## 3 Theoretical Derivation of Models and Generalized Modeling for Rapid Prediction
### Theoretical Derivation of Agent Model Based on PSO-SVR
#### 3.1.3 SVR Forecasts  
SVR prediction refers to extrapolating the angular coefficients using the SVR model built during the training phase. The SVR training is completed to obtain the objective function:  
\[f(\mathbf{x})=(\mathbf{w}\cdot\varphi(\mathbf{x}))+b \tag{8}\]  
The results of the solution can be brought in to obtain a specific expression for \(f(x)\) as  
\[f(\mathbf{x})=\sum_{i=1}^{l}\beta_{i}K(\mathbf{x},\mathbf{x}_{i})+b \tag{9}\]  
where \(\beta_{i}\) is the coefficient and the correspondence with the training stage parameters is \(\beta_{i}=\alpha_{i}-\alpha_{i}^{*}\). The input to the SVR model for prediction is the infrared heating cage parameter, and the output is the angular coefficient corresponding to this parameter. Generalized Model for Rapid Prognostics of Heat Flow for Different Infrared-Heated Cage Configurations  
After determining the theoretical expression of the rapid prognostic model, it is necessary to train the model with known samples to determine the relevant parameters of the model that can eventually be used for the rapid prognostic of the heat flow of the infrared heating cage. To this end, the Monte Carlo method is used to simulate the infrared cage base model described in Figures 1 and 3, calculate the angular coefficients of typical spacecraft structures, and train the prediction model with the relevant parameters and angular coefficients under different configurations as inputs. Both the Monte Carlo simulation and the PSO-SVR method were written using MATLAB, and the tests were carried out on the same computer, which was configured with a CPU of i5 12400F and 32 G of RAM. Combined with the actual use of an infrared heating cage and to ensure the generality of the rapid prognostic model, the relevant parameters of the three base models shown in Figures 1 and 3, namely, plane, cylinder, and circular truncated cone, are taken as follows:  
* The spacing \(L\) takes the values [100, 120, 150, 180, 200, 250, 300, 350, 400] mm;
* The height \(H\) takes the values [200, 400, 600, 800, 1000] mm;
* The small surface arc length \(S1\) and the large surface arc length \(S2\) take the values [200, 400, 600, 800, 1000] mm and \(S1\leq S2\);
* The small-surface curvature \(K1\) and the large-surface curvature \(K2\) take values of [0, 0.001, 0.002, 0.003, 0.004, 0.005] mm\({}^{-1}\);
* The circumferential bias \(B\)s takes the values [0-2] and the radial bias Bh takes the values [0-2]. The Monte Carlo heat flow simulation was carried out by selecting the relevant parameters in the above data according to the three configurations of the planar, cylindrical, and circular truncated conic; a total of 12,000 combinations were solved, and their corresponding results were obtained. The simulation results of heat flow distribution in typical partitions under the three configurations are shown in Figure 5, and the angular coefficients of the infrared heating cage partition and the spacecraft surface partition can be obtained by calculating the ratio of the cumulative value of the heat flow distribution in the partition and the value of the heat flow emitted. Figure 5: _Cont._  
The 12,000 sets of data are randomly arranged, and 850 sets of them are randomly selected as the training set, 2000 sets of data as the validation set, and 10,000 sets of data as the test set. The test set is used to evaluate the generalization ability of the die final model. The process shown in Figure 4 is carried out for the rapid prognostic model training. The support vector regression method performs parameter learning on the training set and evaluates the learning effect on the validation set. The particle swarm optimization algorithm generates new parameter combinations based on the learning effect where the learning effect is evaluated using Root Mean Square Error (RMSE) as an evaluation index.