# A Data-Driven Model for Predictive Modeling of Vortex-Induced Vibrations of a Long-Span Bridge
## 4 Experimental Results and Analysis
In addition, the frequency-domain characteristics of the predicted response are consistent with those of the measured response, which indicates that the trained model can also predict the frequency-domain characteristics accurately. By predicting the time history of the VIV response of a full-scale bridge, the proposed model can provide more details about the VIV event than predicting the RMS value or the envelope of the response. It should be noted that the acceleration response is predicted piecewise, which is 30 s by 30 s. The trained model can be utilized to perform a long-time prediction, but the error accumulation may degrade the accuracy. A future improvement is to enhance the long-time prediction ability. Due to the fact that there are no similar approaches for predicting the time history of VIV response, the novelty of the proposed model is discussed as follows. First of all, the interpretability of the proposed model is discussed by analyzing the middle output of the trained model. The middle outputs of typical prediction I are selected as an example and are presented in Figure 9. As shown in Figure 9, the middle outputs have a sudden change. The main reason behind this phenomenon is that the generated initial hidden-state vectors are not close enough to the ideal initial hidden-state vectors, and the inaccurate initial vector may cause the deviation of the modal characteristics and normalized force from the real ones. However, with the trained model handling the sequence, the generated modal characteristics and normalized force stabilize and represent some patterns. For the predicted frequency, it can be seen that it lies in the range between 0.2 and 0.3 Hz, which is deviated from the real modal frequency of the vortex-induced vibration. For the predicted  
Figure 8: Typical predictions: (**a**) typical prediction I; (**b**) typical prediction II; (**c**) typical prediction III. damp ratio, it is convergent to zero, which is deviated from the real damp ratio. For the predicted force, it oscillates near zero. With the predicted frequency and predicted damp ratio deviating from the real values, it is likely that the predicted force also deviates from the real value. It can be concluded that the physical meaning of the middle output is not consistent with its pre-designed role. The reason behind this phenomenon is that no physical restrictions other than response measurements are put on these middle outputs. Therefore, the individual role of these middle outputs can be mispositioned. More physical restrictions are needed to enhance the interpretability of the proposed model and help the understanding of the mechanism of the vortex-induced vibration of full-scale bridges, which is another future improvement. Secondly, the effectiveness of the weighted mean square error loss function is validated by comparing the prediction accuracy of the model trained by optimizing WMSE and that of the model trained by optimizing mean square error (MSE). It should be noted that these two models are of the same structure, and the key difference is the loss function. The summary of accuracies is shown in Table 3 and a comparison of typical VIV events is shown in Figure 10.  
Figure 9: Middle outputs of typical prediction I: (**a**) predicted force; (**b**) predicted frequency; (**c**) predicted damp ratio. As shown in Table 3, the prediction accuracy of the model trained by optimizing WMSE is higher than that of the model trained by optimizing MSE, especially for WMAPE which put more emphasis on the amplitude. As shown in Figure 1, the model trained by optimizing WMSE can predict the VIV event of large amplitude more accurately than the model trained by optimizing MSE. Therefore, it can be concluded that the WMSE can aid in the accurate prediction of the amplitude of a VIV event. The advantages of the proposed model are as follows. First of all, the proposed model can predict the time history of VIV response, with which both the time-domain characteristics and the frequency-domain characteristics can be predicted and analyzed.