whale optimization algorithm

The simulation of river discharge at hydrometric stations to predict future flow discharge over specific time periods is an important issue typically addressed using hydrological time series associated with the respective station. To predict river discharge with the hights accuracy, three major groups of methods are commonly utilized: empirical and statistical methods, conceptual methods, and process-based approaches. Among data-driven methods, those based on artificial intelligence-based are prominent. The aim of this study was evaluating the performance of machine learning models, including SVM, ANFIS, and ANN, and assessing the performance of a neural network model trained with Whale Optimization Algorithm (WOA) and Particle Swarm Optimization (PSO) to predict flow discharge. In addition to evaluating the intelligent models, the impact of using optimization algorithms on the accuracy of river discharge predictions was also examined. Since input data have a significant impact on the performance of data-driven models, the criteria influencing the river discharge were identified, and the best combination of input variables for each model was determined.

In this study, to predict the daily discharge at the Poldokhtar hydrometric station located on the Kashkan River, discharge and precipitation data from 1971 to 2018 were collected, and Intelligent models, including Adaptive Neuro-Fuzzy Inference System (ANFIS), Support Vector Machine (SVM), Artificial Neural Network (ANN), and the hybrid model of Artificial Neural Network with Particle Swarm Optimization (ANN-PSO) and the hybrid model of Artificial Neural Network with Whale Optimization Algorithm (ANN-WOA) were employed. In the two hybrid models, efforts were made to adjust the criterias of the artificial neural network using metaheuristic algorithms, and their impact on the performance of the ANN model was examined. Additionally, this study investigated the impact of river discharge and precipitation data, along with their time lags (data from previous days), and combinations of these metrics as input variables for the models. To determine the best combination of input variables, statistical methods such as the Autocorrelation Function (ACF), Partial Autocorrelation Function (PACF), and Pearson Correlation Coefficient (PCC) were employed. After applying the effective inputs and training the mentioned intelligent models, their performance in predicting river discharge was evaluated by comparing RMSE, R², and NE metrics.

The evaluation of all models in this study showed that the river discharge of one day (Q-1), two days (Q-2), and three days (Q-3) ago, along with the precipitation of one day ago (P-1), exhibited the highest correlation with the river’s daily discharge. Overall, all models demonstrated acceptable accuracy in modeling the river discharge in the Kashkan watershed. According to the results, the highest accuracy in predicting daily discharge was achieved by the ANN-WOA model, with the highest coefficient of determination (R² = 0.896), Nash-Sutcliffe efficiency (NE = 0.803), and the lowest error (RMSE = 0.0186). Subsequently, the SVM model, using a radial basis kernel function with parameters C=4, γ=1, and ϵ=0.001 demonstrated superior performance, with a coefficient of determination (R² = 0.895), Nash-Sutcliffe efficiency (NE = 0.801), and an error (RMSE = 0.0187). Then, the ANN-PSO and ANN models ranked third and fourth, respectively. The results indicated that using metaheuristic optimization algorithms significantly improved the accuracy of the ANN model, making it a suitable tool for neural network training. The evaluation of different ANFIS structures revealed that triangular and Gaussian functions performed better for modeling river discharge in the study area. On the other hand, the error of this model, with values of RMSE=0.023 and NE=0.76 was higher compared to the other models. 

This study demonstrated that machine learning models, such as SVM, ANFIS, and ANN, exhibited acceptable accuracy in predicting river discharge. Adjusting neural network parameters using optimization algorithms like WOA and PSO significantly enhanced the performance of the ANN model. Finally, it can be concluded that these models can serve as suitable alternatives to conceptual and hydrological models for addressing hydrological and discharge-related issues. It is recommended to train the SVM and ANFIS models using the PSO and WOA algorithms and then compare the results with the findings of this study.