## 2 Materials and Methods
### Construction Method of Combined Model
#### 2.2.3 Modification Model Training  
In this step, the training set is classified into different operational data categories, resulting in \(k\) distinct datasets that do not overlap. The impact of ambient temperature on gas turbine performance is well-defined and significant. This correlation stems from fundamental mechanisms: as inlet temperature increases, air density decreases, resulting in reduced mass flow rate through the compressor; simultaneously, compressor work requirements increase with temperature, further reducing the net power output of the gas turbine [30]. Similarly, when the IGV angle changes, the characteristic map of the gas turbine shifts, necessitating adjustments to the characteristic map for different IGV angles during mechanistic modeling. It is observed that gas turbines display varying performance under different operating conditions, exhibiting certain non-linear characteristics. Employing a unified model across the entire range often limits model precision. Therefore, modification models for the basic model will be established for each operational dataset. This paper utilizes the k-means clustering algorithm from clustering algorithms to perform the classification of operating conditions. K-means clustering, an unsupervised learning technique, categorizes data into \(k\) predefined clusters based on sample distances. This algorithm benefits from rapid convergence and clear interpretability. A limitation is the necessity to predefine the cluster count \(k\), which, for this research, is advantageous as it allows selection of modification model numbers to suit computational demands. For \(m\) data points divided into \(k\) clusters, the process is as follows:  
1. Centroids Initialization: Select \(k\) random points as the initial cluster centers.
2. Distance Calculation: During iteration \(t\), compute each point's distance to the \(k\) centroids using Equation (5), assigning points to the closest cluster. \[distance_{i,j}^{t}=\left\|x_{i}-\mu_{j}^{t}\right\|^{2},\ i=1,\ldots,m,j=1, \ldots,k\] (5)
3. Centroid Update: Determine each cluster's mean and update centroids according to Equation (6). \[\mu_{j}^{t}=\sum_{x_{i}\in C_{j}}\frac{x_{i}}{n},\ n\ is\ the\ number\ of\ points\ in\ C_{j}\] (6)
4. Convergence: Repeat the distance calculation and centroids update until centroids calculated by Equation (6) fall below a certain threshold or the maximum number of iterations is reached. The cluster centroids calculated using Equation (6) will be retained as model information. When encountering new prediction points, the operating condition of the point will be determined based on the distance between the prediction point and the cluster centroids.  
After establishing the basic model and categorizing the operating conditions, modification models for each specific condition are constructed. In this step, the residuals, which are the differences between the basic model's predictions and the actual training data for each condition, served as the new prediction targets, with the aim of minimizing these residuals. Assuming the data are divided into \(k\) conditions through clustering algorithms and that the \(i\)th condition dataset contains \(m\) data points, the training sample for the \(i\)th modification model is represented as \(\{(x_{i,1},y_{i,1}-\overline{y}_{i,1}),\ldots,(x_{i,m},y_{i,m}-\overline{y}_{ i,m})\}\). The \(i\)th modification model is trained on this sample set, and modification models are trained for all other conditions, forming a set of modification models \(\{M_{1},\,M_{2},\,\ldots,\,M_{k}\}\).