Accurately estimating the discharge coefficient (Cd) in spillways remains a complex challenge, critical to hydraulic engineering. Recent advancements suggest that hybrid Machine Learning (ML) models offer significant potential for improving Cd predictions. This study explores the application of four novel hybrid ML models to estimate Cd in Hydrofoil-Crested Stepped Spillways (HCSSs): Light Gradient Boosting Machine with Pelican Optimization Algorithm (LightGBM-POA), Neural Gradient Boosting with Osprey Optimization Algorithm (NGBoost-OOA), Tabular Neural Network with Moth Flame Optimization (TabNet-MFO), and Support Vector Regression with Improved Whale Optimization Algorithm (SVRIWOA). Outlier detection was performed using the Isolation Forest algorithm, and dimensional analysis identified the hydrofoil formation index (t) and the ratio of upstream flow depth to total spillway height (yup/P) as the most influential parameters for Cd estimation. The parameters were validated through ANOVA, while SHapley Additive exPlanations (SHAP) and Explainable Boosting Machine (EBM) quantified their contributions to Cd modeling, highlighting the dominant influence of t. Data normalization employed the StandardScaler method, with the dataset split into training (75%; 342 records) and testing (25%; 115 records) subsets. Model performance was assessed using metrics such as R², RMSE, SI, WMAPE, and sMAPE, and further evaluated using Taylor diagrams and a performance index (PI). During training stage, NGBoost-OOA achieved the highest accuracy, followed by LightGBM-POA, TabNet-MFO, and SVR-IWOA, with centered root mean square error (E’) values of 0.0057, 0.0064, 0.0067, and 0.0068, and PI scores of 165.5, 165.17, 123.25, and 123.25, respectively. In testing stage, TabNet-MFO and SVR-IWOA outperformed the other models, achieving equal E′ values of 0.0060 and PI scores of 165.34, ranking first. NGBoost-OOA and LightGBM-POA ranked third and fourth, respectively. These findings demonstrate the potential of hybrid ML models in accurately predicting Cd for complex hydraulic structures like HCSSs, offering valuable insights for future engineering applications.

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