## 2 Methodology of SMPO  
The methodology of SMPO involves building a surrogate model using GAALBP and using a GA to find the best parameters, as shown in Figure 1.  
Part I: GAALBP network training  
This paper proposes an improved BP neural network. The improved network uses a GA to optimize the initial weights and thresholds of the network, the best individuals in the population are selected in a winner-takes-all manner [22], and the coding information of the best individuals is used for the initial weights and thresholds of the network. Therefore, before the training of the network, the GA coding length and fitness value need to be calculated and data preprocessing conducted. The calculations are as follows:  
(1) Calculation of the encoding length. The length \(S\) of the GA encoding is derived from the network topology determined by the feature dimensions of the input and output and the numbers of layers and nodes of the hidden layer. The calculation is  
\[S=(n_{in}+1)\times n_{1}+\left(\sum\limits_{i=1}^{N-1}((n_{i}+1)\times n_{i+1} )\right)+(n_{N}+1)\times n_{out} \tag{1}\]  
where \(n_{in}\) is the number of neurons in the input layer, \(N\) is the number of layers in the hidden layer, \(n_{i}\) is the number of neurons in the hidden layer (\(i\) = 1, 2,..., \(N\)), and \(n_{out}\) is the number of neurons in the output layer. (2) Calculation of the fitness value. In the training phase of the network, the mean square deviation of the predicted values fitted to the network from the true values is taken as the fitness value of the GA. The maximum fitness value is zero when \(T_{real}\) and \(T_{pre}\) are equal. The calculation is  
\[fitness=-\frac{\sum\limits_{i=1}^{N}\left|T_{reali}-T_{prei}\right|}{N} \tag{2}\]  
where \(T_{prei}\) is the predicted temperature, \(T_{reali}\) is the true temperature, and \(N\) is the number of test samples. Figure 1: SMPO flow chart. (3) Data preprocessing. The input data are normalized to eliminate the effect of variables of different orders of magnitude on the network training. Here, the optimal normalization method [23] is used to map the data to the range \([-1,1]\). The calculation is  
\[y=\frac{2\times(x-x_{\text{min}})}{x_{\text{max}}-x_{\text{min}}}-1 \tag{3}\]  
where \(x_{\text{min}}\) is the minimum value for the same dimension, and \(x_{\text{max}}\) is the maximum value for the same dimension. The optimized weights and thresholds are assigned to the BP network, and the input data are used to train the network. During the training process, the learning rate changes adaptively with the relative change in the error, bearing the aim of keeping the learning stable while maintaining the largest possible learning step. If the error increases, a smaller learning rate is used in continuously searching for the direction of the gradient descent, and if the error increases by more than a certain percentage, the weights and thresholds of the previous round are discarded and the learning rate is reduced. This process improves the learning rate, however, when the learning rate is too high and the error reduction is not guaranteed, the learning rate is reduced until stable learning is restored. The improved BP network compensates for drawbacks such as the fixed learning rate of the BP network and can be trained to obtain a better surrogate model. The rule for correcting the learning rate with error is determined by manual debugging, and the method of adaptively adjusting the learning rate is shown in Figure 2. If the error in the current round increases by more than a factor of 1.04 relative to that in the previous round, the weights and thresholds of the current round are discarded, and the weights and thresholds in the next round are calculated using the values of the previous round, and the learning rate is reduced by a factor of 0.7. If the error in the current round is higher than that in the previous round but by less than a factor of 1.04, then the current weights, thresholds, and learning rate are retained.