## 4 The Polydisperse Drag Model
### Drag Model with OCRA  
So far, we have established an ANN drag model of a polydisperse granular system in Section 4.1. In order to verify the feasibility of OCRA in the ANN drag model, the ANN drag model with OCRA will be established, and its predicted results will be analyzed in this section. First, the prediction accuracy of ANN with the OCRA needs to be explored. We use 1290 random training samples and 1000 adaptive training samples to train the ANN and ANN with OCRA, respectively. Then, the two drag models will be employed to predict the dimensionless drag force of a polydisperse granular system. The experimental conditions are the same as in Section 4.1, and the weight matrices and biases from the input layer to \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Training Samples** & **1000** & **1300** & **1600** & **1000 Adaptive** \\ \hline
**CPU Time (s)** & 207.5 & 291.3 & 662.6 & 230.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: CPU time for different number of training samples. Figure 5: Different training samples in ANN prediction. the hidden layer and the hidden layer to the output layer of the trained neural network are illustrated in Appendix C.  
After comparisons of the predicted results in a different void fraction of the granular system, the results demonstrated that in a relatively dilute granular system, the ANN with OCRA outperforms the ANN in the situation with a wide range of granular sizes. The representative predicted results of the granular system, which are \(\epsilon_{g}=0.7\), are demonstrated in Figure 6. In a relatively dense granular system, ANN with OCRA and ANN have a similar prediction accuracy in general (see Table 8); both have their own advantages. Figure 7 shows the drag force predictions of the granular system when \(\epsilon_{g}=0.35\  
As shown in Figure 7, when ANN and ANN with OCRA predict the drag force of medium-sized granules in \(\varepsilon_{g}=0.35\), the overall prediction capability of the two drag models are approximately the same. Specifically, considering the granule size of \(d=3d_{0}\), the prediction accuracy of ANN with OCRA is slightly better than ANN. For the granule size of \(d=4d_{0}\), ANN with OCRA and ANN have a similar prediction accuracy in general. However, in smaller and larger granules (\(d=2d_{0}\), \(d=5d_{0}\)) ANN gives better results than the ANN with OCRA drag model. Generally, the prediction accuracy of ANN with OCRA is better than ANN in the polydisperse granular system. The results show that, although a certain portion of samples is excluded in the ANN with OCRA model, the prediction accuracy still increases. It can be inferred that OCRA plays an active role in preventing over-fitting problems and repeated ineffective training for the ANN drag model. Meanwhile, compared to the DNS results, ANN with OCRA has more prediction deviations than ANN for the granular size \(d=2d_{0}\) and \(d=5d_{0}\) generally (see Figure 7). The reason may be that, due to the filtering of OCRA, the information loss of larger and smaller granules in the training process will produce a certain prediction deviation. However, the loss of information for medium-sized granules in the ANN with OCRA training does not  
Figure 7: Dimensionless drag force prediction when \(\varepsilon_{g}=0.35\). significantly affect the predicted results, which is probably because the large number of granules will lead to the information loss of medium-sized granules easily compensated for by other similar medium-sized granules. On the other hand, the smaller and larger granules have difficulty achieving this compensation because of their smaller amount, and this phenomenon will become worse in denser granular systems that are dominated by the heterogeneous configuration of a granular cluster. Nevertheless, considering that such a large part of the flow in nature is relatively dilute flow, ANN with OCRA can be a suitable tool to capture the characteristics of flow in nature.