# Vehicle Interior Noise Prediction Based on Elman Neural Network
## 6 Analysis of Prediction Results  
The Elman neural network predicts the maximum sound pressure level of the target point in a vehicle, the body mass, the first-order global modal of the body-in-white, and the maximum intrusion of the B-pillar P4 and P5 points. Figure 12 shows the training process of the Elman neural network, and Figure 13 shows the prediction results for the maximum sound pressure level of the target point in a vehicle. To compare the prediction accuracy of the proposed algorithm with that of other neural network algorithms, the generalized regression neural network (GRNN), BP neural network, and GA-BP neural network were used to predict the maximum sound pressure level of the target point in a vehicle. The prediction results are shown in Figures 14-16. Figure 16: GA-BP neural network prediction results. Figure 14: GRNN neural network prediction results. Figure 15: BP neural network prediction results. As shown in Figure 12, at the beginning of training, the minimum error decreased sharply, and then the rate of decrease of the minimum error slowed down. At step 82,822, the minimum error reached \(5\times 10^{-5}\), which met the preset termination conditions. It can be seen from Figure 13, the predicted value of the Elman neural network was highly consistent with the finite element calculation value, and the prediction error was smaller than that of other neural networks. The Elman neural network reported the highest degree of agreement between predicted values and finite element calculations, followed by GA-BP, GRNN, and BP. The mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) [45] were used to evaluate the level of prediction error of these models as shown in Table 4.  
The total number of samples, predicted value, and actual value are denoted by \(n\), \(\hat{y}_{i}\), and \(y_{i}\), respectively. MSE represents the sum of squares of the distance between each predicted value and the actual value. A smaller MSE corresponds to a smaller error and higher prediction accuracy. \[MSE=\frac{1}{n}\sum_{i=1}^{n}\left(\hat{y}_{i}-y_{i}\right)^{2} \tag{18}\]  
RMSE represents the root sign of MSE and is used to measure the deviation between the observed and true values. \[RMSE=\sqrt{MSE} \tag{19}\]  
Given that dispersion is represented as an absolute value, no positive or negative offset can be obtained. MAE can better reflect the actual situation of the predicted value error. \[MAE=\frac{1}{n}\sum_{i=1}^{n}\left|\hat{y}_{i}-y_{i}\right| \tag{20}\]  
MAPE represents the ratio of the difference between the predicted and actual values, which can intuitively indicate prediction accuracy. \[MAPE=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{\hat{y}_{i}-y_{i}}{y_{i}}\right| \tag{21}\]  
The prediction errors of the body mass, the first-order global modal of the body-in-white, and the maximum intrusion of the B-pillar P4 and P5 points as predicted by the Elman neural network algorithm are shown in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Arithmetic** & **MSE** & **RMSE** & **MAE** & **MAPE** \\ \hline Elman & 0.004056 & 0.063685 & 0.047954 & 0.000772 \\ GRNN & 0.382991 & 0.618863 & 0.401589 & 0.006679 \\ BP & 0.636089 & 0.797552 & 0.433544 & 0.007105 \\ GA-BP & 0.105924 & 0.325459 & 0.159235 & 0.002559 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of prediction errors of neural network algorithms.