# Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher
## 5 Multi-Objective Optimization Algorithm Example
Firstly, the greedy unsupervised learning algorithm is used to initialize the DBN. The input vector, as the input of the first restricted Boltzmann machine (RBM) visual layer in the DBN network, is trained by the contrastive divergence (CD) algorithm to obtain the connection weight and node bias vector of RBM1. After the training of RBM1, the output of its hidden layer is used as the input of the visible layer of RBM2 for training. It is carried out in sequence, i.e., starting from the input layer, each two layers is trained as an RBM, layer by layer. After all RBM training is complete, the entire DBN initialization is complete. Compared with the traditional method of the random initialization of artificial neural network weight parameters, this training method can effectively solve the problems of low training efficiency and falling into local optimal. After layer by layer training, the DBN output feature vector is used as the input of the deep neural network, so that the deep neural network can be trained quickly, effectively, stably, and reliably. Finally, the BP algorithm is used to train the DNN supervised to fit labeled data, and a DBN-DNN model is obtained. The Adam optimizer combines the advantages of Adagrad in dealing with sparse gradients and RMSprop in dealing with non-stationary targets, and it can calculate different adaptive learning rates for different parameters. In this article, the Adam optimizer is selected in the PyTorch library and the learning rate is set to 0.0001. The activation function employed here is the Rectified Linear Unit (ReLU). The prediction error for the outer layer proxy model can be found in Table 3.  
Upon testing, 100 data groups are randomly selected for a comparative analysis. The predicted outcomes of the outer proxy model are juxtaposed with the results derived from the optimization method. The comparison reveals a high degree of consistency between the predicted and actual values, with negligible instances of marked deviation. This robust correlation indicates the efficacy and predictive accuracy of the established outer proxy model. The prediction effect of the outer proxy model is shown in Figures 7 and 8. The single-objective PSO algorithm is deployed for internal optimization, while the multi-objective particle swarm optimization (MOPSO) algorithm is employed for external optimization. Here, the number of iterations is fixed at 200. Due to the nature of multi-objective optimization, it is difficult to find a single global optimal solution. More commonly, the process culminates in the acquisition of a series of Pareto optimal solution sets. The computational results corresponding to 50, 100, 150, and 200 iterations are visually represented in Figure 9. Notably, the disparity between the optimization values at 150 and 200 iterations and those at 100 iterations is minuscule. Additionally, the optimization values at 150 and 200 iterations are practically identical. This consistency suggests that the optimization results reach a state of convergence, indicative of the algorithm's effective convergence performance. Upon inspecting the distribution of points within the solution set, it appears to be quite uniform. The multi-objective evaluation function \(f_{1d}(X)\) reaches a minimum value of \(-1.71869\), concurrently with the multi-objective evaluation function \(f_{2d}(X)\) achieving a  
Figure 8: \(f_{2}(X,U)\) lower bound and \(f_{2}(X,U)\) upper bound. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Objective Function** & **MAPE/(\%)** & **MSE/(\(\times\)10)\({}^{-3}\)** & \(\mathbf{R^{2}}\) \\ \hline \(f_{1}\) lower bound & 1.04 & 2.11 & 0.99 \\ \(f_{1}\) upper bound & 1.12 & 2.28 & 0.99 \\ \(f_{2}\) lower bound & 1.36 & 3.21 & 0.99 \\ \(f_{2}\) upper bound & 1.27 & 3.04 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Outer proxy model prediction error. Figure 7: \(f_{1}(X,U)\) lower bound and \(f_{1}(X,U)\) upper bound. maximum value of 0.70619.