Weighted regularized extreme learning machine to model the discharge coefficient of side slots

Highlights

• Coefficient of Discharge.

• Regularization Parameter (PR).

• Weighted Regularized Extreme Learning Machine (WRELM).

• Extreme Learning Machine (ELM).

• Comparison of WRELM with ELM.

Abstract

In the current research, a modern learning machine algorithm named “Weighted Regularized Extreme Learning Machine (WRELM)" is implemented for the first time for the simulation of the coefficient of discharge of side slots. For this purpose, an effective variable on the coefficient of discharge of side slots is firstly introduced, then five distinctive WRELM models are produced by it for the estimation of the coefficient. In the next stage, a database is created for verification of WRELM results. it should be mentioned that 70% of the data are utilized for training the WRELM models, while the rest (i.e. 30%) for testing them. After that, the optimal number of hidden layer neurons as well as the best activation function of the WRELM algorithm are chosen. In addition, the best regularization parameter and also the weight function of the WRELM are achieved. By conducting a sensitivity analysis, the most effective variable for the simulation of the coefficient of discharge along with the WRELM superior model is introduced. The WRELM superior model estimates values of the coefficient of discharge with the maximum exactness and the highest correlation. For instance, the estimations of the correlation coefficient and scatter index for this model are computed to be 0.930 and 0.051, respectively. The sensitivity analysis shows that the ratio of the side slot crest height to its length and the Froude number should be considered as the most important input variables. A comparison between the WRELM with the ELM displays that the former works much better. Furthermore, an uncertainty analysis is executed for both models. Eventually, an equation is suggested for the estimation of the coefficient of discharge and a partial derivative sensitivity analysis is performed on it.

Introduction

A side slot is installed on the sidewall of main channels to control and measure the flow. The flow within the inlet of a side slot is a complex, complete three-dimensional flow, while the governing flow in the main channel is spatial variable with flow reduction. It should be noted that the coefficient of discharge of the side slot is considered as the most important variable for the design of this type of hydraulic structures.

Owing to the importance of side slots in irrigation networks, drainage systems and water and wastewater treatment plants, many experimental, analytical and numerical studies have been focused on the pattern of the flow within them. Ojha and Subbaiah [1] managed to investigate the flow field and the coefficient of discharge of side slots via an experimental program. By presenting an analytical approach, they successfully measured side slot flow rates and compared the findings with laboratory data to show that the results obtained by this analytical method are virtually equal to 10%. Furthermore, Hussain et al. [2] studied discharge values of rectangular side orifices via an experimental research. The authors examined the flow pattern around side orifices and put forward an equation for the estimation of the coefficient of discharge of such installations.

In recent years, machine learning based approaches have been extensively utilized for the estimation and simulation of the real world problems [5] and especially coefficient of discharge of flow divert facilities [3,4] such as group method of data handling (GMDH) [6], gene expression programming (GEP) [7]. A hybrid neuro-fuzzy model was developed by Khoshbin et al. [8] for the prediction of the coefficient of discharge of rectangular side weirs. By combining the genetic algorithm, adaptive neuro-fuzzy inference system (ANFIS) and singular value decomposition (SVD), they managed to produce the ANFIS-SVD-GA model. They compared the results of the mentioned hybrid model with the neural network and declared that the hybrid model is considerably more precise.

Azimi et al. [9] employed the linear regression approach to provide a relationship in terms of hydraulic and geometric variables of the laboratory model. They validated the results achieved by the linear regression model by the computational fluid dynamics (CFD) model. Akhbari et al. [10] estimated the coefficient of discharge of labyrinth triangular weirs using two artificial intelligence models, Neural Networks (RBNN) and M5’. By executing an analysis of the simulation outcomes, they indicated that the Froude number and ratio of the head above the weir to the rectangular channel width are identified as the most crucial input variables. Azimi et al. [11] simulated the coefficient of discharge of rectangular side orifices through the development of a hybrid neuro-fuzzy model. Using ANFIS and the genetic algorithm, they put forward this hybrid model and compared its outcomes with those obtained from a CFD model and showed that the hybrid model (i.e. ANFIS-GA) had a better performance. Furthermore, Azimi et al. [12] forecasted the discharge capacity of side weirs on trapezoidal canals using the support vector machine (SVM). They introduced six different SVM models for the simulation of the coefficient of discharge and managed to provide the superior SVM model via a sensitivity analysis. These researchers proved that the ratio of the weir length to the channel width is the most important input variable.

One of the most popular machine learning techniques is the feed-forward neural network (FFNN) whose efficiency in different routines especially water and hydraulic engineering has been confirmed by researchers [13]. The learning algorithm used in this method is generally back-propagation (BP). The most important drawbacks of this method are the high modeling time to determine the FFNN parameters during a process based on iteration [14], entrapment in local optima and low generalization [15]. Huang et al. [16] has recently presented a new algorithm entitled “extreme learning machine (ELM)" for training the single layer FFNN (SLFFNN) which overcomes to many issues of classical FFNN trained with BP. This approach benefits from advantages such as high modeling speed, minimal user intervention, so that only the type of activation function and the number of hidden neurons need to be determined, and remarkable generalizability. Although the ELM solves BP problems and performs modeling in a very short time, this method itself has disadvantages, the most important of which are: (1) the random determination of two matrices of input weights and bias of hidden neurons, which may affect the final results, and (2) its poor performance in the presence of outliers.

The main of the current study is developing weighted regularized ELM (WRELM) as a new schem of classical ELM to overcome the main limitation of this approach. Indeed, using developed WRELM, two major drawbacks of this method are addressed in order to not only show good performance in the presence of outliers, but also use many iterations to solve the problem caused by the random determination of the two mentioned matrices. It should be noted that the WRELM learning machine has not yet been used to simulate the coefficient of discharge of side slots, and the present study is the first application of this algorithm in this field. To do this, the used algorithm is introduced first. The used laboratory models are then described. In the next section, the results of the simulations are presented. Finally, the most important results of the present study are summarized.