Improvement of extreme learning machine using self-adaptive evolutionary algorithm for estimating discharge capacity of sharp-crested weirs located on the end of circular channels

Weirs are used in different shapes such as rectangular, triangular and circular to control and measure the flow in open channels. To design a weir properly, determining its discharge coefficient is very important. In this study, the discharge coefficient of sharp-crested weirs located on circular channels is modeled by the self-adaptive optimization of extreme learning algorithm using the differential evolutionary algorithm (SaDE-ELM). Also, Monte Carlo simulations (MCs) are used to study the compatibilities of SaDE-ELM models. However, the k-fold cross validation method (k=5) is used to investigate the abilities of the used numerical models. According to the input parameters, four models of SaDE-ELM are introduced. Study of the numerical results shows that the superior model simulates the discharge coefficient as a function of the Froude number (Fr) and the ratio of the circular channel diameter to the weir crest height (D/P) and a relationship is provided for the superior model. The values of mean absolute relative error (MARE), root mean square error (RMSE) and correlation coefficient (R2) for the superior model are calculated 0.184, 0.002 and 0.997, respectively. However, the maximum error value for this model is less than 3%. Also, the results of the uncertainty analysis show that the superior model has an underestimated performance which its 95% prediction error interval is simulated between 0.000314 and −0.000314. In other words, the width of uncertainty band for the superior model is calculated equal to −0.000314.