## 3 ANN Drag Model and Training Strategy
### The Optimal Contribution Rate Algorithm (OCRA)  
The ability of ANN to explore the correlation between variables is not limitless [45], and the predicted results depend significantly on the correlation between inputs and outputs. When the input samples corresponding to the output variable are incomplete and  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline Number of neurons & 7 & 8 & 9 & 10 \\ \hline Number of iterations & 94 & 91 & 105 & 119 \\ Correlation coefficient & 0.9337 & 0.9415 & 0.9593 & 0.9621 \\ CPU time (s) & 141.2 & 155.2 & 172.2 & 184.6 \\ \hline Number of neurons & 11 & 12 & 13 & 14 \\ \hline Number of iterations & 112 & 123 & 126 & 102 \\ Correlation coefficient & 0.9733 & 0.9736 & 0.9824 & 0.9861 \\ CPU time (s) & 203.3 & 219.3 & 231.5 & 234.6 \\ \hline Number of neurons & 15 & 16 & 17 & 18 \\ \hline Number of iterations & 101 & 135 & 143 & 166 \\ Correlation coefficient & 0.9956 & 0.9953 & 0.9951 & 0.9950 \\ CPU time (s) & 243.1 & 272.7 & 342.1 & 473.6 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of the number of neurons in single hidden layer.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Exp.** & **Number of Hidden Layers** & **1** & **2** & **3** & **4** \\ \hline \multirow{3}{*}{1} & Number of iterations & 93 & 121 & 135 & 232 \\  & Correlation coefficient & 0.9699 & 0.9603 & 0.9110 & 0.6673 \\  & CPU time (s) & 195.7 & 219.2 & 269.1 & 306.3 \\ \hline \multirow{3}{*}{2} & Number of iterations & 95 & 129 & 133 & 193 \\  & Correlation coefficient & 0.9716 & 0.9515 & 0.9035 & 0.7018 \\  & CPU time (s) & 172.5 & 213.3 & 255.6 & 272.3 \\ \hline \multirow{3}{*}{3} & Number of iterations & 88 & 135 & 157 & 229 \\  & Correlation coefficient & 0.9798 & 0.9709 & 0.9512 & 0.6365 \\  & CPU time (s) & 173.9 & 217.1 & 273.2 & 315.2 \\ \hline \multirow{3}{*}{\(E(X)\)} & Number of iterations & 92 & 128 & 142 & 218 \\  & Correlation coefficient & 0.9738 & 0.9609 & 0.9219 & 0.6685 \\ \cline{1-1}  & CPU time (s) & 180.7 & 216.5 & 266.0 & 297.9 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of the number of hidden layers for ANN’s experimental results.  
the relationship between them is not clear, it will bring great challenges to the ANN. Thus, an additional preprocessing method may be required when the correlation between the inputs and outputs is found to be ambiguous.  
In the applications of the ANN-machine learning model, various fitting problems are encountered due to diverse functional realizations. In order to complete the prediction of some complex nonlinear relationships, a huge training sample set is required to construct a precise ANN prediction model. However, the huge training sample set will bring a significant reduction in the training and computing efficiency of the ANN. The reason is that according to the characteristics of the ANN-machine learning model, it is prone to result in repeated and inefficient training for ANN by using the linearly correlated training samples (i.e., inefficient training samples). Meanwhile, the training samples with the property of mutual independence (i.e., efficient training samples) are more preferable [45]. Considering that the huge training sample set for the complex nonlinear relationships contains the efficient and inefficient training samples, it is necessary to design an algorithm that can identify the different types of training samples and search for the most efficient one for training. For this purpose, we will propose an adaptive data filtering algorithm, i.e., the OCRA. The implementation of the algorithm can be divided into two steps. The first step is to analyze the contribution rate of the training samples inspired by the idea of the kernel principal component analysis (KPCA) [58]. The second step is to search for an optimal training sample set adaptively with the best contribution rate according to the certain tolerance of the training error.