## 2 Numerical Method for Dataset Generation
### Numerical Method
An in-house C++ LBM solver developed by our research group was used, and the parameters of direct numerical simulation (DNS) [52] are listed in Table 1.  
Given the advantages of a simple calculation process, ease of programming, and good prediction accuracy in particle-suspended flow, the momentum exchange method [53] is adopted to calculate the drag force. Specifically, the drag force that acts on the solid surface is obtained by calculating the momentum exchange on each boundary link between each two opposing directions of the neighboring nodes as:  
\[\mathbf{F}_{d}=\sum_{i\
eq 0}\left[\mathbf{e}_{i}f_{i}(\mathbf{x}_{f},t)-\mathbf{e}_{i}f_{i}( \mathbf{x}_{f}+\mathbf{e}_{i}\delta_{t},t)\right], \tag{7}\]  
where \(\mathbf{x}_{f}\), \(\mathbf{x}_{f}+\mathbf{e}_{i}\delta_{t}\) are the fluid node and its neighboring solid node, respectively, and \(\mathbf{e}_{i}=-\mathbf{e}_{i}\). For training the ANN, the particle Reynolds number \(Re_{p}\)[20] is calculated by  
\[Re_{p}=\frac{\rho_{g}(1-\varepsilon_{s})\Big{|}\mathbf{u}_{slip}\Big{|}d_{p}}{\mu _{g}}, \tag{8}\]  
where  
\[f_{\alpha}^{eq}=\omega_{\alpha}\rho\left(1+\frac{\mathbf{e}_{\alpha}\cdot\mathbf{u}}{c _{\rm s}^{2}}+\frac{\left(\mathbf{e}_{\alpha}\cdot\mathbf{u}\right)^{2}}{2c_{\rm s}^{4 }}-\frac{\mathbf{u}^{2}}{2c_{\rm s}^{2}}\right),\ \alpha=0,1,\ldots,n-1 \tag{2}\]0  
with \(\mathbf{v}_{g}\) is the gas instantaneous velocity in the position of an individual granule, \(\mathbf{v}_{s}\) is the solid velocity; in the simulation, the solid velocity \(\mathbf{v}_{s}\) was set to zero. \(\varepsilon_{s}\) is the solid volume fraction. \(<d>\) is the Sauter mean diameter [54], which is defined as:  
\[f_{\alpha}^{eq}=\omega_{\alpha}\rho\left(1+\frac{\mathbf{e}_{\alpha}\cdot\mathbf{u}}{c _{\rm s}^{2}}+\frac{\left(\mathbf{e}_{\alpha}\cdot\mathbf{u}\right)^{2}}{2c_{\rm s}^{4 }}-\frac{\mathbf{u}^{2}}{2c_{\rm s}^{2}}\right),\ \alpha=0,1,\ldots,n-1 \tag{2}\]1  
The solid volume fraction [20] is the ratio of the volume of the granular cluster \(V_{gc}\) to the grid volume occupied by the granular cluster \(V_{grid}\) and can be represented as:  
\[f_{\alpha}^{eq}=\omega_{\alpha}\rho\left(1+\frac{\mathbf{e}_{\alpha}\cdot\mathbf{u}}{c _{\rm s}^{2}}+\frac{\left(\mathbf{e}_{\alpha}\cdot\mathbf{u}\right)^{2}}{2c_{\rm s}^{4 }}-\frac{\mathbf{u}^{2}}{2c_{\rm s}^{2}}\right),\ \alpha=0,1,\ldots,n-1 \tag{2}\]2  
and the void fraction \(\varepsilon_{\text{g}}\) as:  
\[f_{\alpha}^{eq}=\omega_{\alpha}\rho\left(1+\frac{\mathbf{e}_{\alpha}\cdot\mathbf{u}}{c _{\rm s}^{2}}+\frac{\left(\mathbf{e}_{\alpha}\cdot\mathbf{u}\right)^{2}}{2c_{\rm s}^{4 }}-\frac{\mathbf{u}^{2}}{2c_{\rm s}^{2}}\right),\ \alpha=0,1,\ldots,n-1 \tag{2}\]3 \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Domain size & 0.15 m \(\times\) 0.15 m \(\times\) 0.45 m \\ Grid size \(\Delta_{x}\) & 0.0015 m \\ Time step size \(\delta_{t}\) & 0.0015 s \\ Characteristic velocity \(U\) & 0.1 m/s \\ Characteristic length \(L\) & 0.15 m \\ Smallest granular diameter \(d_{0}\) & 4\(\Delta_{x}\) \\ Granular diameters & \(d_{0}\), 2\(d_{0}\), 3\(d_{0}\), 4\(d_{0}\), 5\(d_{0}\) \\ Percentage of different size granules & 0.096, 0.221, 0.372, 0.212, 0.099 \\ Gas density \(\left(\text{kg}/\text{m}^{3}\right)\) & 1.28 \\ Number of granules & 450 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The DNS parameters. Table 2 presents the average drag force of the granules calculated by the LBM model when the void fraction \(\varepsilon_{g}=0.5\). Moreover, the scaling of the dimensionless average drag coefficient is the drag force acting on a single granule in the flow field under \(Re_{p}=1\). The diameter of a single granule is \(12\Delta_{x}\). After comparison, the simulation results are consistent with the published literature [19].