# A Data-Driven Model for Predictive Modeling of Vortex-Induced Vibrations of a Long-Span Bridge
## 1 Introduction
[12] proposed a machine learning-based method to predict the root-mean-square (RMS) value of the VIV response of prototype bridges where the support vector machine and a nonlinear autoregressive exogenous (NARX) architecture are employed. [PERSON] et al. [18] then proposed a recursive recurrent neural network model to predict the envelope of the VIV response of on-site long-span bridges. The aforementioned research focuses on the statistics of the VIV response (root-mean-square value and the envelope), where the prediction of the time history of the VIV response is lacking. The prediction of the time history can complement the frequency-domain characteristics and the phase characteristics, which can help the understanding of the mechanism of VIV events of full-scale bridges. Therefore, there is an urgent need to propose a model to predict the time history of VIV response. There are two main limitations of the existing machine learning-based models for handling the task. One is that existing models are black-box models, which lack interpretability, and the physical meaning of the middle output is often unclear. The black-box models have little effect on uncovering the underlying mechanisms hidden in the monitoring data. There are mainly three different approaches to embedding physical information in the machine learning model, which are introducing observational, inductive and learning biases [29; 30]. Introducing observational biases relies on sufficient data to cover the input domain, introducing learning biases is usually achieved by optimizing the loss functions of relevant physics equations and introducing inductive biases is achieved through the architecture design of the model. Given a full-scale bridge, sufficient input about the wind load and accurate modal and aerodynamic characteristics are not available. Therefore, introducing inductive biases is the viable approach. In this paper, we embed the Newmark-beta method inside the model to enforce the physical meaning of the middle output to enhance interpretability. The other limitation is that mean square error (MSE) is often employed to train the models, which treats all the responses as equally important. With the amplitude occupying a small part of VIV response, the model trained with MSE may not pay enough attention to amplitude and can have difficulty in predicting the amplitude accurately. In this paper, we propose a machine learning-based model for predicting the time history of vortex-induced vibration response, where gated recurrent unit (GRU) neural network is employed for its outstanding capacity to handle sequences. To enhance the interpretability of the model, the Newmark-beta method is introduced as a physics-based module inside the GRU neural network. With the introduced module, the physical meaning of the middle output can be predefined and discussed. A novel weighted mean square error (WMSE) loss function is employed to help the model put emphasis on the large vibrations to better predict amplitude. The remainder of this paper is as follows. The theoretical background of vortex-induced vibration, the Newmark-beta method and gated recurrent unit neural network are presented in Section 2, and details of the proposed model are also presented in Section 2. In Section 3, a description of the data that consist of VIV events of a long-span bridge is given. Experimental results and then the analysis and conclusion are presented in Sections 4 and 5, respectively.