## Case Study
### The Influence of Measurement Noise  
As mentioned in the previous sections, the established LSTM model maps the one-to-one relationship between train loads and bridge responses. The train load information is usually obtained through the installed sensors, which may inevitably include measurement noise [32]. If the established LSTM model is sensitive to the measurement noise, the accuracy and robustness will be weakened. Therefore, this section explores the influence of noise level of monitored axle load and wheelbase on the prediction accuracy. Four testing cases are designed and tabulated in Table 3.  
The evaluation metrics of the predicted displacement and acceleration under different noise cases are shown in Figures 20 and 21. The results show that the Pearson correlation coefficient and \(\mathrm{R}_{\mathrm{W}}^{2}\) decrease obviously with the noise included. The decrease of the evaluation metrics of displacement is smaller that of the acceleration. Specifically, the Pearson correlation coefficient and \(\mathrm{R}_{\mathrm{W}}^{2}\) of displacement prediction stay above 0.999, while those of acceleration decreased by a large margin, but still remained above 0.8. This indicates that the predicted values of displacement and acceleration with limited input noise always have a strong correlation with the theoretical values.  
Figure 19: The influence of the carriage number on prediction performance.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Description of the Noise Level** \\ \hline Case 0 & Both axle load and axle spacing adopt the optimal values identified by load recognition. \\ \hline Case 1 & Axle spacing adopts the optimal value identified by load recognition, while axle load values are randomly generated based on their uncertainty results. \\ \hline Case 2 & Axle load adopts the optimal value identified by load recognition, while axle spacing adds 5\% noise. \\ \hline Case 3 & Axle load values are randomly generated based on their uncertainty results, while axle spacing adds 5\% noise. \\ \hline \hline \end{tabular} The specific method of adding 5\% noise involves adding random errors with a maximum absolute error of 5\% based on the optimal axle spacing.  
\end{table}
Table 3: The design of testing cases with different noise levels.  
Regarding APVE, both the evaluation metrics for displacement and acceleration increase after adding noise, with the maximum increase being approximately 0.1% and 2%, respectively. In terms of dimensional indicators, each evaluation metric shows an increase after noise addition, but the increase in acceleration evaluation metrics is greater than that of displacement. This suggests that acceleration prediction is more sensitive to noise compared to displacement prediction. Furthermore, transitioning from Case 0 to Case 1, or from Case 2 to Case 3, results in minimal changes in each metric. However, there is a noticeable variation trend in the evaluation indices between Case 2 and Case 3 compared to Case 0 and Case 1.  
In summary, the noise associated with axle load has minimal impact on response prediction, whereas noise related to wheelbase has a more pronounced effect. Nonetheless, overall, there remains a strong correlation between predicted response values and theoretical values, with errors consistently falling within an acceptable range. Particu  
Figure 21: The impact of input noise on acceleration prediction accuracy.  
Figure 20: The impact of input noise on displacement prediction accuracy.  
larly noteworthy is the robust performance of displacement prediction, which maintains accuracy even when influenced by noise.