Prognostication of discharge coefficient of labyrinth weir using outlier robust extreme learning machine

In this paper, for the first time, a robust artificial intelligence (AI) technique called "outlier robust extreme learning machine (ORELM)" is used to estimate the discharge coefficient of labyrinth weirs. At the beginning, the number of the hidden layer neurons initials from 5 and continues to 45 and the most optimal number of the hidden layer neurons are taken into account equal to 5. Then, the numerical model is verified by the k-fold cross-validation approach in which k is considered equal to 5. After that, different activation functions are evaluated to detect the most accurate one for the numerical model. Subsequently, six different ORELM models are developed using the parameters affecting the discharge coefficient of labyrinth weirs. Also, the superior model and the most effective input parameters are identified through a sensitivity analysis. For example, the values of R2, RMSRE and NSC for the superior model are calculated 0.943, 5.224 and 0.940, respectively. Furthermore, the ratio of the head above the weir to the weir height (H-T/P) and the ratio of the width of a single cycle to the weir height (w/P) are introduced as the most important input parameters. Also, the results of the ORELM superior model are compared with the artificial intelligence models including the extreme learning machine, artificial neural network and the support vector machine and it is concluded that ORELM has a better performance. Then, an uncertainty analysis is conducted for the ORELM, ELM, ANN and SVM models and it is proved that the ORELM has an overestimated performance. Eventually, a practical formula is presented to engineers without knowledge of artificial intelligence for approximating the discharge coefficient of labyrinth weirs. Additionally, a partial derivative sensitivity analysis is carried out for this equation.