Prediction of coefficient of discharge (CPK) of A-type piano key weir using ANN and hybrid ANFIS models

Piano key weirs are a special type of labyrinth weir and have a higher discharge capacity than linear weirs. Accurate prediction of the coefficient of discharge (CPK) of Piano Key (PK) weirs is essential for their effective design, planning, and optimization of their hydraulic performance. Various regression models available in the literature, based on limited experimental data sets, often may not be able to capture the complex nonlinear relationships governing CPK. To address these limitations, the present study utilizes soft computing techniques, specifically Artificial Neural Networks (ANN), the Adaptive Neuro-Fuzzy Inference System (ANFIS), and two hybrid ANFIS modeling techniques to develop accurate and effective predictive models. This study utilizes extensive data sets available in the literature to model CPK under free-flow conditions. Relative crest length (L/W), relative head (H/P), inlet and outlet key width ratio (Wi/Wo), relative weir height (P/Wu), and relative weir width (B/P) are used as predictor variables. A non-linear regression-based model was developed for comparison purposes. ANN models such as Feed-forward back propagation (FFBP), radial basis function (RBF), and generalized regression neural network (GRNN), along with a traditional ANFIS model, were developed. Additionally, two hybrid ANFIS models were designed and optimized using biogeography-based optimization (ANFIS-BBO) and the cultural algorithm (ANFIS-CA). Standard statistical metrics, such as the coefficient of determination (R2), mean percentage error (MPE), mean absolute deviation (MAD), and root mean square error (RMSE), mean absolute relative error (MARE) and percent bias (PBIAS) were used to evaluate the predictive performance of the models. The results indicated that the ANN and ANFIS models outperformed the traditional regression model in predicting CPK, whereas Hybrid ANFIS models producing the highest predicting accuracy. Among ANN models, the FFBP model produced superior predictive capability compared to RBF and GRNN models. A comparison of the predictive performance of the present study's models with previously existing regression models was also conducted. The results showed that all present study models predicted CPK more accurately than existing regression models. The results of sensitivity analysis showed that relative head (H/P) is the most sensitive parameter, followed by inlet and outlet key width ratio (Wi/Wo) and relative crest length (L/W). Relative weir height (P/Wu) and relative side wall crest length (B/P) moderately affect CPK. The present study shows that the hybrid ANFIS models were more efficient in CPK prediction, which the hydraulic engineering community can utilize to improve PK weirs' design and efficiency.