Machine learning-based prediction of scour depth evolution around spur dikes

Spur dikes represent crucial components within river training systems, which are designed to address the dynamic alterations induced by river degradation and aggradation (Kothyari & Ranga Raju 2001). These structures are strategically placed either perpendicular or at an angle to the riverbank, extending from one end into the river while securely anchored at the other to protect against erosion and regulate various hydraulic processes (Duan et al. 2009; Choufu et al. 2019; Pourshahbaz et al. 2022; Saikumar et al. 2022). Spur dikes serve multifaceted purposes, including diverting flow and safeguarding riverbanks from flood events, establishing their pivotal role in hydraulic engineering (Kuhnle et al. 2002; Shah et al. 2023). The construction of spur dikes introduces complexities in the flow path, causing shifts in hydrostatic pressure upstream and downstream (Zhang et al. 2009; Yazdi et al. 2010; Nayyer et al. 2019; Pandey et al. 2019; Farshad et al. 2022). Spur dikes have proven to be highly effective in various river management projects, playing a crucial role in stabilizing riverbanks and managing sediment transport (Noret et al. 2013; Bigham 2020). The key applications of spur dikes are in addressing river bank protection, river degradation and aggradation by preventing excessive erosion along riverbanks, which poses a threat to both infrastructure and ecosystems. Real-world case studies further highlight the importance of spur dikes. For instance, along the Mississippi River in the USA, spur dikes are strategically placed to prevent bank erosion while maintaining navigation channels (Klingeman et al. 1984; Barnett 2017). Their controlled positioning manages sediment movement, ensuring riverbank stability and reducing the frequency and cost of dredging operations. In the Rhine River, Europe, they regulate sediment transport, preventing unwanted accumulation in key areas, thus maintaining navigability and protecting riverbanks from erosion (Habersack et al. 2016; Havinga 2020). In China's Yellow River, where high sediment loads cause severe aggradation, spur dikes guide sediment-laden waters into predetermined areas, lowering the risk of catastrophic floods (Peng et al. 2010). In Pakistan's Indus River, where seasonal monsoons and snowmelt lead to severe erosion, the Water and Power Development Authority implemented spur dikes to reduce bank erosion, safeguarding agricultural lands and infrastructure from flood damage (Atta-Ur-Rahman & Shaw 2015). Additionally, in the Nile River, Egypt, where aggradation has threatened irrigation systems, spur dikes manage sediment flow, preventing blockages in irrigation channels and ensuring a stable water supply for agriculture. These examples demonstrate the versatility and critical role of spur dikes in river engineering, not only for stabilizing riverbanks and managing sediment transport but also for mitigating flood risks and supporting agricultural productivity through improved flow management. This alteration initiates the formation of intricate vortex areas, particularly marked by substantial vortices at the head of the spur dikes, thereby constituting the principal local scour mechanism. Local scour poses a substantial threat to the structural integrity of these river training elements, potentially leading to catastrophic failures (Pandey et al. 2018; Gu et al. 2023).

Researchers have employed various techniques to reduce the scour depth around spur dikes (Garde et al. 1961; Gu et al. 2020; Guguloth & Pandey 2023b). These techniques can be broadly categorized into direct and indirect methods. Direct methods involve the utilization of construction materials, such as revetments and riprap, positioned on spill slopes to resist erosion and directly protect structures against flow attack (Lauchlan & Melville 2001; Yılmaz 2014; Gupta et al. 2023). In contrast, indirect methods modify flow patterns through the implementation of specialized structures, such as protective spur dikes, guide banks, or collars, to induce a reduction in local scour (Karami et al. 2011; Pandey et al. 2016; Delavari et al. 2022; Guguloth & Pandey 2023a). Recent attention has been directed toward the use of a protective spur dike as an indirect method. Given that spur dikes are typically constructed consecutively, the first spur dike (SD1) upstream, often referred to as ‘the first spur dike,’ experiences the most destructive flow influence (Pandey 2014). As a result, reinforcing this initial dike becomes imperative in an effort to diminish local scour depth. The introduction of a protective spur dike upstream, especially in a series of parallel spur dikes, alters the flow direction significantly, leading to a substantial reduction in scour depth around the main spur dike (Gu et al. 2023). This reduction is particularly crucial for safeguarding the SD1, which directly faces the oncoming flow (Shah et al. 2023). The temporal variation of scour and the reduction of maximum scour depth around the SD1, place paramount importance on understanding and optimizing the main parameters of a protective spur dike, specifically its length and spacing between protected spur dikes (Pandey et al. 2016; Delavari et al. 2022). In recent years, numerous researchers have conducted both laboratory and field experiments aimed at mitigating scour around spur dikes through various methods, while also examining how scour varies over time (Iqbal et al. 2022; Aung et al. 2023; Gu et al. 2023; Tabassum et al. 2024). Nayyer et al. (2019) explored the flow dynamics near spur dikes of three distinct shapes – namely I, T, and L shapes – using experimental methods and numerical simulations via Flow 3D software. Their investigation analyzed three combinations, namely (ILI), (TLI), and (LTT), revealing that the (LTT) series was most effective in reducing factors such as flow velocity, shear stress, pressure, and turbulence near the spur dikes.

Recent studies have focused on applying AI models to predict temporal variation in scour depth and maximum scour depth around hydraulic structures (Sreedhara et al. 2021; Pandey et al. 2022; Guguloth et al. 2024). Pandey et al. (2020) employed genetic algorithms to estimate time-dependent scour depths around circular bridge piers. Similarly, Azamathulla & Wu (2011) utilized various soft computing techniques to predict scour depths beneath river pipelines, showcasing good agreement with observed data points. Pandey et al. (2022) developed three novel machine-learning techniques, including Gradient Boosting Decision Tree, Kernel Ridge Regression, and Cascaded Forward Neural Network for predicting the depth of the scour hole around the spur dike. Aamir & Ahmad (2019) employed the artificial neural network (ANN) and ANFIS models to predict the non-dimensional maximum scour depth under wall jets downstream of a rigid apron, showcasing superior performance compared to previously proposed empirical and regression methods. Ahmadianfar et al. (2021) developed an artificial intelligence approach for predicting wave-induced local scour depth near circular piles.

The objective of this study is to develop novel hybrid machine-learning models aimed at predicting the temporal variation and maximum scour depth around the SD1 within a series of spur dikes. The study intends to explore four machine-learning models, including eXtreme Gradient Boosting (XGBoost), random forest (RF), convolutional neural network–long short-term memory (CNN–LSTM), and ANN for analyzing the temporal variation of scour depth and non-dimensional maximum scour depth around the SD1 in a series of spur dikes. Utilizing a dataset comprising 465 data points on temporal scour depth variation and 27 experimental data points on maximum scour depth, the study will compare and evaluate the performances of these machine-learning models using graphical tools and statistical metrics. Overall, the study aims to contribute a new approach to quantifying the temporal variation of scour depth around the SD1 within a series of spur dikes through the application of advanced machine-learning techniques.

The experimental study was carried out in a precisely controlled rectangular hydraulic flume designed to replicate natural flow conditions. The flume had a total length of 10.30 m, a width of 0.8 m, and a height of 0.5 m, with a longitudinal slope of 0.0004 to simulate natural riverbed gradients. The working section was strategically placed 4 m downstream from the inlet section to ensure a fully developed flow before the measurements were taken. The test section itself measured 2.3 m in length, 0.8 m in width, and 0.5 m in depth, providing ample space for accurate flow and scour observations. The flume bed was filled with non-cohesive, uniform sediments characterized by a median particle diameter (D50) of 0.32 mm and a geometric standard deviation (σg) of 1.31, indicative of sediment homogeneity according to the classification by Dey et al. (1995). The relative density of the sediment particles was recorded as 2.65, meaning the sediment was considerably denser than the water, promoting realistic scour processes under controlled conditions.

Three impermeable spur dikes were installed at predetermined intervals within the test section. These dikes were designed with a uniform thickness of 4 mm and varied lateral lengths of 0.15, 0.12, and 0.10 meters, all standing at a height of 0.55 m. They were aligned perpendicular to the primary flow direction to create intentional flow disturbances and induce local scour. The experiments were conducted under three different flow intensities: 23.11, 19.26, and 15.4 l/s, corresponding to average flow velocities of 24.07, 20.6, and 16.05 m/s, respectively. All trials maintained clear water conditions (U/U_c < 1), where the approach velocity (U) remained below the critical velocity (U_c) for sediment movement, ensuring that the scour observed was due solely to the interaction between the flow and spur dikes. Three distinct spacing configurations between the spur dikes, referred to as L, 2L, and 3L, were tested. These varying configurations were critical in studying the influence of dike spacing on flow interference, sediment transport, and the formation of scour holes.

Throughout the early stages of each experimental trial, the scour process exhibited similar patterns across all three spur dikes. However, as the experiments progressed, sediment that had been eroded from the scour hole at the base of SD1 was transported downstream, forming a ridge of deposited material. This ridge-shaped sediment formation subsequently migrated downstream, progressively filling the scour hole at the second spur dike (SD2). Initially, this process of sediment transport between dikes occurred rapidly, but after 2–3 h, the rate of transfer slowed considerably. This reduction in sediment transport is a clear indication of the protective, or ‘shielding,’ effect exerted by SD1 on the downstream spur dikes. Essentially, the presence of SD1 reduced the erosive forces acting on SD2, thereby limiting further scour development at SD2. Additionally, the third spur dike exhibited unique scour patterns, where sediment deposits were observed both upstream and downstream of its position. These deposits were particularly concentrated at the junction between the spur dike and the flume walls, further illustrating the complex interaction between flow structures and sediment transport in the vicinity of spur dikes.

The key hyperparameters (Pannakkong et al. 2022) that impact the performance of the algorithm are presented in Table 2.

Table 2

Hyperparameter information on ANN

Notation .	Hyperparameter .	Inference .
	Learning rate	It controls the size of the step taken during optimization and affects the speed and quality of convergence during training. Too high of a learning rate can speed, while too low of a learning rate can result in slow convergence
	Number of hidden layers	It determines the depth of the network and its capacity to learn complex relationships in the data. g more hidden layers can increase the model's capacity to capture intricate patterns but also increases the risk of overfitting
	Number of neurons	More neurons can lead to a higher capacity to learn complex patterns, but it also increases the computational complexity of the model
	Batch size	Larger batch sizes can lead to faster convergence but may require more memory, while smaller batch sizes may lead to more noise in the gradient estimates but can sometimes generalize better
	Epochs	Increasing number of epochs can improve the model's performance up to a certain point, but may also increase the risk of overfitting. Proper tuning the number of epochs is crucial for achieving the right balance between underfitting and overfitting
	Optimizer	It impacts the convergence speed and quality of the trained model. Choosing appropriate optimizer aids the network's ability to navigate the loss and identify the optimal parameter values
	Regularization parameters	Regularization parameters control overfitting by penalizing large weights in the network. Lasso and Ridge type regularization improve generalization handling the noisy data

Notation .	Hyperparameter .	Inference .
	Learning rate	It controls the size of the step taken during optimization and affects the speed and quality of convergence during training. Too high of a learning rate can speed, while too low of a learning rate can result in slow convergence
	Number of hidden layers	It determines the depth of the network and its capacity to learn complex relationships in the data. g more hidden layers can increase the model's capacity to capture intricate patterns but also increases the risk of overfitting
	Number of neurons	More neurons can lead to a higher capacity to learn complex patterns, but it also increases the computational complexity of the model
	Batch size	Larger batch sizes can lead to faster convergence but may require more memory, while smaller batch sizes may lead to more noise in the gradient estimates but can sometimes generalize better
	Epochs	Increasing number of epochs can improve the model's performance up to a certain point, but may also increase the risk of overfitting. Proper tuning the number of epochs is crucial for achieving the right balance between underfitting and overfitting
	Optimizer	It impacts the convergence speed and quality of the trained model. Choosing appropriate optimizer aids the network's ability to navigate the loss and identify the optimal parameter values
	Regularization parameters	Regularization parameters control overfitting by penalizing large weights in the network. Lasso and Ridge type regularization improve generalization handling the noisy data

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RF (Pham et al. 2021), an ensemble learning bagging algorithm, is extensively utilized for both classification and regression tasks due to its robustness and versatility across various applications. As a non-parametric method, it operates without making any assumptions about the data distribution, making it suitable for diverse datasets (Liu et al. 2020). However, relying on a single decision tree with only one split may not yield reliable estimates. To address this limitation, it builds multiple decision trees of varying complexity during training, comprising roots, nodes and aggregates predictions from individual trees to generate final predictions at the leaves.

It ensures diversity and improves generalization ability in the algorithm, which are crucial aspects for reducing overfitting. Their effectiveness stems from its incorporation of randomness through two primary mechanisms: bootstrapping, involving random sampling, and the consideration of random subsets of features at each split within the trees. The key hyperparameters (Rehman et al. 2022) that impact the performance of the algorithm are presented in Table 3.

Table 3

Hyperparameter information on RF

Notation .	Hyperparameter .	Inference .
	Number of estimators (n_estimators)	Increasing the number of trees generally improves predictive performance, but may lead to longer training times and higher memory consumption
	Maximum number of features (max_features)	Determining the optimal number of features can help control overfitting. A smaller value can increase model diversity and reduce correlation among trees
	Maximum depth of each tree (max_depth)	Constraining tree depth can prevent overfitting but may also lead to underfitting if set too low
	Minimum number of samples required to a split an internal node (min_samples_split)	Increasing this value can make the model more robust to noise but may also lead to underfitting
	Minimum number of samples required at a leaf node. (min_samples_leaf)	Similar to min_samples_split, increasing this value can prevent overfitting but may also lead to underfitting

Notation .	Hyperparameter .	Inference .
	Number of estimators (n_estimators)	Increasing the number of trees generally improves predictive performance, but may lead to longer training times and higher memory consumption
	Maximum number of features (max_features)	Determining the optimal number of features can help control overfitting. A smaller value can increase model diversity and reduce correlation among trees
	Maximum depth of each tree (max_depth)	Constraining tree depth can prevent overfitting but may also lead to underfitting if set too low
	Minimum number of samples required to a split an internal node (min_samples_split)	Increasing this value can make the model more robust to noise but may also lead to underfitting
	Minimum number of samples required at a leaf node. (min_samples_leaf)	Similar to min_samples_split, increasing this value can prevent overfitting but may also lead to underfitting

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XGBoost (Vogeti et al. 2022) is an ensemble boosting algorithm that constructs decision trees (base learners based on similarity scores), features, and an additive model with the aim of loss function minimization. These are constrained using the number of leaves, nodes, splits, or layers. The additive trees are introduced without replacing the existing trees use a gradient descent procedure to minimize losses. The separation that yields the highest loss reduction is selected. The constructed tree begins at node ‘i’, which is divided into either left or right branches depending on the chosen separation criteria. Now, reduction in loss can be calculated, and the branch having the highest loss reduction is preferred.

CNN–LSTM (Khorram & Jehbez 2023) is a hybrid algorithm that utilizes the potential of both CNN and LSTM algorithms for establishing relationships between and output variables. Initially, elements of the input data matrix are navigated through the convolution layer which performs matrix multiplication between input and filter matrices, resulting in a convoluted matrix. This matrix is dimensionally reduced using pooling operations capturing the most pertinent information captured from the input layer. The pooled feature map is converted into a one-dimensional vector in this layer. Diverse features learned by the convolution pooling and flattening layers are transformed into a dense vector. The final layer of the workflow is the output layer and is passed as input to the LSTM. Furthermore, this information is passed through three gating units, respectively. The key hyperparameters (Lilhore et al. 2023) that impact the performance of the algorithm are presented in Table 4.

Table 4

Hyperparameter information on CNN–LSTM

Notation .	Hyperparameter .	Inference .
	Number of kernels	These aid in efficient feature extraction from the given input data
	Pooling	Pooling layers are the layers that can be used in truncating the spatial dimensions of the input data in a network
	LSTM layer node	They are a hidden layer nodes that assists in the transmission of data between gating units
	Number of neurons	The performance of a model is significantly influenced by the number of neurons present in a dense model layer. Overfitting occurs when very high neuron values are considered
	Batch size	The number of samples that are simultaneously transmitted to the network is set by the batch size. Larger batch size captures the intricate patterns of the data incorrectly.
	Learning rate	The learning rate characterizes the size of the changes made to the weights in order to achieve the minimum loss function. Lower learning rates are preferable as they ensure better feature extraction resulting in improved model performance
	Epochs	An epoch refers to a single pass through the entire training process of a neural network. The higher number of epochs ensures better simulation accuracy increasing the overall computation time
	Dropout	It is discarding of a noisy data from a neural network. Removal of noisy data improves performance

Notation .	Hyperparameter .	Inference .
	Number of kernels	These aid in efficient feature extraction from the given input data
	Pooling	Pooling layers are the layers that can be used in truncating the spatial dimensions of the input data in a network
	LSTM layer node	They are a hidden layer nodes that assists in the transmission of data between gating units
	Number of neurons	The performance of a model is significantly influenced by the number of neurons present in a dense model layer. Overfitting occurs when very high neuron values are considered
	Batch size	The number of samples that are simultaneously transmitted to the network is set by the batch size. Larger batch size captures the intricate patterns of the data incorrectly.
	Learning rate	The learning rate characterizes the size of the changes made to the weights in order to achieve the minimum loss function. Lower learning rates are preferable as they ensure better feature extraction resulting in improved model performance
	Epochs	An epoch refers to a single pass through the entire training process of a neural network. The higher number of epochs ensures better simulation accuracy increasing the overall computation time
	Dropout	It is discarding of a noisy data from a neural network. Removal of noisy data improves performance

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To enhance the predictive accuracy of our machine-learning models – RF, XGBoost, ANN, and CNN–LSTM – we employed a systematic approach for hyperparameter tuning. Hyperparameters play a critical role in the performance of machine-learning models, and optimizing them is essential for achieving robust and accurate predictions. In this study, we utilized the genetic algorithm (GA) to efficiently search the hyperparameter space and identify optimal configurations for each model. GA is a metaheuristic optimization technique inspired by the process of natural selection, which mimics the evolutionary principles of selection, crossover, and mutation. This algorithm proves particularly effective in exploring complex and nonlinear parameter spaces, making it well-suited for enhancing the performance of machine-learning models.

The hyperparameters considered for tuning varied across the different models that were presented in the preceding sections. Our objective was to find the set of hyperparameters that maximizes the predictive performance of each model. The GA implementation involved the generation of initial populations of hyperparameter sets, subsequent evaluation of each set using a fitness function based on model performance metrics, and the iterative evolution of populations through genetic operations. This process continued until convergence to an optimal set of hyperparameters.

The study employed six key statistical performance evaluation metrics to rigorously assess the predictive capabilities of the machine-learning models. The metrics include R² (coefficient of determination), providing insights into the proportion of variation in the observed data captured by the model; Kling–Gupta efficiency (KGE), offering a comprehensive measure of goodness-of-fit by evaluating the model's ability to reproduce observed variability, correlation, and bias; root mean square error (RMSE), quantifying the model's average prediction error; mean absolute error, representing the average absolute difference between observed and predicted values; mean absolute percentage error (MAPE), expressing prediction accuracy as a percentage of the observed values; and percentage bias (PBIAS), indicating the model's tendency to systematically overestimate or underestimate the observed values. These metrics collectively provided a robust framework for assessing the models' performance across different combinations of predictors during both training and testing periods, enabling a comprehensive understanding of their accuracy, precision, and generalization capabilities.

Mathematical formulations of these indices are defined as follows:

• (1) coefficient of determination (R²):

  

  (17)
• (2) KGE:

  

  (18)
• (3) RMSE

  

  (19)
• (4) (MAE)

  

  (20)
• (5) MAPE, %

  

  (21)
• (6) PBIAS:

  

  (22)

  where X_i denotes the observed non-dimensional scour depth, Y_i represents the corresponding predicted non-dimensional scour depth, and indicate the averages of the observed and predicted values, and n is the total number of observations. stand for the standard deviation of the simulated and observed time series, while, represent the mean of the simulated and observed time series, respectively.

This study focused on predicting the temporal variation of scour depth by employing non-dimensional time ⁠, non-dimensional spacing ⁠, channel width ⁠, and time-averaged velocity as predictors. A preliminary correlation analysis between and the four predictors revealed noteworthy associations, with non-dimensional time exhibiting a particularly strong correlation (>0.8), followed by ⁠, ⁠, and in descending order. Motivated by these findings, we established four machine-learning models – RF, XGBoost, ANN, and CNN–LSTM. Each model was constructed by considering all four predictors simultaneously (referred to as Combo-1) and subsequently removing the least correlated predictor in three additional combinations (Combo-2, Combo-3, and Combo-4). This iterative approach aimed to differentiate the impact of individual predictors on the predictive performance of the models. Similarly, we extended our predictive analysis to non-dimensional maximum scour depth ⁠, utilizing predictors such as ⁠, ⁠, ⁠, and densimetric Froude number ⁠. Correlation analysis underscored robust associations, particularly between and with (>0.95), while and exhibited insignificant correlations. Subsequently, machine-learning models were developed for using the same iterative approach with four predictor combinations, highlighting the influential factors governing maximum scour depth dynamics. The data were split into 75% for training and 25% for testing, and the hyperparameters of the ML models were optimized using GA. Subsequently, the performance of these models in both training and testing periods was rigorously assessed through graphical representations and a set of six performance evaluation indicators, providing a comprehensive understanding of the predictive capabilities and generalization of the models.

Furthermore, the transferability of accuracy from the training to the testing period is notable. RF, XGBoost, and ANN exhibit high transferability, with XGBoost demonstrating the most consistent performance across all four combinations. In contrast, CNN–LSTM shows poor results during the testing period, indicating a limited ability to generalize its predictions. It is noteworthy that XGBoost maintains stable performance in all four combinations, while the performance of ANN, although strong in Combo-1, experiences a decline in Combo-2 to Combo-4. These graphical illustrations provide valuable insights into the predictive capabilities and generalization performance of the machine-learning models in forecasting the temporal variation of scour depth, emphasizing the role of predictor combinations and the varying effectiveness of different models in capturing the underlying dynamics.

Furthermore, the comprehensive statistical evaluation of model performance for predicting the temporal variation of scour depth reveals notable distinctions among different combinations and models. The statistical performance evaluation measures computed for the predictions were tabulated in Tables 5 and 6 for the training and testing periods, respectively. In Combo-1, XGBoost demonstrates exceptional predictive accuracy with R² = 0.997, the lowest RMSE of 0.012, and MAE of 0.008, showcasing its superior ability to capture the intricate dynamics of scour depth variation. This outperformance is evident when compared to other models, such as RF with R² = 0.985 and ANN with R² = 0.976. Moving to Combo-2, XGBoost maintains its stellar performance, outperforming other models with R² = 0.966 and RMSE of 0.039, reinforcing its robustness and reliability. As the number of predictors decreases in Combo-3, XGBoost continues to perform well with R² = 0.947 and the lowest RMSE of 0.049, showcasing its adaptability and effectiveness. The ability of XGBoost to consistently outperform others is emphasized when compared to RF with R² = 0.939 and ANN with R² = 0.907. In Combo-4, while XGBoost experiences a slight decrease in performance, it remains the top-performing model with R² = 0.945 and the lowest RMSE of 0.05, reinforcing its superiority in capturing the temporal variation of scour depth.

Even though RF consistently demonstrates solid performance across all combinations, with R² ranging from 0.959 to 0.944, indicating commendable accuracy. However, it falls slightly short of XGBoost in terms of the chosen performance evaluation measures. Similarly, ANN initially performs well in Combo-1 (R² = 0.976) but experiences a decline in subsequent combinations, with R² dropping to 0.883 in Combo-4. This sensitivity to changes in predictor combinations underscores the importance of model robustness. CNN–LSTM, on the other hand, consistently lags behind other models, demonstrating a decline in performance across predictor combinations and notably poor results in Combo-4 (R² = 0.44).

The evaluation of model performance during the testing period provides insights into their generalization capabilities. In Combo-1, during testing, XGBoost once again exhibits superior predictive accuracy with R² = 0.959, the lowest RMSE of 0.044, and MAE of 0.032. This reinforces XGBoost's robustness, as it outperforms other models, including RF, which shows R² = 0.926, and ANN with R² = 0.952. Moreover, XGBoost achieves the highest KGE at 0.979, indicating its excellence in capturing both the variability and pattern of the observed data during the testing period. However, it is worth noting that CNN–LSTM continues to struggle, with R² = 0.667, and KGE at 0.669, indicating limitations in capturing the temporal variation of scour depth during the testing period. The performance in the remaining combinations follows a similar pattern to their training period results. In summary, during both training and testing periods, XGBoost consistently outperforms other models across all combinations, showcasing its superior predictive capabilities, stability, and adaptability. Its ability to achieve the highest R² values, lowest RMSE, and competitive values for other metrics in both periods underscores its robustness in capturing the complex dynamics of scour depth variation. RF consistently performs well, demonstrating commendable accuracy in both training and testing. However, it falls slightly short of XGBoost in terms of R², RMSE, and KGE. ANN exhibits good performance during training but experiences a decline during testing, highlighting potential challenges in generalization. CNN–LSTM consistently lags behind other models, struggling to capture the intricate patterns of scour depth variation, particularly during testing.

The performance assessment metrics computed against the observed versus predicted non-dimensional maximum scour depth were tabulated in Tables 7 and 8 for training and testing periods, respectively. During the training period, XGBoost consistently outshone other models across various predictor combinations, achieving exceptional R² values, notably R² = 0.999 in Combo-1 and R² consistently above 0.98 in subsequent combinations. XGBoost demonstrated minimal RMSE and MAE, showcasing its robust fit to the observed non-dimensional maximum scour depth. RF also performed commendably, maintaining good accuracy and stability, although slightly below XGBoost in terms of precision. In contrast, both ANN and CNN–LSTM struggled to generalize effectively, with CNN–LSTM consistently exhibiting challenges in capturing the dynamics of scour depth.

The testing period further confirmed XGBoost's superiority, with consistent R² values above 0.99 and low RMSE across all combinations. RF also performed well during testing, closely following XGBoost. ANN showed reasonable accuracy but experienced a decline in precision during testing. CNN–LSTM continued to face challenges, with lower R² values and higher error metrics, indicating limitations in capturing non-dimensional maximum scour depth dynamics. The comprehensive analysis across training and testing periods underscores XGBoost as the preferred model for predicting non-dimensional maximum scour depth. Its consistent excellence, stability, and adaptability make it a robust choice for hydraulic engineering applications. While RF also demonstrated commendable performance, the marginal edge of XGBoost in accuracy and stability positions it as the most effective model for this specific prediction task. The challenges faced by ANN and CNN–LSTM in capturing the dynamics persist during both training and testing periods, emphasizing the critical importance of model selection for accurate predictions in hydraulic engineering scenarios.

In this study, the channel shape is rectangular, though in practice, channels can vary in shape or be compound. The experiments used uniform sand with a median size of 0.32 mm and a standard deviation of 1.31 mm, conducted under clear water conditions with steady, uniform flow. The water depth was consistently maintained at 12 cm for all tests. Scour due to contraction effects was neglected when the width of channel (b) was 20% or less of the total channel width (B). Additionally, all models, including RF and XGBoost, consistently overpredicted scour depth at lower values, especially when the observed non-dimensional scour depth was below 0.4, suggesting that the predictors struggled to capture the complexity of scour processes at low depths. CNN–LSTM faced further limitations due to the dataset size, as its complex architecture combining convolutional layers with LSTM units requires larger datasets to accurately capture temporal and spatial patterns, leading to higher errors with limited data.

Future research to enhance scour depth prediction accuracy could focus on expanding datasets to include diverse field data, combining physical-based models with machine learning for improved predictions. Incorporating unsteady flow conditions, refining feature selection, and adding more hydrodynamic variables like turbulence and sediment gradation could also improve model accuracy and generalization. These efforts would lead to more accurate and practical scour depth predictions for real-world applications.

In the present study, the temporal variation of scour depth and non-dimensional maximum scour depth by leveraging machine-learning models. The investigation began with a preliminary correlation analysis, revealing strong associations between scour depth and non-dimensional time ⁠, non-dimensional spacing ⁠, channel width ⁠, time-averaged velocity ⁠, and densimetric Froude number ⁠. Motivated by these correlations, four machine-learning models – RF, XGBoost, ANN, and CNN–LSTM – were established using an iterative approach involving four predictor combinations (Combo-1 to Combo-4). The following conclusions were drawn from the study:

• For temporal variation of scour depth, XGBoost achieved the best performance across all predictor combinations. In Combo-1, XGBoost achieved an R² of 0.997 with the lowest RMSE of 0.012 and MAE of 0.008 during training, and R² of 0.959, RMSE of 0.044, and KGE of 0.979 during testing. RF followed closely with R² of 0.985 in training and R² of 0.926 in testing, but XGBoost's ability to capture intricate dynamics made it superior.

• For non-dimensional maximum scour depth, XGBoost again outperformed the other models. In Combo-1, it achieved R² = 0.999 and RMSE = 0.005 during training, and maintained R² > 0.91 across all combinations during testing, demonstrating its robustness in capturing maximum scour depth dynamics. RF also performed well but slightly lagged behind with R² values consistently lower than XGBoost by a small margin.

• Model performance declined with fewer predictors, as evidenced by the decrease in R² and KGE from Combo-1 to Combo-4 across all models. The R² of XGBoost-based predictions dropped from 0.997 in Combo-1 to 0.945 in Combo-4 for temporal variation prediction, while RF experienced a similar drop from 0.985 to 0.944.

• ANN initially performed well but experienced a decline in subsequent combinations, highlighting the importance of model robustness. Conversely, CNN–LSTM consistently lagged behind, displaying a decline in performance across predictor combinations, particularly poor results in Combo-4 during prediction of both and ⁠.

• In the analysis of relative deviations, XGBoost demonstrated its superiority, particularly at when and ⁠, with a relative deviation below 30%, while other models exhibited overestimation biases at lower magnitudes.

In conclusion, XGBoost stands out as the preferred model for predicting scour depth dynamics, showcasing robustness, stability, and superior performance. The study offers valuable insights into the impact of predictor combinations on model performance and emphasizes the critical role of model selection in achieving accurate predictions of and ⁠.

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declare there is no conflict.