Robust discharge coefficient modeling in triangular weirs via an enhanced Adam-optimized BP network

Abstract

Accurate estimation of the discharge coefficient ( C d ) in triangular weirs is fundamental for data-driven hydrological modeling and the reliable assessment of water resource infrastructure. To achieve robust, high-precision prediction of C d under varying hydraulic conditions, an enhanced Adam-optimized back-propagation neural network (AdaBP-E) was developed. The proposed framework integrates adaptive learning rate adjustment, L2 regularization, Dropout, and an independent early-stopping mechanism to effectively capture the complex nonlinear relationships between weir geometry and flow characteristics. The model was benchmarked against a linear baseline and conventional BP training variants, including gradient descent and Levenberg–Marquardt optimization, to isolate the specific contributions of the proposed enhancements. The AdaBP-E model demonstrated superior predictive performance, achieving a coefficient of determination (R 2 ) of 0.9697 and a root mean square error (RMSE) of 0.0119 on the test set, while also exhibiting the lowest average sensitivity index (19.51), indicative of enhanced robustness to input perturbations. Stratified Bootstrap validation further confirms its superior stability, with a 77.4% reduction in R 2 standard deviation compared to BP-LM. The AdaBP-E framework offers a robust, interpretable data-driven solution for discharge coefficient modeling, thereby supporting broader integration of machine learning techniques into smart water management, hydrological risk mitigation, and sustainable infrastructure planning.