## 2 Materials and Methods
### Methodology  
#### 2.2.1 Physics-Informed Data-Driven Models  
Three ML models and three coupled models including the ELM, SVR, LSTM, HEC-HMS-ELM, HEC-HMS-SVR, and HEC-HMS-LSTM models were used to investigate the ability of ML and coupled methods to simulate and forecast river runoff, respectively. [PERSON] [26] suggested that the ELM technique is more efficient than an ANN model. He found that the ELM technique has some advantages owing to its internal capabilities. There was no requirement for weight adjustment through a feasible generalization ability or a high learning speed. Various studies have revealed that the ELM is superior to an ANN [10; 27; 28]. [PERSON], et al. [29] found that the ELM model performed 26 times better  
Figure 1: Study area with gauge stations in the Voshmgir basin, Iran. than the ANN model during the prediction of solar radiation in Turkey. Similarly, [PERSON] and [PERSON] [30] applied ELM and ANN models to generate a monthly efficient drought index, and found that the prediction accuracy of the ELM model was significantly higher than that of the ANN model. In agreement with previous studies, [PERSON], et al. [31] found that the ELM's prediction ability was superior to that of an ANN during streamflow forecasting. The ELM model is generally an extended version of a single-hidden layer feed-forward neural network (SLFN) [32]. The structure of the model is the same as that of the standard ANN model with three layers: a single input layer, a hidden layer, and an output layer. The hidden biases and weights are chosen randomly by the Moore-Penrose generalized inverse of the hidden output matrix method instead of the fine-tuning method applied in the ANN. Because of this modification, the ELM model is capable of progressing faster than the ANN model while maintaining a sufficient generalization ability. More information on this model can be found in [PERSON], et al. [33]. In the current study, the ELM model was constructed through Python software using the \"HIP-ELM\" package [34]. The support vector regression (SVR) model is an ML model used for classification and regression [35]. It uses structural risk reduction (SRR) rather than empirical risk reduction through self-learning models such as ANNs, as mentioned by [PERSON], et al. [36]. SVRs, which are data-driven, have recently emerged as an alternative to ANNs for hydrological studies [37]. [PERSON] [38] introduced the use of SVR models from support vector machines (SVMs), which are used to solve classification problems. The SVR model has a unique architecture whose main parameters include a kernel, a hyperplane, and boundary lines. The kernel function is the key to determining the right hyperplane, which is used as a baseline to calculate the data distribution trend. SVRs have two boundary lines used to bind the positive and negative distributions. The abovementioned optimal structure and working frameworks lead to greater efficiency by avoiding overfitting. Such characteristics make SVR a popular data-driven model in the rainfall-runoff modeling field [39]. The SVR method reduces the empirical risk of reaching an adequate generalization performance by reducing the generalization error instead of the training error [40]. As mentioned earlier, the SVR model maps data into a higher-dimensional feature space to perform efficient predictions. Kernel functions are essentially tunable hyperparameters for optimal outputs. Several kernels, such as polynomial, linear, sigmoid, and radial basis functions (RBF) have been applied in various studies. However, the RBF was found to be  
Figure 2: Measured discharge and mean TRMM satellite rainfall data from 2002 to 2007.  
the best among them for predicting nonlinear hydrological problems [41]. In addition, the RBF kernel outperformed the other kernels in the rainfall-runoff simulation [42]. Decreasing the calculation time with improved generalization capacity is a significant benefit of the RBF kernel [43]. Therefore, the RBF kernel was used in this study.