## 1 Introduction  
Most granular flows encountered in daily life and industrial processing have polydisperse characteristics [1; 2; 3; 4; 5; 6; 7], for instance, drug production, coal gasification, haze control, and sandstorm prevention [8; 9]. The polydisperse gas-solid system, compared with the monodisperse counterpart, usually contains particles with different sizes and demonstrates heterogeneous characteristics [10; 11; 12]. Generally, the drag force of polydisperse systems, which is very important to the discrete element method (DEM) simulations [13; 14; 15; 16; 17], is substantially different from that of the monodisperse systems [18; 19; 20]. The polydispersity has been recognized as one of the most important factors that greatly affects the fluidization state of the granular system [21]. Accordingly, the drag force model for the monodisperse system is no longer applicable for polydisperse systems [22; 23]. In recent years, there have been many important studies on the drag force models of polydisperse granular systems. In 2005, [PERSON] et al. [24] extended the monodisperse drag force model of [PERSON] et al. [25] to polydisperse granular systems by introducing a corrected volume fraction, a diameter fraction, and the Sauter mean diameter. In 2006, [PERSON] et al. [26] extended the applicability for the [PERSON] et al. formula [25] in a wider range of the Reynolds number. In 2007, [PERSON] et al. [27] indicated that the linear dependence on the Reynolds number in the Ergun drag force model [28] is insufficient, and revised the model subsequently by using the direct simulation data that were obtained by the lattice Boltzmann method (LBM). Subsequently, the drag model of polydisperse granular flow with Gaussian and log-normal distribution was constructed by [PERSON] et al. in 2009 [PERSON] et al. (2009). However, the above models are only applicable to some high and moderate void fraction flow systems. For low void fraction and moderate-to-high Reynolds number flow systems, these models have not been thoroughly verified yet. To improve the applicability of the drag force models in the polydisperse granular systems, [PERSON] and [PERSON] (2007) and [PERSON] et al. (2011) made further corrections based on the work of [PERSON] et al. (2014) and [PERSON] et al. (2017). Compared with [PERSON] et al. 's work [PERSON] et al. (2011), [PERSON] et al. (2014) gave a more concise and unified model by using the data from ([PERSON] et al., 2011; [PERSON] et al., 2011). It has been verified that these improvements can give good predictions for bi-disperse granular systems, but it is not confirmed that these models are applicable to polydisperse granular systems mixed with three or more kinds of granules ([PERSON] et al., 2011). For more general cases, [PERSON] et al. considered the particle size distribution in their model, where the data were obtained by using LBM simulations ([PERSON] et al., 2011). In 2019, [PERSON] et al. (2011) derived a polydisperse energy-minimization multiscale (EMMS) drag model by dividing the systems into three particle phases and considering each phase separately. Although a lot of important progress has been achieved in modeling the drag force for polydisperse granular flows, there are still some drawbacks that need to be addressed and improved. Firstly, the samples used in building the drag models are few and mainly derived from bi-disperse granular systems. Secondly, most drag models are only applicable to dilute phase systems with a large void fraction. Thirdly, these polydisperse drag models are in general not compatible with monodisperse systems. In view of these drawbacks, this paper establishes a new drag model for polydisperse granular flows by using a supervised machine learning approach. More samples will be used in the proposed model, so that it can be applied to dense phase systems and be compatible with monodisperse systems. So far, much attention has been paid to the supervised machine learning approach in modeling the granular drag force. [PERSON] et al.