## 3 Example Applications and Results
### Application of SMPO  
Before being imported into the surrogate model for training, the data need to be normalized to eliminate the effects of variables of different orders of magnitude on the training results. The present paper adopts most-valued normalization to map the original data to the range of \([-1,1]\) according to Equation (2). The data distributions before and after the normalization are shown in Figure 5. It is observed that the normalized data distribution is more concentrated and contains a smaller difference between the variables, which are conducive to accelerating the convergence of the update of the network training weights. The normalized data are imported into the GAALBP network, with 90% of the data used for training and 10% for testing. Eleven parameters are optimized as inputs and the predicted temperature of the CMOS is taken as the output. Hyperparameters of the network are set as presented in Table 4. Hyperparameters are mainly divided into GAALBP network hyperparameters and hyperparameters of the GA that optimize the initialization weights and thresholds of the network. The training process and results are shown in Figures 6 and 7, where it is seen that, the network training regression rate reaches 0.9999 and the maximum prediction error (the difference between the predicted value and true value) of the network is less than \(4\times 10^{-3}\). The adaptive learning rate during training is shown in Figure 8. At the beginning of training, the learning rate continually increases, indicating that the error is gradually decreasing. In the later stages of training, the learning rate fluctuates around a value of 0.2, indicating that the training error has increased. However, the learning rate displays an overall decreasing trend, indicating that the training error decreases. The training effects of the adaptive learning rate and the improved BP network are compared for constant learning rates of 0.01, 0.1, 0.2, and 0.3, as shown in Figure 9. The test error for the adaptive learning rate is two orders of magnitude smaller than the test errors for the constant learning rates when the hyperparameters are the same, demonstrating the superiority of the adaptive learning rate. Figure 4: CMOS heat transfer path. Figure 5: Comparison of the distribution of data before and after normalization ((a). Data before normalization; (b). Data after normalization). Figure 6: Network training regression rate. Figure 7: Network prediction error. The quality of the surrogate model directly determines how well the parameters are optimized in the subsequent step. Comparisons are made with the traditional BP network, GA-optimized BP (GABP) network, and radial basis function neural network [25] to verify the superiority of GAALBP. The training results are presented in Table 5 and Figure 10. The GABP network is a BP network with a GA-optimized constant learning rate; i.e., the only difference between the GABP and GAALBP is the difference in the learning rate variation. It is seen that the training error of GAALBP is 1% of that of the traditional BP network and 40% that of the GABP network whereas the mean square error of GAALBP is smaller than the mean square errors of the other networks. The results thus demonstrate the superiority of GAALBP. After establishing the surrogate model, the GA is used to optimize the input of the surrogate model to meet the thermal control index of the CMOS. The negative of the absolute value of the difference between the target temperature and predicted temperature is taken as the fitness value of GA training:  
\[fit=-\Big{|}T_{goal}-T_{pre}\Big{|} \tag{5}\]  
where \(T_{goal}\) is the target temperature and \(T_{pre}\) is the predicted temperature. A larger fitness value means that the predicted temperature is closer to the target temperature, meaning that the parameter optimization is better.