## 3 Real Application Case
### Model Establishment and Verification  
First of all, this paper obtains 1058 sets of available data. To make the data have same range and promote the network convergence, the original sample data are normalized by \((x-x_{\text{min}})/(x_{\text{max}}-x_{\text{min}})\), where \(x\), \(x_{\text{min}}\), and \(x_{\text{max}}\) denote the original data, the minimum and maximum values of the original data, respectively. Next, the MGWO algorithm is utilized to search the number of the hidden nodes, the learning rate and the momentum coefficient of the DBN in parallel. In the MGWO algorithm, the population size is set to 20 and the maximum number of iterations is set to 50. The DBN network adopts a classic four-layer structure, including an input layer, a first hidden layer (\(H_{1}\)), a second hidden layer (\(H_{2}\)), and an output layer. In the DBN, the maximum number of RBM iterations is set to 50; the maximum number of BP neural network iterations is set to 100; the training error threshold is set to 0.02. The analytic space of the number of hidden layer nodes is set between 0 and 500, and the analytic spaces of the learning rate and the momentum coefficient are set between 0 and 1. Figure 5 shows the distribution of grey wolf population in the MGWO optimization process for the hidden nodes. The grey wolf population is able to obtain the information about the solution during the search process. Through the surrounding, hunting, and attacking operations, the grey wolf population gradually gathers into the optimal solution area. In the experiment, the initial population of the MGWO algorithm is randomly distributed in the analytic space. As the iteration increases, the grey wolves approach the optimal solution step by step. After eight iterations, the MGWO algorithm finds the optimal solution of the DBN, that is, the number of \(H_{1}\) nodes is 6, the number of \(H_{2}\) nodes is 5, the learning rate is 0.077 and the momentum coefficient is 0.807. The DBN with these searched parameters is used to learn the training samples. The establishment process of the DBN can be divided into two steps. The first step is to train each RBM separately. This is an unsupervised process, ensuring that feature information is preserved as much as possible when feature vectors are mapped to different feature spaces. The second step is to fine-tune the weights and biases of the network. The BP neural network takes the output feature vector of the RBM as its input feature vector and trains the whole network in a supervised manner. After the MGWO optimized DBN (MGWODBN) model is established, there are two ways to verify the model, as follows:  
1. MGWODBN predicts all trained data, and the linear fitting equation of the observed and predicted values is obtained as \(y=1.117x-7.947\), where \(y\) is the actual observed value; \(x\) is the predicted value of the model. The root mean square error (RMSE) is 18.532 \(\upmu\)g/m\({}^{3}\), and the coefficient of determination (R\({}^{2}\)) is 0.713. Figure 6a shows the verification results. It can be seen that the sample points are roughly distributed on both sides of the diagonal line and are more aggregated, indicating that the model has a better fitting effect. 2. Cross-validation--90% data are randomly selected to train the model, and the remaining 10% data are used as the verification points. Repeated 10 experiments showed that the linear fitting equation of the observed and predicted values is obtained as \(y=1.200x-11.710\). The RMSE is 19.815 \(\upmu\)g/m\({}^{3}\), and the R\({}^{2}\) is 0.677. Figure 6b shows the verification results. It can be seen that the