# Multi-Point Deformation Prediction Model for Concrete Dams Based on Spatial Feature Vector
## 2 2D-1D-CNN Deformation Prediction Model for Concrete Dams
### 2D-1D-CNN Concrete Dam Deformation Prediction Model Construction Process  
The literature [22] pointed out that the spatial correlation between the measuring point deformation sequences is manifested by the correlation between the change patterns of the displacement values of each measuring point within the same period, and, on this basis, the spatial correlation of the measuring point deformation is quantified by the time series similarity. In this paper, based on work in the literature [22], a 2D-1D-CNN model is constructed by combining the idea of spatio-temporal data modeling and extracting the spatial features among measuring points through convolutional neural networks. The model consists of three parts, 2D-CNN, 1D-CNN and a tensor calculation layer. Among them, the 2D-CNN part extracts the spatial feature vector \(\mu\), which can reflect the spatial features of each moment, from the historical deformation sequence of multiple measuring points, and replaces the spatial coordinates in the traditional multiple measuring point model with it. The tensor calculation layer refers to the expression of the traditional multi-point model and combines the spatial feature factor with the causal factors of the dam deformation to obtain the final input factors. Considering the advantages of 1D-CNN in sequence prediction with fewer parameters and higher prediction accuracy [23], 1D-CNN is selected in this paper to construct the nonlinear relationship between input factors and measuring point deformation to obtain the final prediction results. The specific structure of the 2D-1D-CNN model is shown in Figure 1.  
Figure 1: Schematic diagram of 2D-1D-CNN model. The model expression is shown in Equation (7). \[\delta= f_{1}[f(H),f(\mu)]+f_{2}[f(T),f(\mu)]+f_{3}[f(\theta),f(\mu)] \tag{7}\] \[= \sum_{i=1}^{4}\sum_{p=1}^{k}A_{ilmn}a_{i}H^{i}\mu_{p}+\] \[\sum_{i,j=0}^{1}\sum_{p=1}^{k}B_{ilmn}\sin\frac{2\pi i\Delta t}{36 5}\cdot\cos\frac{2\pi j\Delta t}{365}\mu_{p}+\] \[\sum_{i,j=0}^{1}\sum_{p=1}^{k}C_{ilmn}\theta_{i}\ln\theta_{j}\mu_{p}\]  
where \(k\) is the dimension of the spatial feature vector, which belongs to the hyper parameters of the model. The specific steps for constructing the model are as follows:  
1. Data set construction As shown in Figure 2, each measuring point is constructed into a spatio-temporal panel data in the form of a grid according to its relative spatial position, with each small square representing the relative position of a measuring point, followed by filling the displacement values of each measuring point into the corresponding square, and filling the part of missing measuring points with 0. According to the above process, the deformation sequence of measuring points on the dam is constructed into a three-dimensional matrix of dimension \(X_{h}\in R^{51\times S2\times t}\) as the panel data input of the model, where \(S1\), \(S2\), represent the horizontal and vertical numbers of measuring point distribution, and \(t\) denotes the length of the time window. According to Equation (2), this paper takes \(H,H^{2},H^{3},H^{4}\), \(\sin\frac{2\pi\Delta t}{365}\), \(\cos\frac{2\pi\Delta t}{365}\), \(\sin\frac{4\pi\Delta t}{365}\), \(\cos\frac{4\pi\Delta t}{365}\), \(\theta,\ln\theta\) as the environmental casual factors of dam deformation, and constructs sliding windows in the time dimension for data addition, followed by the REPEAT function. The environment casual factor is replicated and filled n times (\(n\) is the number of measuring points), and, finally, the casual factor input \(X_{f}\)\(\in R^{1\times n\times 10}\) is obtained. 2. Spatial feature vector extraction The spatio-temporal panel data \(X_{h}\in R^{51\times S2\times t}\) in (1) is input to the 2D-CNN, and the spatial feature vector \(\mu\)\(\in R^{t\times n\times k}\), which can reflect the spatial features at the current moment, is finally obtained through convolution calculation, and the calculation process is shown in Equation (8).