Scour depth estimation using standalone metaheuristic algorithms and their combinations with CatBoost

Scouring around bridge piers poses a critical threat to structural integrity, leading to costly damage and safety risks. Traditional equations often fail to accurately predict equilibrium scour depth (S_eq) due to the complexity and nonlinearity of the underlying hydraulic processes. This study proposes two approaches for estimating S_eq: (1) optimizing a Categorical Boosting (CatBoost) machine learning model using five metaheuristic algorithms-Harris Hawk Optimization (HHO), Moth-Flame Optimization (MFO), Whale Optimization Algorithm (WOA), Pelican Optimization Algorithm (POA), and Fox Optimization Algorithm (FOX); and (2) using the abovementioned optimization methods (i.e., HHO, MFO, WOA, POA, and FOX) to derive explicit equations. Among the hybrid models, HHO-CatBoost achieved the highest accuracy, with a root-mean-square error (RMSE) of 0.0286 m, mean absolute error (MAE) of 0.0178 m, and coefficient of determination (R²) of 0.9670 during testing. Among the explicit formulations, the HHO-based model excluding the Reynolds number outperformed 18 existing equations, achieving an RMSE of 0.066 m, an MAE of 0.043 m, and an R² of 0.828. SHapley Additive exPlanations (SHAP) analysis identified pier diameter as the most influential factor and critical velocity as the least, while sensitivity analysis highlighted the ratio of pier diameter to flow depth as most important and Reynolds number as the least significant under turbulent flow conditions.