Derivation of Optimized Equations for Estimation of Dispersion Coefficient in Natural Streams Using Hybridized ANN With PSO and CSO Algorithms

In this paper, a new hybrid model is developed to improve the accuracy in the prediction of the longitudinal dispersion coefficient (
$K_{x}$
) and the derivation of novel optimized explicit equations for natural streams. For this purpose, an artificial neural network (ANN) is hybridized with particle swarm optimization (PSO) and cat swarm optimization (CSO) algorithms. The CSO and PSO are used to find the optimum values of biases and weights in ANN structure and formulate the results as novel explicit predictive equations than the classical black-box methods. The hydraulic parameters of the natural stream and some geometric parameters were utilized for the model developments. Eight different input combinations are used as the input vectors to ANN, ANN-PSO, and ANN-CSO models, whereas the dispersion coefficient (Kx) is the target model output. The developed models are trained and tested by a comprehensive reference data sets measured on streams in the United States, that were used previously by Tayfur and Singh (2005) in ANN models. The main aims, novelty, and contributions of the present study are 1) improving the accuracy of classical ANN-based Kx predictions by hybridizing with CSO and PSO. 2) Performing sensitive analysis of ANN, ANN-CSO, and ANN-PSO based on input combinations 3) derivation of novel explicit optimized ANN-CSO, ANN-PSO, equations for predicting Kx rather than the classical ANN black-box methods. The results depicted that the highest accuracy and superiority were attained by the ANN-PSO model, with input variables of B, H, U,
$\text{U}_{\ast,}$
followed by ANN-CSO and ANN. By using the optimized trained black box ANN models, two novel explicit predictive equations are derived, and their results are compared with the empirical equations. Comparative assessments confirmed significant improvements in the hybrid equations’ results than the classical ANN and previously published equations. The developed novel equations can be used to estimate the Kx in one-dimensional pollutant transfer models that are essential for the pollution studies in environmental river engineering practices.