Discharge coefficient estimation of modified semi-cylindrical weirs using machine learning approaches

Based on the principles design of hydrofoil weirs, Modified Semi-Cylindrical Weirs (MSCWs) incorporate an innovative tangential ramp along the downstream crest contour, thereby significantly enhancing their performance compared to conventional semi-cylindrical weirs. A pivotal parameter in the calculation of flow discharge over the weir is the discharge coefficient (C_d). This study involves a comprehensive comparative analysis of various C_d estimation methodologies for MSCWs, employing a range of machine learning-based models, notably including Artificial Neural Network (ANN), Multivariate Adaptive Regression Splines (MARS), M5 tree, Locally Weighted Polynomial Regression (LWPR), and Support Vector Machine (SVM) models. To begin, a feature selection analysis utilizing the Gamma Test (GT) method was conducted to identify the optimal input configuration for modeling the discharge of MSCWs. The results of the feature selection revealed that the C_d of the MSCWs is primarily influenced by the ratio of upstream flow depth (y_up) to crest radius (R), while showing negligible sensitivity to the slope of the downstream ramp (θ). The dataset was partitioned into two segments: 70% were assigned to the training stage, while the remaining 30% were allocated to the testing stage. The precision of C_d predictions is evaluated through four key statistical metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Square Error (RMSE), Correlation Coefficient (R²), and Nash –Sutcliff Efficiency (NSE). The outcomes reveal that, for the training and testing phases, the R² values for the ANN, MARS, M5 tree, LWPR and SVM models are respectively 0.967, 0.931, 0.974, 0.937, and 0.933, and 0.925, 0.953, 0.953, 0.980, and 0.954. Notably, the LWPR model outperforms the ANN, MARS, M5 tree, and SVM models, boasting MAE, MSE, RMSE, and NSE values of 0.0167, 0.0005, 0.0217, and 0.942 during training, and 0.0107, 0.0002, 0.0136, and 0.949 during testing. As a result, the LWPR model clearly emerges as the superior model, followed by the M5 model tree.