## 2 Methodology
### Surrogate Model Based on the Lifetime Dataset  
In order to realize the on-line RUL prediction, a machine learning surrogate model was established instead of the numerical simulation model. Figure 7 illustrates the framework of the RUL prediction model. Based on the numerical results obtained from the thermal-fluid-structural coupled calculations, several critical operational parameters were selected as inputs for the surrogate model. The inputs were the speed \(n_{1}\), blade inlet pressure \(P_{3}\), blade outlet pressure \(P_{34}\), blade outlet temperature \(T_{34}\), and fuel mass flow rate \(Qm_{t}\). The output was EOH for each combination of operating conditions. Using operational data from the turbine unit, machine learning models like Backpropagation (BP), Long Short-Term Memory (LSTM), and Deep Neural Network (DNN) were used to automatically detect blade life features.  
The BP neural network model is one of the most basic neural network models, and has been proven to be useful in various applications [33, 34]. Nevertheless, some limitations exist with BP, such as susceptibility to overfitting and incapability of handling sequential data. Therefore, it is worth exploring other advanced algorithms such as DNN and LSTM to enhance the performance of lifetime prediction. DNN can explore the complex nonlinear data relationships deeply and autonomously learn to extract crucial features from data. It is particularly suitable for the large-scale datasets [35, 36]. LSTM is good at capturing patterns and trends in time-series data. This is crucial for accurately predicting the lifetime of turbine blades [37, 38]. The MATLAB's general-purpose toolkits were employed to establish the machine learning model. The BP, LSTM, and DNN models were implemented through MATLAB's neural network and deep learning toolbox.  
Figure 6: Calculation flow for the EOH of turbine blades.  
Figure 7: Surrogate model for life prediction.  
To assess the accuracy of the developed machine learning surrogate model, several specific evaluation criteria were introduced. The criteria generally included the coefficient of determination (R\({}^{2}\)), root-mean-square error (RMSE), and mean absolute error (MAE), as shown in Equations (13)-(15), respectively.  
\[R^{2}=1-\frac{\sum_{i=1}^{k}\left(y_{i}-\widetilde{y}_{i}\right)^{2}}{\sum_{i= 1}^{k}\widetilde{y_{i}}^{2}} \tag{13}\]  
\[RMSE=\sqrt{\frac{1}{k}{\sum_{i=1}^{k}\left(y_{i}-\widetilde{y}_{i}\right)^{2}}} \tag{14}\]  
\[MAE=\frac{1}{k}{\sum_{i=1}^{k}\left|y_{i}-\widetilde{y}_{i}\right|} \tag{15}\]  
where \(k\) is the number of samples, \(y_{i}\) is the expected value, and \(\widetilde{y}_{i}\) is the predicted value.  
The methodology proposed in this paper enabled rapid RUL prediction, as well as considering the effect of different operation conditions. It covered the whole process, from the 3D reconstruction of actual turbine blades to the final RUL calculation results, which is of great significance for practical engineering applications and maintenance plans. However, the proposed methodology was highly reliant on offline calculations. Therefore, it is necessary to collect ANSYS calculation results under various working conditions to create reliable datasets for subsequent data-driven analysis.