## 5 Modeling and Analysis by BP Neural Network  
### Prediction of Sectional Radial Deformation  
The topological structure of the BP neural network prediction model is 1-3-1. The input layer has one node receiving the data of radius r; the hidden layer has three nodes; the output layer has one node to output the radial thermal deformation of the section. The single hidden layer is selected. As the input layer and the hidden layer use the sigmoid function, the output layer uses the linear transfer function.  
There are nine situations in total based on the statistics in Table 3. The training group has four training points and one control point in every situation. BP neural network prediction model is trained. The calculated thermal deformation predicted value and fitting residual are presented in Figure 9. Three precision models for predicting the radial deformation of sections are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{_v/m/s_} & \multirow{2}{*}{T\({}^{F}\)C} & \multicolumn{5}{c}{_RE_} \\ \cline{3-8}  & & _r =_ 69 mm** & _r =_ 74 mm** & _r =_ 79 mm** & _r =_ 84 mm** & _r =_ 89 mm** \\ \hline \multirow{4}{*}{0.016} & 300 & 4.09E-04 & \(-\)9.68E-04 & 8.68E-04 & \(-\)3.84E-04 & 7.57E-05 \\  & 400 & 5.53E-04 & \(-\)1.31E-03 & 1.17E-03 & \(-\)5.16E-04 & 1.02E-04 \\  & 500 & 6.89E-04 & \(-\)1.63E-03 & 1.45E-03 & \(-\)6.41E-04 & 1.26E-04 \\  & 300 & 4.11E-04 & \(-\)9.70E-04 & 8.66E-04 & \(-\)3.83E-04 & 7.57E-05 \\  & 400 & 5.36E-04 & \(-\)1.27E-03 & 1.13E-03 & \(-\)4.99E-04 & 9.87E-05 \\  & 500 & 6.52E-04 & \(-\)1.54E-03 & 1.38E-03 & \(-\)6.08E-04 & 1.20E-04 \\  & 300 & 4.16E-04 & \(-\)9.84E-04 & 8.80E-04 & \(-\)3.89E-04 & 7.66E-05 \\  & 400 & 5.12E-04 & \(-\)1.21E-03 & 1.08E-03 & \(-\)4.77E-04 & 9.45E-05 \\  & 500 & 5.77E-04 & \(-\)1.36E-03 & 1.22E-03 & \(-\)5.37E-04 & 1.06E-04 \\ \hline \multirow{4}{*}{0.061} & SSE & 2.60E-06 & 1.45E-05 & 1.16E-05 & 2.26E-06 & 8.80E-08 \\  & MAE & 5.28E-04 & 1.25E-03 & 1.12E-03 & 4.93E-04 & 9.73E-05 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The prediction model accuracy of sectional radial deformation.  
Figure 8: Single hidden layer neural network topology.  
As can be seen from Table 4, when \(r\) = 74 mm, the prediction error is the largest, and the prediction error decreases with the increase or decrease of radius \(r\). When \(r\) increases to 89 mm, SSE is reduced to 8.8007E-08.  
The polynomial model, exponential model and Gaussian model are, respectively, used for curve fitting, according to the principle of least squares. The fitting errors SSE and MAE of the three models are calculated and compared with the SSE and MAE of the BP neural network, which is shown in Table 5.  
It can be seen that the SSE and MAE of the BP neural network are smaller than those of the exponential model and the Gaussian model in Table 5. Therefore, the BP neural network is significantly better than the exponential model and the Gaussian model in terms of prediction accuracy. According to Figure 10, the SSE and MAE of the BP neural network are larger than those of the polynomial model when \(r\) < 83 mm; therefore, it is  
Figure 9: Sectional radial deformation prediction model and residua.  
concluded that the polynomial model is better than the BP neural network under this condition. Because the SSE and MAE of the BP neural network are smaller than those of the polynomial model when \(r\) > 83 mm, thus it is concluded that the BP neural network is better than the polynomial model under this condition.