# A Data-Driven Model for Predictive Modeling of Vortex-Induced Vibrations of a Long-Span Bridge
## 2 Methodology
### The Proposed Model  
With unknown modal and aerodynamic characteristics and the corresponding aerodynamic force of a full-scale bridge, the VIV model cannot be employed to predict the VIV response. Therefore, there raises a need to employ a data-driven model as an alternative. However, the usual black-box data-driven model lacks interpretability, which demands a physics-informed architecture to enhance the physical meaning. In this paper, we propose to embed the Newmark-beta method which is a numerical method for solving dynamic equations inside the recurrent calculation of a GRU neural network, by which the GRU neural network outputs the estimated modal characteristics and the aerodynamic force and the Newmark-beta method calculates the corresponding response according to Equations (2)-(9). The diagram of the proposed model is shown in Figure 2a. As shown in Figure 2a, two GRU neural networks and four fully connected (FC) neural networks are employed in the proposed model. One GRU neural network is employed to generate the hidden-state vector of modal frequency and modal damping, and the other GRU cell is employed to generate the hidden-state vector of normalized aerodynamic force. For the GRU neural network for modal frequency and modal damping (denoted as GRU\({}_{2}\)), only the time histories of displacement and velocity are selected as the input. The reason behind this design is that modal characteristics are not influenced by wind characteristics, and dynamic adjusting is employed to avoid error accumulation. A fully connected neural network (denoted as FC\({}_{2-2}\)) is employed to postprocess the hidden-state vector of GRU\({}_{2}\) to output the modal frequency and modal damping. Another fully connected neural network (denoted as FC\({}_{2-1}\)) is employed to generate the initial hidden-state vector of the GRU neural network for the modal characteristics instead of a zero vector to avoid system error. The input of FC\({}_{2-1}\) is the first few measurements of acceleration, velocity and displacement. For the GRU neural network for normalized aerodynamic force (denoted as GRU\({}_{1}\)), the time histories of displacement, velocity and wind characteristics (such as wind speed at different sections) are selected as the input. The reason behind this design is that normalized aerodynamic force is influenced by not only wind characteristics but also the dynamic response of the bridge girder, and dynamic adjusting is employed to model the dynamic evolution of normalized aerodynamic force. A fully connected neural network (denoted as FC\({}_{1-2}\)) is employed to postprocess the hidden-state vector of GRU\({}_{1}\) to output the normalized aerodynamic force. Another fully connected neural network (denoted as FC\({}_{1-1}\)) is employed to generate the initial hidden-state vector of GRU\({}_{1}\) instead of a zero vector to avoid system error. The input of FC\({}_{1-1}\) is the first few measurements of acceleration, velocity and displacement. Figure 1: The diagram of the GRU cell. The calculation procedure inside the proposed model is illustrated in Figure 2a,b. Firstly, FC1-1 and FC2-1 are used to produce initial states for GRU1 and GRU2. Secondly, GRU1 and GRU2 can be used to produce hidden states, which can be postprocessed with FC2-1 and FC2-2 to obtain estimated normalized force and modal characteristics. Thirdly, the Newmark-beta method is performed to calculate the VIV response. During sequential prediction, GRU1, GRU2, FC2-1 and FC2-2 are all performed multiple times, while FC1-1 and FC2-1 are only performed in the initial state. In the proposed model, the direct output of the neural networks is the modal characteristics and normalized aerodynamic force, and the dynamic response is the indirect output. Instead of outputting the dynamic response directly, the relationship among dynamic responses that can help prevent error accumulation can be utilized, and the interpretability of the deep learning-based model can be enhanced.