## 4 The Polydisperse Drag Model  
### Drag Model without OCR  
The learning process of the ANN can be regarded as the process of finding the optimal solution in the given error tolerance of the system. The better compatibility and self-learning capabilities of ANN, which are different from traditional regression methods, are promising to obtain the implicit connections and relationships among variables under different granular sizes. Because the difference of granular size will bring obvious heterogeneity in the granular flow, the drag force model of a polydisperse system has complex and implicit nonlinear characteristics. Generally, it has great advantages for the application of ANN to establish the polydisperse drag model.  
Since a small training sample set will produce large deviations for ANN training, it is not reasonable to use a small training sample set in the training process. Meanwhile, the training sample set should not be set too large either, considering the computational efficiency and over-fitting problems. In order to find a reasonable training sample set, we start with 990 samples and increase samples with the interval size of \(\delta_{Re_{p}}=100\) and \(\delta_{\varepsilon_{s}}=0.15\) to search for the optimal number of training samples in each interval \([Re_{p,k},Re_{p,k+1}]\times[\varepsilon_{s,l},\varepsilon_{s,l+1}]\) (\(k=1,2,\cdots,11\), \(l=1,2,\cdots,4\)). The specific searching process and results are illustrated in Appendix B.  
After preprocessing the training sample set, a single hidden layer with a 15-neuron BP neural network is used to fit the training data. The LM algorithm and Tanh function are used as the learning method and activation function, respectively, in addition to the input and output variables, which are shown in Figure 2. Since the prediction results are similar, we only show the drag force of granules of different sizes when \(\varepsilon_{\text{g}}=0.35\).  
Figure 4 presents the neural network; in general, it can accurately describe the drag force of the polydisperse granular system. Yet, it can still be shown that there is an accuracy difference in the prediction of different-sized granules. From the prediction results, ANN has good prediction accuracy for larger-sized granules (see in Figure 4c,d), but it produces certain deviations for smaller granules, and these deviations will become more obvious in a larger Reynolds number flow system. The main reason is perhaps that the motion of smaller granules is prone to be influenced by larger granules in polydisperse granular flow, and it will lead to irregular features for a smaller granule's motion that will bring drastic fluctuations in the granular drag force. Furthermore, some smaller granules that are distributed with larger ones will result in heterogeneous distributions. This distribution will cause drastic complex nonlinear relationships in the drag force distribution. Thus, the complicated non-smooth functional relationship between drag force and other variables will appear. On the other hand, the ANN model can achieve more accurate predictions for smooth and flat functional relationships, whereas when ANN encounters the input and output fitting samples with complicated non-smooth functional relationships, it may produce a certain deviation, and this deviation becomes larger as the complex functional relationship becomes stronger. Thus, there will be certain deviations in the prediction of the drag force for smaller granules. Toward these deviations, it is difficult to eliminate them by only considering commonly used inputs in computation, including the mathematical expectations and variance of granular diameters. Thus, it is worth conducting more in-depth studies in searching for the more efficient inputs of ANN in future work.