## 1 Introduction
Therefore, effective methods are required to improve the ELM performance for daily R-R modeling [28; 29]. In recent years, evolutionary algorithms have been widely used to train neural networks because of their better global searching ability that enhances the model performance [30; 31; 32; 33]. For example, particle swarm optimization (PSO), biogeography-based optimization (BBO) and grey wolf optimization (GWO) were used to enhance the performance of ELM models in several engineering applications [34; 35; 36; 37; 38]. Recently, [PERSON] et al. [39] proposed a novel optimization algorithm called equilibrium optimizer (EO), inspired by accounting mass balance mathematical equations within a control volume. The mass balance equations are based on the conservation of mass law in physics. The mass balance equations estimate the dynamic and equilibrium state of object of interest within a defined system boundary. EO has several dominance propertiesfor qualifying a good optimization algorithm. It is easy to implement, proposes good balance between exploration and exploitation tasks and population diversity in individuals. Therefore, it may be used to solve various optimization problems. This study integrated EO with ELM and validated it by R-R modelling. On the other hand, deep neural networks (DNNs) are successfully used for solving complex problems using high-dimensional data [40]. DNN approach has been successfully applied in fields like natural language processing [41], speech recognition [42], bioinformatics [43] and image recognition [44]. Deep learning has shown promise for modeling time-series data through techniques like Conditional RBM (restricted Boltzmann machine), RBM, recurrent neural network (RNN), and auto encoder [45]. In this study, a multiple layer of neural network, called DNN model, is developed for R-R modelling. This paper proposes EO-based optimization of ELM learning parameters (EO-ELM) to avoid a trap into local optima and enhance the generalization capability of the network and a DNN model for R-R modelling. The two different benchmark stations of the catchments in the United Kingdom (UK) are used to validate the performance efficacy of the proposed models. The catchments were the river Teifi at Glanteifi and the river Fal at Tregony. The river gauges of the two catchments were from the UK benchmark network stations and were suitable for high, low, and medium flow analysis [46]. The predictors are daily rainfall and runoff time series datasets. The preceding days of rainfall and runoff were selected as the input features for the ML models. A partial autocorrelation function (PACF) was used for optimal number of correlated preceding days. Furthermore, the performance of the proposed models was compared with the performance of other well-known models, called ELM, kernel (radial basis function (RBF)) ELM (KELM), PSO optimized ELM (PSO-ELM), support vector regression (SVR), artificial neural network (ANN) and gradient boosting machine (GBM)  
Furthermore, to enhance prediction accuracy of the proposed models, discrete wavelet transform (DWT)-based pre-processed rainfall and runoff time series data have been used in this study. Recently, DWT-based time series forecasting has been used in several engineering fields to enhance the prediction performance of several machine learning models [47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. All of the previous studies have proved that wavelet-based hybrid models performed with better efficacy compared to single models. Therefore, the paper proposes hybrid wavelet-based R-R models, namely hybrid DWT-based EO-ELM (WEO-ELM) and DWT-based DNN. The performance of WEO-ELM and WDNN is compared with DWT-based ELM (WELM), DWT-based KELM (WKELM), DWT-based hybrid PSO-ELM (WPSO-ELM), DWT-based support vector regression (WSVR), DWT-based artificial neural network (WANN) and DWT-based gradient boosting machine (WGBM) using two-time series datasets.