## 5 Conclusions  
A reasonable ANN model has been successfully established for describing the drag force in polydisperse granular flows. To accomplish this task, the 3D LBM for a polydisperse granular system is used to obtain diverse training data, and then the structures of ANN, including the network layer, learning algorithm, neurons of hidden layers, activation function, and input variables, were comprehensively discussed regarding the computational efficiency and Spearman correlation coefficient between the outputs and expected ones. For the training of ANN, a new adaptive algorithm for a polydisperse drag model \begin{table}
\begin{tabular}{c c} \hline \hline
**Authors** & **Formulas** \\ \hline [PERSON], [PERSON], & \(F_{i}=\left((1-\Phi)y_{i}+\Phi y_{i}^{2}+0.064(1-\Phi)y_{i}^{3}\right)\cdot F (\Phi,<\ Re>)\), \\ [PERSON] (BVK) [27] & \(F_{i,i}=\beta_{i}\overline{f}3\pi\mu_{D}\mu\), \\  & \(\beta_{i}=y_{i}+\frac{(1-\epsilon)}{\epsilon}\left(\frac{1-\epsilon-0.27}{1-0. 27}\right)\frac{(y_{i}^{2}-y_{i})}{\chi_{i}y_{i}}\) \\ [PERSON] et al. [21] & \(\left\{\begin{array}{l}\overline{f}=K_{1}+K_{2}\epsilon^{4}+K_{3}(1-\epsilon ^{4})\\ K_{0}=\frac{1-\epsilon}{1+3\epsilon^{4}};K_{1}=\frac{1+128K_{0}+715K_{0}^{2}}{ \epsilon^{4}(1+49.5K_{0})}\\ K_{2}=\frac{1-1+0.130K_{0}+6.66-10^{-1}\epsilon^{2}}{1+3.42\cdot 10^{-2}\overline{K}+6. 92\cdot 10^{-4}\overline{K}}-1\\ K_{3}=\left(\frac{2\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{ \overline{   }}}}}}}}}}}}}}} \right)\left(\frac{440}}{1+\frac{1}{6600\omega+4.92\cdot 10^{-4} \overline{K}\cdot K_{0}^{20}}{\epsilon^{600+4.92\cdot 10^{-4}\overline{K}-4.30\cdot 10^{ -4}}}\\ -\frac{1900\omega^{2}-6.60\cdot 10^{-2}\overline{K}}{-1.31\cdot 10^{-4}\cdot K_{0}^{2 }+7.38\cdot 10^{-4}\cdot s})\end{array}\right.\) \\ [PERSON] \& [PERSON] (HS) [30] & \(F_{D-\text{fixed}}^{*}=\left(\frac{1}{1-\Phi}\right)+\left(F_{D-\text{fixed}}^{*} -\frac{1}{1-\phi}\right)\left(ay_{i}+(1-a)y_{i}^{2}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Empirical drag formulas for comparison. was proposed to improve the training efficiency and avoid the over-fitting problem, where redundant and singular samples can be automatically eliminated from the training set. By comparing and analyzing the simulation results, the ANN drag model has good prediction ability in the range of \(\varepsilon_{s}\in[0.05,0.65]\), \(Re_{p}\in[0.008,980]\), \(U_{slip}\in[0.1\text{m/s},0.5\text{m/s}]\), and \(<d>\in[4.97\Delta_{x},16.83\Delta_{x}]\). Finally, there are several conclusions that we have obtained. 1. A single-hidden-layer BP neural network with 4 input neurons and 15 hidden neurons, with the activation function of Tanh and the network established by the LM learning method, can effectively realize complex nonlinear mapping in a polydisperse drag model. 2. Using the OCRA to train the ANN drag model can effectively improve the computational efficiency of ANN, which does not significantly increase the system errors. 3. Due to ANN having advantages of self-learning ability and good compatibility, the drag model based on ANN can realize the good prediction of both the polydisperse and monodisperse granular systems. 4. The proposed kernel function based on the exponential form can process training samples of polydisperse drag force effectively for the contribution rate analysis. In brief, ANN has obvious advantages in the construction of the drag model of the polydisperse granular system, considering the complex non-linear relationship between the variables. The verified structure of the ANN drag model can effectively realize the drag prediction both in monodisperse and polydisperse granular systems. Moreover, using the OCRA to train the ANN drag model can obviously improve the computational efficiency without significantly increasing the system errors.