# Long-Term Structural State Trend Forecasting Based on an FFT-Informer Model
## 1 Introduction  
Infrastructure structures play a vital role in maintaining the well-being of people, protecting significant capital investments, and promoting regional and national prosperity [1; 2]. Structural health monitoring (SHM) [3; 4; 5], which collects a large amount of data that can reflect the states and changes of a structure, has attracted increasing attention in both academic and industrial communities to ensure the safe and reliable operation of infrastructure systems. Compared with short-term forecasting, long-term structural state forecasting is more focused on disaster early warning and damage repair for large infrastructures that actually is an LSTF task. However, infrastructure structures suffer from diverse types of potential damages during their service lives [6; 7; 8]. The nonlinear change process of structural state greatly increases the difficulty of structural state prediction. Most machine learning forecasting methods have been widely applied in structural state forecasting. [PERSON] et al. [9] established an incremental Bayesian matrix/tensor learning scheme to achieve efficient prediction of structural response. [PERSON] [10] applied ConvLSTM for three-year structural response forecasting. [PERSON] [11] predicted the structural response using a backpropagation neural network (BPNN) method. [PERSON] [12] applied an encoder-decoder architecture with gated recurrent unit (GRU) and long short-term memory (LSTM) for bridge response forecasting. [PERSON] [13] proposed a novel deep recurrent neural network(RNN) model for long-term prediction. [PERSON] [14] proposed a Bayesian modeling approach embedding model class selection for dynamic forecasting purposes. However, the trend of structural state forecasting is an LSTF task, while these methods in a step-by-step manner cannot predict the long-term changing of structural state efficiently. Although some statistical regression models [8; 15; 16; 17] fully consider various constraints in specific situations, in order to achieve good results, these methods must fully consider the professional knowledge and influence factors of the prediction object, which is impossible to comprehensively consider under limited data collection conditions. As the application of various prediction models continues in various domains, such as energy [18], economics [19], and disease propagation analysis [20], researchers are painy more attention to the LSTF problems, which analyze a large amount of time series data and capture its potential relevance to make long-term forecasts. In order to successfully apply machine learning to an LSTF task, it is crucial to capture long-term dependence between input and predicted values. Recently,a Transformer model [21] with an encoder-decoder system has shown better performance in obtaining long-range dependency than recurrent neural network (RNN) models [22; 23]. It brings the analysis, processing, and prediction of sequential data into a new stage. The Informer model [24], a new kind of LSTF model based on Transformer, is able to forecast long sequences in a one-forward operation instead of in a step-by-step manner, which achieves higher inference speed of prediction by introducing an encoder-decoder system. Meanwhile, the Informer model has lower computational complexity with the Multi-head ProbSparse self-attention mechanism compared to Transformer, which has been applied for LSTF tasks in many domains, such as wind power [25] and load [26]. By simply employing the Informer model in long-term structural state forecasting, however, it is easy to fall into the problem of invalid attention mechanism and an unobvious forecasting trend because of the complex characteristics of sensing data. Especially, due to the error of data collection and the influence of external factors, the deviation of some data will not affect the development of the data overall trend. Because the Multi-head ProbSparse self-attention focuses on the deviation point, this particularly affects the efficiency and accuracy of prediction.