## 4 Calculus Analysis
### Projected Results
To verify the effectiveness and sophistication of the proposed cascaded residual smoothing-MSSAN, this paper also compares it with similar techniques, which include the BiLSTM, the CNN-GRU, and CNN-Attention-LSTM [22]. The results of the comparison experiments are presented in Table 3, where MAE, RMSE, and \(R^{2}\) are used as indicators for validation, where smaller RMSE and MAE are better, larger \(R^{2}\) is better, and bolding represents the optimal value of the indicator for the position of the static level. As can be seen from Table 3, although our proposed method does not work best on the static level 5, it is optimal in most cases. This paper also statistically averages the metrics of these methods at the overall static level, as shown in Figure 14. In terms of MAE, the mean MAE of the cascaded residual smoothing-MSSAN is 2.4525, which is 28.17% lower compared to the mean MAE of the BiLSTM of 3.4143, 37.37% lower compared to the mean MAE of the CNN-GRU of 3.9158, and 25.79% lower compared to the mean MAE of CNN-Attention-LSTM of 3.3047. In terms of RMSE, the average RMSE of the cascaded residual smoothing-MSSAN is 3.7556, which is 27.14% lower compared to the average RMSE of the BiLSTM of 5.1543, 38.41% lower compared to the average RMSE of the CNN-GRU of 6.0979, and 24.69% lower compared to the average RMSE of CNN-Attention-LSTM of 4.9869. In terms of \(R^{2}\), the \(R^{2}\) mean of the cascaded residual smoothing-MSSAN is 0.9384, which is 7.29% higher compared to the \(R^{2}\) mean of the BiLSTM of 0.8746, 9.59% higher compared to the \(R^{2}\) mean of the CNN-GRU of 0.8563, and 2.36% higher compared to the \(R^{2}\) mean of CNN-Attention-LSTM of 0.9168. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Static Level Position** & **Methodology** & **MAE** & **RMSE** & \(R^{2}\) \\ \hline \multirow{8}{*}{1} & Cascade residual smoothing-MSSAN & **9.6932** & **15.0834** & **0.9775** \\  & BiLSTM & 13.3812 & 20.5796 & 0.9581 \\  & CNN-GRU & 15.6298 & 24.3279 & 0.9281 \\  & CNN-Attention-LSTM & 13.5982 & 21.0110 & 0.9477 \\ \hline \multirow{8}{*}{2} & Cascade residual smoothing-MSSAN & **0.1173** & **0.1674** & **0.9868** \\  & BiLSTM & 0.3329 & 0.3781 & 0.8923 \\  & CNN-GRU & 0.8363 & 1.6303 & 0.7839 \\  & CNN-Attention-LSTM & 0.1764 & 0.2003 & 0.9637 \\ \hline \multirow{8}{*}{3} & Cascade residual smoothing-MSSAN & **0.6403** & **0.9998** & **0.9467** \\  & BiLSTM & 1.0948 & 1.6285 & 0.8586 \\ \cline{1-1}  & CNN-GRU & 0.8853 & 1.4356 & 0.8902 \\ \cline{1-1}  & CNN-Attention-LSTM & 0.8005 & 1.095 & 0.9361 \\ \hline \multirow{8}{*}{4} & Cascade residual smoothing-MSSAN & **0.8013** & **1.0842** & **0.9277** \\  & BiLSTM & 1.2577 & 1.7821 & 0.8047 \\ \cline{1-1}  & CNN-GRU & 1.0098 & 1.416 & 0.8751 \\ \cline{1-1}  & CNN-Attention-LSTM & 0.9637 & 1.116 & 0.8978 \\ \hline \multirow{8}{*}{5} & Cascade residual smoothing-MSSAN & 1.0105 & 1.4434 & 0.8533 \\ \cline{1-1}  & BiLSTM & 1.0049 & **1.4034** & **0.8597** \\ \cline{1-1}  & CNN-GRU & 1.2181 & 1.6798 & 0.8046 \\ \cline{1-1}  & CNN-Attention-LSTM & **0.9848** & 1.5126 & 0.8390 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of experimental results. This paper argues that the poor performance of the BiLSTM in bridge deflection prediction stems from its inability to comprehensively model nonlinear relationships and complex spatial features. While the CNN-GRU excels in extracting local features, bridge deflection prediction requires capturing multiscale features and modeling long-term dependencies. The CNN-GRU struggles to effectively capture these complex relationships, limiting its predictive accuracy. CNN-Attention-LSTM incorporates an attention mechanism to focus on important features but fails to preprocess the raw data effectively, leading to input data containing high-frequency noise and abnormal fluctuations. Compared to these methods, the method based on cascaded residual smoothing and an MSSAN proposed in this paper predicts better.