# A Data-Driven Model for Predictive Modeling of Vortex-Induced Vibrations of a Long-Span Bridge
## 4 Experimental Results and Analysis  
A machine learning-based model that is presented in Section 2.4 is built for predicting the dynamic response of vortex-induced vibrations of the investigated bridge. The specifics of the employed model are presented as follows. For gated recurrent unit neural networks, the dimensions of the hidden-state vectors are both 32. For fully connected neural networks for generating initial hidden-state vectors, the first five responses (including acceleration, velocity and displacement) are selected as the input. Therefore, the input dimensions of these two fully connected neural networks are 15, and the corresponding output dimensions are 32. For fully connected networks for generating normalized force and modal characteristics, their output dimensions are 1 and 2, respectively. The choice of the length of the predicted dynamic response is a tradeoff between error accumulation and capturing the underlying patterns of the evolution of vortex-induced vibration. With a larger length, the model may suffer error accumulation, and the computational consumption can be demanding. With a small length, the model may find it difficult to capture the evolution of vortex-induced vibration over time. In this paper, the length of the predicted dynamic response is 60 (30 s). The model is trained using the adaptive moment estimation [35], while L2 regularization is introduced to help prevent overfitting. It should be noted that because of the proposed loss function, the model may put more emphasis on the acceleration, and the prediction accuracies of velocity and displacement response may be degraded. However, with the accurate prediction of the acceleration, accurate predictions of velocity and displacement can be obtained by integration in the frequency domain. Therefore, the acceleration response is the main focus of this paper, and the prediction results of the acceleration response are presented to illustrate the effectiveness of the proposed model. The learning curve of the proposed model is shown in Figure 7. As shown in Figure 7, the training curve is convergent, which indicates that the training process is accomplished. The performance of the trained model on the training set and the testing set is summarized in Table 2. The values of MAE, RMSE and WMAPE indicate that the prediction error is small compared to the amplitude of the acceleration response of the vortex-induced vibration, which validates that the trained model has the capacity to predict the amplitude accurately. Both the PCCs on the training set and the testing set are close to 1, which indicates that the trained model has the capacity to predict the phase accurately. Typical predictions are presented in Figure 8. As shown in Figure 8, acceleration responses of vortex-induced vibration events of various shapes and amplitudes can be predicted accurately, which indicates the effectiveness and the practicability of the trained model. Take typical prediction III as an example: the trained model can predict the amplitude of the vortex-induced vibration of any stage accurately. The reason behind this phenomenon is the introduction of the weighted loss function. By introducing the weight, the model can put more emphasis on the large amplitude and prevent the problem of failing to predict the vortex-induced vibration of large amplitude inaccurately. It can also be seen that the evolution pattern of the phase of the measured acceleration is consistent \begin{table}
\begin{tabular}{c c c c} \hline  & **Training Set** & **Validation Set** & **Testing Set** \\ \hline MAE & 1.715 & 1.941 & 1.567 \\ RMSE & 2.819 & 2.791 & 2.708 \\ WMAPE & 0.148 & 0.137 & 0.144 \\ PCC & 0.981 & 0.987 & 0.982 \\ \hline \end{tabular}
\end{table} 
Table 2: Performance of the proposed model on the training set and the testing set. Figure 7: The learning curve of the proposed model. with that of the predicted acceleration.