## 4 Experiment
### Results and Discussion  
First, to verify the real-time performance of the 3D non-rigid transformation derivation model based on multi-control point perception data, we deduced all samples in the dataset and compared the times spent by the FEM method and the proposed model. The manual secondary scanning time was considered to be at least 20 min. According to the statistical results presented in Figure 5 and Table 3, the FEM method required an average of 4.155 s compared to the 0.021 s average deduction time of the proposed model, nearly 200 times faster than the FEM method. In addition, this real-time performance is vastly superior to that of manual secondary scanning. Secondly, to verify the feasibility and optimality of the CNN-GRU-SA network architecture, we compared it with CNN-GRU [16], CNN-GRU-CBAM [17], and CNN-LSTM-SA [18] through comparative experiments. Among them, CNN-GRU-CBAM combines the Convolutional Block Attention Module (CBAM) [19], and CNN-LSTM-SA combines Long Short-Term Memory (LSTM) [20]. The comparison results of the training and testing are shown in Figure 6.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Data Size** & \begin{tabular}{l} **Upper** \\ **Limit** \\ **(mm)** \\ \end{tabular}  & \begin{tabular}{l} **Q3** \\ **(mm)** \\ \end{tabular}  & \begin{tabular}{l} **Q2** \\ **(mm)** \\ \end{tabular}  & \ \begin{tabular}{l} **Q1** \\ **(mm)** \\ \end{tabular}  & \begin{tabular}{l} **Lower** \\ **Limit** \\ **(mm)** \\ \end{tabular}  \\ \hline \(1\times 10^{4}\) & 0.430 & 0.210 & 0.110 & 0.060 & 0.003 & 0.165 \\ \(2\times 10^{4}\) & 0.397 & 0.195 & 0.095 & 0.061 & 0.008 &  
Figure 5: (**a**) Statistical results of FEM deduction time; (**b**) statistical results of neural network deduction time. Figure 6: (**a**) Loss drop comparison of the training set; (**b**) statistical results of the RMSE of the training set; (**c**) loss drop comparison of the test set; (**d**) statistical results of the RMSE of the test set. The results indicate that the CNN-GRU-SA proposed in this paper has the following advantages:  
* In Figure 6a,b, compared with the other three network architectures, the loss and RMSE of CNN-GRU-SA have a faster convergence speed on 80% of the training set. * In Figure 6c,d, in the 20% test set, the fluctuation of the loss and RMSE of the network is also smaller than that of the other comparison networks. * To further verify the performance of the CNN-GRU-SA, we provide the final results of the network training after completing the iterations. The results listed in Table 4 suggest that the CNN-GRU-SA network architecture proposed in this study has a maximum decrease of 39% in the training set loss, 49% in the training set RMSE, 48% in the testing set loss, and 29% in the testing set RMSE compared to the minimum values of other networks. Finally, to show the results of network deduction more intuitively, we selected a set of data in the dataset, performed deduction using the trained network model, and developed a deviation map of the deduction results and label data. Figure 7a-c shows the situation before and after the control point changes and the appearance before the entity changes. Figure 7d shows the changed appearance, which requires the network architecture to obtain the label data through deduction. Figure 7e-h depicts the deviation graphs of the results of the four network architecture simulations and the label data. These four figures indicate that outliers with deviations of approximately 0.4 mm remain in the extrapolation results of CNN-GRU and CNN-GRU-CBAM. The inference results of CNN-LSTM-SA and CNN-GRU-SA outperform those of the other two networks. The proposed CNN-GRU-SA performs better at the edge of the tip than CNN-LSTM-SA. Moreover, our method has better bias results in areas far from the tip. In addition, we conducted statistical analysis on the deviation, as shown in Table 5. The results indicate that the difference between CNN-GRU-SA and the minimum value of all networks is less than \(10^{-3}\) mm.