# Real-Time State Evaluation System of Antenna Structures in Radio Telescopes Based on a Digital Twin
## 4 Real-Time Simulation Construction of Antenna Physical Field
### Surrogate Model Training  
The structural performance of a radio telescope antenna has a large amount of data and takes a long time, which cannot meet the needs of real-time calculation and analysis. To solve the above problems and meet the real-time requirements of DT technology, the present study utilizes the structural performance data sample \(P2\) of the antenna's primary reflector, which was obtained in Section 4.2, as the data support. The machine learning RF regression model training data are employed to establish the surrogate model, thereby forming the mapping relationship between the input and output of the structural performance. The surrogate model is used to directly predict its structural performance, to improve the computational efficiency, and to realize the structural performance prediction of the antenna primary reflector under any working conditions. RF is an ensemble learning technique that consists of several decision trees, which can effectively capture the complex relationship between input and output responses. In this paper, the regression tree is utilized to achieve high-precision prediction by learning the mapping relationship between the input features and the mechanical response of the antenna model [23]. Figure 8: Point cloud data matching update flow chart. The flow of the RF regression algorithm is illustrated in Figure 9. Taking the structural deformation value of the primary reflector of the antenna as an example, based on the Section 4.2 data set, the bootstrap sampling method is used to form the subsequent training set through N times of sampling. The training set is set to \(D=\{(\mathbf{x}_{1},\mathbf{y}_{1}),(\mathbf{x}_{2},\mathbf{y}_{2}),\cdots,( \mathbf{x}_{N},\mathbf{y}_{N})\}\), where \(x_{i}=\{x_{i}^{(1)},x_{i}^{(2)},\cdots,x_{i}^{(n)}\}\) represents the input feature vector, and \(N\) is the sample size and represents the deformation value of each node of the antenna model under \(N\) working conditions. For the above data sets, the key is to select the appropriate partition points. The deformation value in the data set is divided into m nodes \(R_{m}\), and each node corresponds to an output prediction deformation value \(c_{m}\). The regression tree model can be represented as  
\[f(x)=\sum_{m=1}^{M}c_{m}I(x\in R_{m}) \tag{11}\]  
where \(I(x\in R_{m})\) represents the indicator function; that is, when \(x\) falls in the region of the node \(R_{m}\), the value is 1, otherwise it is 0. At each node, the split is chosen to minimize the sum of squared residuals. Therefore, the best \(c_{m}\) is the average of \(y_{i}\) in region \(R_{m}\). \[c_{m}=ave(y_{i}\mid x_{i}\in R_{m}) \tag{12}\]  
The selection of the optimal partition point is as follows: choose the \(j^{th}\) variable \(X^{(j)}\) and its value \(s\) as the segmentation variable and point, and define two regions as follows. Region 1: all data points satisfying \(X^{(j)}\leq s\), denoted as \(R_{1}\); region 2: all data points satisfying \(X^{(j)}>s\) are denoted as \(R_{2}\). Therefore, the sum of squared errors (SSE) for the candidate segmentation is calculated according to the following equation:  
\[SSE_{\text{total}}(j,s)=\min_{j,s}\left[\min_{c_{1}}\sum_{x_{j}\in R_{1}(j)}{(y _{i}-c_{1})}^{2}+\min_{c_{2}}\sum_{x_{j}\in R_{2}(j,s)}{(y_{i}-c_{2})}^{2}\right] \tag{13}\]  
Figure 9: Flow chart of the RF algorithm. the values \(j\) and \(s\) that minimize the sum of square errors of the two regions can be obtained. Here, \(c_{1}\) and \(c_{2}\) represent the established output values for the two regions after the division.