# Performance Optimization Design of Diagonal Flow Fan Based on Ensemble of Surrogates Model
## 4 Ensemble of Surrogate Model and Multi-Objective Optimization
### Ensemble of Surrogate Model  
We built the sample space using optimal Latin hypercube sampling. In order to complete the fitting of the approximate model at a lower number of samples and form the response function of design variables and target parameters of the diagonal flow fan blades, an ensemble of surrogates model was formed by the combination of the radial basis function (RBF) and Kriging (KRG). The Ensemble of surrogates, also known as the weighted average surrogate model, is generally expressed as:  
Figure 8: Analysis of arc fitting error in blade. \[\hat{y}_{e}(x)=\sum\limits_{i=1}^{M}\omega_{i}(x)\hat{y}_{i}(x) \tag{10}\]  
In the formula, \(\hat{y}_{e}(x)\) is the predicted value of the Ensemble of surrogates model; \(\hat{y}_{i}(x)\) and \(\omega_{i}(x)\) are the predicted value and weight coefficient of the \(i\)th single model, respectively, and \(\sum\limits_{i=1}^{M}\omega_{i}(x)=1\). First of all, the weight coefficient of a single model needs to be determined to construct an Ensemble of surrogates model. By assigning weights to each single model, the adverse effects of poor-performance models can be further reduced. The heuristic weight coefficient algorithm proposed by [PERSON] et al. (2017) is introduced here, and its operation formula is:  
\[\omega_{i}=\frac{\omega_{i}^{*}}{\sum\limits_{j=1}^{n}\omega_{j}^{*}} \tag{11}\] \[\omega_{i}^{*}=\left(E_{i}+\alpha E_{avg}\right)^{\beta}\] \[E_{avg}=\frac{\sum\limits_{j=1}^{n}E_{j}}{n}\]  
where \(E_{i}\) is the generalized mean square error of the \(i\)th surrogate model, \(\alpha\) (\(\alpha\) < 1) controls the mean value of the surrogate model, \(\beta\) (\(\beta\) < 0) is used to adjust the influence of the single surrogate model on the Ensemble of surrogates model, and the recommended values are \(\alpha\) = 0.05, \(\beta\) = \(-1\). We used the optimal Latin hypercube experimental design to obtain leaf sample space over a range of design variables. We set an appropriate number of training samples and test samples by considering the prediction accuracy of the approximate model and computation time cost. The root mean square error between the number of samples of different surrogate models and the corresponding output values is shown in Table 1.  
Generally, there is a great relationship between the fitting accuracy of the surrogate model and the number of samples. More samples can ensure higher fitting accuracy. By comparing the prediction mean square error of the corresponding output values of different surrogate models under different training sample numbers, it can be seen that the fitting accuracy of the Kriging model increases with increasing sample size. The RBF model has obvious advantages in the lower number of samples. The Ensemble of surrogates model has the advantages of both models and has higher prediction accuracy under the same number of samples. Figure 9 shows the Ensemble of surrogates model's test set error. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Delegation Model** & **Number of Training Samples** & **Efficiency MSE** & **MSE Flow MSE** \\ \hline RBF & 40 & 50.183 & 12.542 \\ RBF & 50 & 40.127 & 10.229 \\ RBF & 60 & 34.982 & 7.732 \\ RBF & 70 & 28.212 & 4.908 \\ Kriging & 40 & 70.234 & 19.127 \\ Kriging & 50 & 35.235 & 8.394 \\ Kriging & 60 & 12.175 & 2.598 \\ Kriging & 70 & 4.536 & 0.740 \\ ES & 40 & 17.236 & 10.332 \\ ES & 50 & 4.556 & 3.194 \\ ES & 60 & 1.742 & 0.992 \\ ES & 70 & 0.125 & 0.032 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of sample size and output response MSE for different surrogate models. The model prediction deviations of the Kriging model, RBF model, and Ensemble of surrogates model under 70 training samples are shown in Figure 10. The mean square error MSE of each model under test samples (five groups) is selected to characterize the overall error.