## 1 Introduction
(2013) scaled the drag force applied in a filtered two-fluid model (TFM) by using the slip velocity predicted by an artificial neural network (ANN). Only the datasets obtained from the simulation of the bubbling fluidized beds containing bubble-dominated meso-scale structures were employed in training the network; therefore, the resulting model may not be suitable for flows in other regimes. [PERSON] and [PERSON] [PERSON] and [PERSON] (2014) used ANN to build the drag force model for polydisperse spherical granules; besides a solid fraction and Reynolds number, the relative neighboring granule locations are also considered as new inputs in the ANN to improve the predicted capability of the model. Yet, the model's applicable range is only in a low-intermediate Reynolds number and low solid fraction in a polydisperse granular system. Further, [PERSON] et al. (2015) compared the performance of two different machine learning models, ANN and eXtreme gradient boosting (Xgboost), in predicting the drag force between interphases in a filtered sub-grid model (SGM). They found that the Xgboost model is more suitable for fewer training samples, while the ANN is more suitable for adequate training samples. In 2020, [PERSON] et al. (2016) applied the convolutional neural network (CNN) to predict the drag force in a filtered TFM. They found the information (e.g., Reynolds stress and solid phase stress) from the neighboring coarse grids is of great significance in improving the accuracy. [PERSON] et al. (2017) modeled the drag force in deep bubbling beds by using ANN with the data obtained from the immerse boundary method. Besides the Reynolds number and solid volume fraction, two new inputs, the velocity fluctuation and position fluctuation of granules, were introduced to the ANN to improve the accuracy of the model. Compared to [PERSON]'s drag model [PERSON] et al. (2016), their model can give better predictions for monodisperse granular systems. In 2022, [PERSON] et al. (2019) used CNN to study three types of ellipsoidal granules; the training data are obtained from particle resolved simulations (PRSs), but the solid fraction of the drag model is 0.1-0.3, and the applicable range of the model is limited in bubbling fluidized beds. In 2023, [PERSON] et al. (2023) used deep learning techniques to study the drag force of polydisperse irregular-shaped granules in particle-laden flow; the applicable range of the model was expanded to an intermediate Reynolds number incompressible flow compared to their previous work [41]. Whereas their drag model still mainly focused on dilute polydisperse flows, the applicability in a dense monodisperse granular flow has not been verified yet. Beyond that, there are still other works associated with the drag force based on ANN [42; 43; 44]. The above studies took full advantage of ANN in self-learning and high compatibility during modeling of the drag force. Compared with the classical mathematical analysis approaches, the ANN can learn complex nonlinear relationships from the datasets spontaneously. It has the advantages of flexible structure, simple algorithm, self-learning ability, and good compatibility [45]. The strong predictive capabilities of ANN in modeling the drag force have been well demonstrated in the present studies. However, the models given by these studies still have three imperfections. First, these polydisperse models are applicable in the relatively dilute low-intermediate Reynolds number flow systems; they can scarcely be adopted in a relatively dense granular system. Second, these polydisperse drag models are hardly compatible with a monodisperse granular system since the training samples were mainly from the direct simulations of polydisperse granular systems. Third, the polydisperse drag model suffers from low computational efficiency and over-fitting problems because of the huge training sample set; these studies rarely focus on the training strategy to improve the computational efficiency and avoid over-fitting problems.