Simulation of labyrinth weir discharge coefficient by modern artificial intelligence models

Generally, labyrinth weirs pass more water compared to their equivalent rectangular weirs. Thus, these types of weirs are popular amongst hydraulic and environmental engineers. In this paper, for the first time, a novel artificial intelligence (AI) technique called "outlier robust extreme learning machine (ORELM)" is used to estimate the discharge coefficient of labyrinth weirs. The ORELM method has been proposed in order to overcome the difficulties of the classical ELM in predicting datasets with outliers. In this method, the concept of “sparsity characteristic of outliers” is used. Also, in this study, to verify the results of the numerical models the experimental measurements conducted by Kumar et al. (2011) and Seamons (2014) are employed. The experimental model established by Kumar et al. (2011) is composed of a rectangular channel with a length of 12m, a width of 0.28m and a depth of 0.41m. The weir is made of steel sheets and placed at an 11m distance from rectangular channel inlet. Also, Seamons (2014) experimental model has been set up in a rectangular channel with the length, width and height of 14.6m, 1.2m and 0.9m, respectively. First, the number of the hidden layer neurons initials from 5 and continues to 45 and the most optimal number the hidden layer neurons are taken into account equal to 5. In this study, the Monte Carlo simulations are used for examining the abilities of the numerical models. The main idea of this method is based on solving problems which might be actual in nature using random decision-making. The Monte-Carlo methods are usually implemented for simulating physical and mathematical systems which are not solvable by means of other methods. In this paper, the K-fold cross validation method is employed for validating the results of the numerical models. To this end, the observational data are divided into five equal sets and each time one set of these data is used for testing the numerical model and the rest for training it. This procedure is repeated five times and each test is used exactly once to train and once to test. This method increases the flexibility of the numerical model when dealing with the observational data, and it can be said that the numerical model has the ability to model a greater range of laboratory data. For instance, the maxim value of R2 is obtained for the K=4 case (R2=0.954), while for the K=5 case the values of RMSE and MARE are estimated 0.034 and 4.408, respectively. After that, different activation functions are evaluated in order 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 (HT/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 ORELM has an overestimated performance.