Hybrid Machine Learning Models for Discharge Coefficient Prediction in Hydrofoil-Crested Stepped Spillways

Abstract

Accurately estimating the discharge coefficient (C_d) in spillways remains a complex challenge, critical to hydraulic engineering. Recent advancements suggest that hybrid Machine Learning (ML) models offer significant potential for improving C_d predictions. This study explores the application of four novel hybrid ML models to estimate C_d 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 (SVR-IWOA). 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 (y_up/P) as the most influential parameters for C_d estimation. The parameters were validated through ANOVA, while SHapley Additive exPlanations (SHAP) and Explainable Boosting Machine (EBM) quantified their contributions to C_d 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 C_d for complex hydraulic structures like HCSSs, offering valuable insights for future engineering applications.