GWO-CatBoost based approach for prediction of live-bed scour around circular bridge pier

Prediction of scour depth under live bed conditions is becoming a major challenge in the field of hydraulic engineering due to the complex interaction between turbulent flow and sediment transport. Empirical equations that are developed for specific site conditions or particular range of data will struggle to achieve high accuracy when applied to the different site or broad range of data. Especially in the case of live bed scour, empirical models will suffer due to the inter-parameter correlations of physical variables, necessitating advanced hybrid models that follow hydraulic laws. To address these limitations, the present study proposes a systematic comparison of ten computational models, including eight hybrid architectures and two standalone models with four established empirical formulas. Rigorous dimensional analysis has been performed using Buckingham \(π \) theorem to reduce the variables into a functional relationship \(( {{d}_{{{se}}} /{b}} ) = {f}( {{u}/{u}_{{c}} ,\;{y}/{b},\;{b}/{d}₅₀ ,\;σ ,\;{Fr}_{{b}} } )\) . To check the influence of each input parameter, the present study evaluated the correlation and feature importance analysis along with feature selection techniques across 448 rows of dataset. The unique contribution of the present study is to develop a data-driven empirical equation using Gene Expression Programming (GEP) to capture real hydraulic trends. This equation serves as a physics constraint in the PINN framework. The results demonstrate that the GWO-CatBoost model achieved superior predictive accuracy (R 2 = 0.869, RMSE = 0.139, Rel % = 95.556%), compared to all other models and also serves as a reliable tool for engineering applications. The findings suggest that GEP-PINN model successfully bridges the gap between hydraulic laws and machine learning models. Together, these methodologies provide a reliable, high-fidelity tool for enhancing bridge pier design and flood-risk assessment.