## 4 Case Analysis
### Case 1  
The Okaloosa Armature Tester (OAT) and MCA 103 armature published by the Florida laboratory were taken as the research objects [25; 26]. The geometric dimensions and material property parameters of the model are shown in Table 4.  
The extrapolation prediction performance of the model was tested using the launch data of SLK 018. The armature has a mass of 249.72 g and flies out after accelerating in the bore for 1.0 ms. The waveform diagram of excitation current is shown in Figure 10.  
Establishing the finite element simulation model of the electromagnetic railgun. Based on the armature conductivity (1.86 \(\times\) 10\({}^{7}\) S/m) in Table 3, the numerical simulation results of the armature electromagnetic force under different armature conductivities are obtained, as shown in Figure 11. Because of the short acceleration time and low flight velocity of the armature in this case, there is no \"pseudo-oscillation\" in the numerical solution of the armature electromagnetic force under each conductivity. Under the condition of low armature conductivity, the current distribution of the conductor is more uniform, and the armature electromagnetic force is larger. Through the above numerical simulation, 4000 groups of required sample data are obtained. In order to eliminate the influence of the magnitude and dimension of data on the prediction accuracy and convergence speed, the sample data are normalized and divided into a training set and a test set. Among them, the training set is divided into two kinds: the sample data obtained under the standard armature conductivity and the sample data obtained under different armature conductivity. The test set is the sample data in 0.87-1 ms time under the standard armature conductivity. \begin{table}
\begin{tabular}{c c c} \hline
**Model Parameters** & **Symbol** & **Values** \\ \hline Rail length/mm & \(l_{\text{r}}\) & 220.00 \\ Rail height/mm & \(h_{\text{r}}\) & 31.75 \\ Rail width/mm & \(w_{\text{r}}\) & 6.35 \\ Rail conductivity/(S/m) & \(\sigma_{\text{r}}\) & \(5.80\times 10\) \\ Rail permeability/(H/m) & \(\mu_{\text{r}}\) & \(4\pi\times 10^{-7}\) \\ Armature length/mm & \(l_{a}\) & 28.59 \\ Armature height/mm & \(h_{a}\) & 25.00 \\ Armature width/mm & \(u_{a}\) & 25.00 \\ Armature conductivity/(S/m) & \(\sigma_{a}\) & \(1.86\times 10^{7}\) \\ Armature permeability/(H/m) & \(\mu_{a}\) & \(4\pi\times 10^{-7}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of armature and rail model. Figure 10: Waveform diagram of excitation current in case 1. The above two training sets are used to train four extrapolated prediction models: Support Vector Regression (SVR), Random Forest (RF), DNN, and DBN-DNN. Then, the test set is used to test the extrapolated prediction model after training. The Mean Absolute Percentage Error (MAPE) is selected as the standard to measure the evaluation performance of the extrapolated prediction model. The value range of MAPE is \([0,+\infty)\). The smaller the value is, the higher the accuracy of the prediction model is. The calculation method is  
\[MAPE=\frac{1}{N}\sum_{i=1}^{N}\frac{|y_{i}-y_{i}|}{y_{i}}\times 100\% \tag{9}\]  
In the formula, \(N\) is the number of samples, and \(\hat{y}_{i}\) and \(yi\) are the predicted and simulated value of armature electromagnetic force of the sample, respectively. Considering the randomness of the performance evaluation of each extrapolation prediction model, the four extrapolation prediction models are trained and tested five times, and then the average value of MAPE is taken, as shown in Table 5.  
Compared with the training set under the standard armature conductivity, the sample data in the training set under a different armature conductivity are more sufficient, which enables the extrapolated prediction model to better grasp the overall characteristics of the data. It is more helpful to learn the mapping relationship between the excitation current, time, velocity, armature conductivity variables, and armature electromagnetic force, and the prediction effect is better.