# Multi-Point Deformation Prediction Model for Concrete Dams Based on Spatial Feature Vector
## 1 Introduction  
Following the development of engineering technology, the number of dams in the world has gradually increased; meanwhile, the risk of dam failure has also been increasing. Deformation is the effect quantity that best reflects the operational safety status of dams. By analyzing the historical monitoring data of dams and establishing an accurate dam deformation prediction model, we can intuitively grasp the safety status of dams, which is important to ensure the stable operation of dams and is one of the research hotspots in the field of dam safety monitoring [1; 2; 3]. Currently, the commonly used dam deformation prediction methods can be classified into deterministic models, statistical models, and machine learning models. Deterministic models combine the actual structure and physical parameters of dams and dam foundations through the finite element method to obtain the evolution of dam deformation, which has high prediction accuracy and interpretability, but has the problem that some parameters are difficult to determine. The traditional statistical regression model is highly interpretable and has high operational efficiency, but it has insufficient ability to fit nonlinear relationships and is susceptible to multi-collinearity among its factors, resulting in poor model fitting [4]. Therefore, machine learning models that are more suitable for nonlinear relations and have stronger generalization ability are gradually introduced into the fieldof dam deformation prediction, such as [PERSON] et al. [5], who combined wavelet analysis, a particle swarm optimization algorithm and support vector machine (SVM) to construct a ReP-WSVM model for dam deformation prediction, which has a greater improvement in prediction accuracy compared with the traditional SVM model. [PERSON] et al. [6] proposed a Hydrostatic-Air temperature-Time (HT\({}_{\rm A}\)T) concrete dam prediction model based on Kernel Extreme Learning Machines (KELM); [PERSON] et al. [7] developed a dam deformation prediction model considering chaotic residuals using a BP neural network (BPNN) and frog jump algorithm. Compared with the statistical regression model, the machine learning model can better extract the nonlinear relationship between the causal factors and the deformation of the measuring point, and improve the prediction accuracy [8]. In addition, most of the traditional dam deformation prediction models only model the deformation of a single measuring point and do not consider the spatial correlation of the deformation of different measuring points, which makes it difficult to reflect the overall deformation state of the dam [9]. As the study progressed, the traditional single-point model is gradually developed into a multi-point model that can take into account the spatial and temporal features. For example, [PERSON] et al. [10] proposed a deterministic multi-point model for concrete arch dams using the finite element method; [PERSON] et al. [11] proposed a wavelet-based SSA-ELM dam deformation spatio-temporal prediction model to address the problem that multi-point statistical models are susceptible to multiple covariance and noise; [PERSON] et al. [9] combined the Finite Element Method(FEM) and SVM of particle swarm optimization to construct a hybrid multi-point model for concrete dams. Most of the existing multi-point models express the spatial characteristics between measuring points by introducing the spatial coordinates of measuring points, which has the following main problems: (1) Due to the complex causes of dam deformation, the spatial distribution pattern of the displacement values of each measuring point is often complex and uneven [12], which is difficult to accurately express by fitting multiple terms of spatial coordinates only. (2) The spatial correlation between multiple measuring points is not constant, but varies from moment to moment [13], and it is difficult to reflect this variability using static spatial coordinates of measuring points.