## 3 ANN Drag Model and Training Strategy
### The Construction of ANN
#### 3.1.3 The Structure of Hidden Layers  
The structure of the hidden layers is an important factor that influences the ANN performance. [PERSON] et al. [45] have proved that a one-hidden-layer back-propagation (BP) neural network can realize arbitrary nonlinear mapping. Thus, we will first discuss the influence of the number of hidden layers on the convergence speed and accuracy of the neural network with a multi-hidden-layer structure, as shown in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Exp.** & **Activation Function** & **Sigmoid** & **Leaky-LU** & **Tanh** & **Gauss** \\ \hline \multirow{2}{*}{1} & Number of iterations & 369 & 43 & 295 & 132 \\  & Correlation coefficient & 0.9946 & 0.9519 & 0.9961 & 0.9849 \\ \hline \multirow{2}{*}{2} & Number of iterations & 286 & 49 & 312 & 682 \\  & Correlation coefficient & 0.9953 & 0.9282 & 0.9952 & 0.9416 \\ \hline \multirow{2}{*}{3} & Number of iterations & 332 & 51 & 299 & 413 \\  & Correlation coefficient & 0.9952 & 0.9207 & 0.9968 & 0.9412 \\ \hline \multirow{2}{*}{\(E(X)\)} & Number of iterations & 329 & 48 & 302 & 409 \\  & Correlation coefficient & 0.9950 & 0.9336 & 0.9960 & 0.9559 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of activation functions of 1 hidden layer with 15 neurons.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Exp.** & **Training Algorithm** & **GDM** & **LM** \\ \hline \multirow{4}{*}{1} & Termination rule & A & B \\  & Correlation coefficient & 0.7851 & 0.9366 \\  & CPU time (s) & 192.9 & 217.1 \\ \hline \multirow{2}{*}{2} & Termination rule & A & B \\  & Correlation coefficient & 0.8179 & 0.9253 \\  & CPU time (s) & 215.5 & 246.2 \\ \hline \multirow{2}{*}{3} & Termination rule & A & B \\  & Correlation coefficient & 0.8223 & 0.9321 \\  & CPU time (s) & 187.3 & 205.3 \\ \hline \multirow{2}{*}{\(E(X)\)} & Termination rule & A & B \\  & Correlation coefficient & 0.8084 & 0.9313 \\ \cline{1-1}  & CPU time (s) & 198.6 & 222.9 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Spearman correlation coefficient and CPU time for two training algorithms.  
The independent experimental results of an ANN with one, two, three, and four hidden layers are given in Table 5. It is found that the calculation time and number of iterations increase as the number of hidden layers increases. Accordingly, considering the prediction accuracy and efficiency of ANN, the single hidden layer is preferred in modeling the drag force.  
Next, we will discuss the influence of the structure and the number of neurons in the single hidden layer of the ANN on the convergence speed, correlation coefficient, and CPU time. We took the average results of three independent experiments and recorded them in Table 6. It can be found that considering the influence of the number of neurons on the correlation coefficient and CPU time, a single-hidden-layer BP neural network with 15 neurons is suggested to be implemented for the current study. Moreover, the impacts of input and output correlations for ANN are illustrated in Appendix A.  
Finally, the conclusion can be drawn that the Levenberg-Marquardt (LM) algorithm is recommended as the learning method, and a single hidden layer with a 15-neurons back-propagation (BP) neural network attached to the Tanh activation function is suggested during the construction of the polydisperse granular drag force model.