datasets, which suggests that the present model would be robust to overfitting due to a noiseless dataset containing relevant input features.

#### Extreme gradient boosting

However, XGboost differs from GBDT techniques in specific ways. First, the GBDT technique only uses the first-order Taylor expansion, whereas XGboost extends the loss function with a second-order Taylor expansion. Second, normalization is used in the target function to avoid overfitting and lower the model's complexity. It is a method of selection that includes both embedded and filtered features.

#### Light gradient boosting machine

LightGBM technique is based on GBDT algorithms, which combine weak and strong learners to form a strong learner. The GBDT method uses a DT that is slightly different from a standard DT. The previous trees' results and residuals are recorded here and utilized in the following learning stage. The final output is calculated by combining the findings of multiple trees (Friedman 2001). The GBDT is extensively used worldwide due to its prediction capabilities. However, its accuracy efficiency has recently suffered from the massive data increase. The benefit of implementing LightGBM Regressor is that it enhances forecasting performance while also lowering memory usage without sacrificing prediction power. Unlike standard GBDT, it also employs a better histogram method. The algorithm development of a DT requires more computing time than traditional GBDT. The developed DT is utilized to locate the optimal segmentation point. The basic idea is to sort feature values and enumerate all accessible feature points, which uses a lot of memory and high computation time. The continuous eigenvalues are partitioned into *k* intervals, with *k* values chosen as division points. The LightGBM algorithm employs the leaf-wise generation approach to decrease the training data. Compared to other methods, such as depth or level-wise traditional techniques, the leaf wise can reduce losses when growing the same leaf. Moreover, an additional parameter is employed to restrict the depth of the DT, preventing overfitting.

#### Custom ensemble model

*N*is the number of individual predictors,

*ŷ*(

*x*) is the final prediction,

*y*(

_{i}*x*) is the

*i*th prediction, and

*w*is the weight associated with the

_{i}*i*th prediction.

Further, the weights were estimated by running the Voting Regressor in a loop with different weights assigned to each predictor, and the corresponding *R*^{2} scores were calculated. The combination that produced the highest score was selected, resulting in a weight of 4 for Gradient Boosting and 1 for AdaBoost, ANN, and XGBoost models. The dataset was fitted to the ensemble model with these weights, and predictions were obtained.

The Ensemble ML models are used in the present study to predict abutment scour in clear-water conditions because they offer several advantages over traditional ML models (ANN and SVM). Ensemble models combine multiple models, making them more robust and reducing the risk of overfitting. They can also handle complex data sets and provide more accurate predictions than a single model. In addition, ensemble models can incorporate different algorithms and techniques, such as DT, random forests, and gradient boosting, to leverage the strengths of each model and improve the overall prediction accuracy. Furthermore, ensemble models can identify the essential features that contribute to the prediction of abutment scour, which can be used to enhance the design and maintenance of hydraulic and coastal structures. The Ensemble ML models have been widely used in hydraulics and coastal engineering to predict several hydraulics and coastal engineering parameters. These models can provide valuable insights into the behavior of hydraulic and coastal systems, informing the safe design and operation of coastal and hydraulic structures.

### Tuning of Ml models

The ML models in the present study predicted the non-dimensional abutment scour depth with low accuracy with their default hyperparameters. Thus, tuning the hyperparameters was required to improve accuracy by boosting their performance. All the models were tuned to get the best hyperparameters. Tuning of the models is performed using the GridSearchCV function. A set of hyperparameters are sent to the function. This is done by passing a dictionary containing several values of the hyperparameters. The function then tries all combinations of the hyperparameters by cross-validation and checks the performance of each set of hyperparameters by fitting them to the model. Finally, the set of values of hyperparameters, which yields the best performance, is selected. The default and the tuned hyperparameters are presented in Table 2.

Models . | Default hyperparameters . | Tuned hyperparameters . |
---|---|---|

DT | criterion = ‘ squared_error' | criterion = 'friedman_mse’ |

AdaBoost | loss = 'linear’ n_estimators = 50 | loss = 'square’ n_estimators = 100 |

XGBoost | learning_rate = 0.3 max_depth = 6 min_child_weight = 1 | learning_rate = 0.1 max_depth = 4 min_child_weight = 2 |

Models . | Default hyperparameters . | Tuned hyperparameters . |
---|---|---|

DT | criterion = ‘ squared_error' | criterion = 'friedman_mse’ |

AdaBoost | loss = 'linear’ n_estimators = 50 | loss = 'square’ n_estimators = 100 |

XGBoost | learning_rate = 0.3 max_depth = 6 min_child_weight = 1 | learning_rate = 0.1 max_depth = 4 min_child_weight = 2 |

### Model performance assessment

*R*

^{2}, MAE, and RMSE, are computed to examine the correctness or performance of the ML techniques (Kumar

*et al.*2022). The coefficient of determination (

*R*

^{2}) is the statistical performance metric that examines the strong linear relationship between dependent and independent variables. The RMSE represents the difference between actual and predicted values. However, MAE represents the magnitude of the mean error. The mathematical formulation of these performance matrices is described in the following equations:where

*M*is the number of observations,

*z*is the actual value and is the predicted value

_{i}## RESULTS AND DISCUSSION

The tree-based ML techniques such as DT, Adaboost, XGBoost, and LightGBM and Custom ensemble model were developed to estimate abutment scour depth in clear-water conditions. The ML models' performance in predicting the abutment scour depth was examined using the dataset of Dey & Barbhuiya (2004).

### Sensitivity analysis

*Fe*is the most sensitive parameter in estimating abutment scour depth under clear-water conditions. It has the highest correlation coefficient of 0.97, followed by

*d*

_{50}

*/l*. On the other hand, the parameter

*d*/

_{50}*h*is the least sensitive parameter, with a correlation coefficient of 0.47. Similarly, the parameter

*h/l*has the second-lowest correlation coefficient of 0.53 in predicting abutment scour depth under clear-water conditions.

### Abutment scour depth prediction using ML techniques

*Fe*), relative flow depth (

*h/l*), relative submergence (

*d*

_{50}

*/h*), and relative sediment size (

*d*

_{50}

*/l*) to predict the abutment scour depth (

*d*

_{s}

*/h*) in clear-water condition. The comparison of observed and predicted pier scour depths for all ML models through line and scatter plots is shown graphically in Figure 5.

The study compared the performance of several ML models in predicting abutment scour depth in clear water conditions. The results showed that the novel-tuned Custom ensemble model outperformed other models in terms of accuracy. The DT, AdaBoost, XGBoost, and LightGBM models offer adequate abutment scour prediction accuracy. The scatter plots indicate that the novel-tuned Custom ensemble and DT models predict the abutment scour depth with the least and maximum scatter, respectively. The Custom ensemble model scatter plot also showed that bias and slope are close to 0 and 1, respectively. The developed Custom ensemble model showed the highest coefficient of determination (*R*^{2} = 0.9593). However, the DT has the lowest coefficient of determination (*R*^{2} = 0.8963) for predicting abutment scour depth in clear water conditions. The other ensemble model also provides comparable results to the developed Custom ensemble model. The XGBoost and AdaBoost model predicts abutment scour depth with a coefficient of determination of 0.9391 and 0.9355, respectively, followed by the LightGBM model, which predicts abutment scour depth with a coefficient of determination of 0.9291.

### Abutment scour depth estimation using empirical formulations

It can be observed that the Muzzammil (2010) formulation predicted value of abutment scour depth shows good agreement with the experimental results. The Muzzammil (2010) formulation predicts abutment scour depth with an accuracy of 0.7279, lower than all the ensemble models used in the present study. The scatter plot of Muzzammil (2010) formulation also shows that maximum scatter in comparison to other existing ML model

### Comparison of ML models with Muzzammil (2010) formulation

The performance metrics of ML techniques and Muzzammil's (2010) formulation are compared to each other for training and testing datasets, shown in Table 3.

Model . | Training . | Testing . | ||||
---|---|---|---|---|---|---|

R^{2}
. | RMSE . | MAE . | R^{2}
. | RMSE . | MAE . | |

DT | 0.9644 | 0.1247 | 0.0718 | 0.8963 | 0.2099 | 0.1497 |

AdaBoost | 0.9533 | 0.1429 | 0.1216 | 0.9355 | 0.1656 | 0.1434 |

XGBoost | 0.9679 | 0.1185 | 0.0888 | 0.9391 | 0.1609 | 0.1306 |

LightGBM | 0.9721 | 0.1105 | 0.0801 | 0.9291 | 0.1735 | 0.1319 |

Custom ensemble | 0.9898 | 0.0668 | 0.0536 | 0.9593 | 0.1318 | 0.1073 |

Muzzammil (2010) | 0.7605 | 0.3236 | 0.2576 | 0.7229 | 0.3432 | 0.2761 |

Model . | Training . | Testing . | ||||
---|---|---|---|---|---|---|

R^{2}
. | RMSE . | MAE . | R^{2}
. | RMSE . | MAE . | |

DT | 0.9644 | 0.1247 | 0.0718 | 0.8963 | 0.2099 | 0.1497 |

AdaBoost | 0.9533 | 0.1429 | 0.1216 | 0.9355 | 0.1656 | 0.1434 |

XGBoost | 0.9679 | 0.1185 | 0.0888 | 0.9391 | 0.1609 | 0.1306 |

LightGBM | 0.9721 | 0.1105 | 0.0801 | 0.9291 | 0.1735 | 0.1319 |

Custom ensemble | 0.9898 | 0.0668 | 0.0536 | 0.9593 | 0.1318 | 0.1073 |

Muzzammil (2010) | 0.7605 | 0.3236 | 0.2576 | 0.7229 | 0.3432 | 0.2761 |

Note: Bold indicates the best performance for each model.

Table 3 provides a comparison of different ML models' performance metrics in predicting abutment scour depth. The models' performance is evaluated based on two evaluation metrics: coefficient of determination (*R*^{2}), root mean squared error (RMSE), and mean absolute error (MAE) for both training and testing datasets. The DT, AdaBoost, XGBoost, LightGBM, Custom ensemble, and Muzzammil (2010) prediction results show that the novel Custom ensemble model achieved the best performance with the highest *R*^{2} value of 0.9898, lowest RMSE value of 0.0668, and lowest MAE value of 0.0536 for the training dataset. For the testing dataset, the novel Custom ensemble model also achieved the highest *R*^{2} value of 0.9593, lowest RMSE value of 0.1318, and lowest MAE value of 0.1073. These results indicate that the Custom ensemble model has the best predictive performance for abutment scour depth compared to the other models.

The DT, XGBoost, and LightGBM models also performed well in predicting abutment scour depth and were comparable to the results of the novel Custom ensemble model. However, the AdaBoost model's performance was slightly lower than the other models, with *R*^{2} values below 0.95 for both the training and testing datasets. The Muzzammil (2010) model performed the worst among all the models, with the lowest *R*^{2}, highest RMSE, and highest MAE values for both the training and testing datasets. This indicates that the ML models outperform the traditional model in predicting abutment scour depth.

The abutment scour dataset had been divided into two parts to validate the ML model. The first set (70% of the data) was used to train the ML model. After that, the remaining 30% of the abutment scour dataset was used as validation data against the predictions done by the developed ML model. This is a standard practice in ML (i.e., testing), considered the same as validation of the numerical model (Afzal *et al.* 2023). Further, the results are compared with Muzzammil (2010) empirical formulation, which found that the ML model outperforms Muzzammil (2010) empirical formulation with higher accuracy.

## CONCLUSION

The bridge piers and abutments often interact with approaching flow and cause scour around it that may lead to the failure of the bridge structure. Several experimental investigations are performed to estimate bridge abutments. However, the accuracy of the estimated abutment scour depth may be affected by introducing assumptions in the experimental study. The empirical formulation developed using conventional regression methods has limitations and may not accurately capture the scour phenomenon's complex nature. Therefore, the ML approach is introduced to determine accurate abutment scour depth estimation in clear water conditions.

This study uses DT, Adaboost, XGboost, LightGBM, and a novel Custom ensemble technique to estimate abutment scour depth. The performance metrics results of Ensemble models were compared with a single DT and the Muzzammil (2010) formulations. All the ensemble models used in the present study predict with higher accuracy than a single DT and the Muzzammil (2010) formulation. The highest value of *R*^{2} (0.9593) for the testing purpose signifies that the developed Custom ensemble model outperforms other models used in the present study. The Custom ensemble model also has the least RMSE and MAE value of 0.1318 and 0.1073, respectively. Therefore, it can be concluded that the Custom ensemble model and ML algorithms can be used as reliable design tools for predicting abutment scour depth in clear water conditions.

Using a Custom ensemble ML model to predict abutment scour depth has multiple advantages, such as improved accuracy, robustness to data variations, faster analysis, identification of potential risks, and incorporation of multiple data sources for scalability. This can lead to improved public safety, reduced infrastructure failures, and proactive measures to prevent damage.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*.*

*Scour at Bridge Piers*

*PhD thesis*

*.*

*.*

*.*