## 5 Modeling and Analysis by BP Neural Network
### Prediction Results and Analysis for Magnitude of Interference Fit and Relevant Factors  
In fact, during the assembly the magnitude of interference fit between the steel joint and the aluminum alloy drill pipe is equal to the steel joints' thermal deformation. Factors affecting the magnitude of interference fit are the cooling water flow velocity and the initial heating temperature of the steel joint. According to the discussion in Section 3, the relation between different cooling water flow velocities, initial heating temperatures and the magnitudes of interference fit are shown in the Table 6. The data of Table 6 are used as the BP neural network training group. This BP neural network prediction model is a 2-7-1 topological structure. The input layer has two nodes for receiving the flow rate of cooling water and the initial heating temperature of the joint; the hidden layer has seven nodes, and the output layer has one node to the output thermal deformation of the steel joint; the hidden layer is individual. As the input layer and the hidden layer use the sigmoid function, the output layer uses the linear transfer function. After training, the prediction model of the relation between the interference of the steel joint, the flow rate of cooling water and the initial heating temperature of the steel joint is obtained, as shown in Figure 11.  
Figure 10: Error comparison between BP neural network and polynomial model. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Error** & **Method** & \(r\) = 69 mm & \(r\) = 74 mm & \(r\) = 79 mm & \(r\) = 84 mm & \(r\) = 89 mm \\ \hline \multirow{4}{*}{SSE} & BP & 2.60E-06 & 1.45E-05 & 1.16E-05 & 2.26E-06 & 8.80E-08 \\  & Polynomial & 2.43E-06 & 1.05E-05 & 1.14E-06 & 7.85E-06 & 3.66E-06 \\  & Exponential & 2.60E-03 & 3.86E-04 & 1.54E-03 & 3.73E-04 & 8.89E-04 \\  & Gaussian & 3.14E-03 & 2.86E-04 & 1.21E-03 & 2.78E-04 & 1.72E-03 \\  & BP & 5.28E-04 & 1.25E-03 & 1.12E-03 & 4.93E-04 & 9.73E-05 \\ \multirow{2}{*}{MAE} & Polynomial & 4.02E-04 & 1.03E-03 & 3.02E-04 & 8.15E-04 & 5.33E-04 \\  & Exponential & 1.67E-02 & 6.43E-03 & 1.28E-02 & 6.32E-03 & 9.76E-03 \\ \multirow{2}{*}{MAE} & Gaussian & 1.82E-02 & 4.73E-03 & 1.08E-02 & 4.35E-03 & 1.29E-02 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of prediction accuracy for different methods. As can be seen from in Table 7, in terms of heating temperature, through the analysis of SSE and MAE, it can be seen that the prediction model error increases with the increase of the heating temperature. In terms of the cooling water flow rate, \(SSE_{0.061}>SSE_{0.016}>SSE_{0.038}\), \(MAE_{0.061}>MAE_{0.016}>MAE_{0.038}\). Therefore, when the flow rate is 0.038 m/s, the prediction accuracy gets the highest, and the prediction error increases with the increase or decrease of the flow rate; especially when the flow rate increases, the trend of the error increase is more obvious. For a comparative analysis, a polynomial model based on the least squares method is used in this paper for surface fitting. The fitting errors RE, SSE and MAE of the polynomial model are calculated separately as shown in Table 8. The RE, SSE and MAE of the polynomial model are compared with the prediction errors SSE and MAE of the BP neural network, as shown in Tables 9 and 10. Based on the statistics in Tables 9 and 10, it can be seen that the errors of the BP neural network are smaller than the errors of polynomial model for different flow velocities and different initial heating temperatures. Therefore, the performance of the BP neural network is better than the polynomial model. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{_v/m/s_} & \multicolumn{3}{c}{**T/\({}^{C}\)C**} \\ \cline{2-4}  & **300** & **400** & **500** \\ \hline
0.016 & 0.2229 & 0.2992 & 0.3797 \\
0.038 & 0.2218 & 0.2939 & 0.3533 \\
0.061 & 0.2205 & 0.2865 & 0.3168 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Thermal deformation of steel joint when all threads are screwed (mm). Figure 11: Magnitude of interference fit and relevant factors prediction model.