# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 4 Implementation Process  
1. Implementation process of the Gappy-POD model With the Gappy-POD multi-fidelity model framework and high-fidelity samples selection methods, the implementation process of the adaptive multi-fidelity model is introduced in the following.  
The modeling process, as shown in Figure 1, begins with the space of the uncertain model parameters and is carried out in six steps. The first four steps involve generating initial samples and constructing the initial model, while the last two steps focus on incrementally adding samples to improve the model.  
1. The inputs for \(n_{train}\) training samples are generated using Latin hypercube sampling in the input parameter space.
2. The low-fidelity responses for all training samples are obtained through low-fidelity CFD calculations.
3. The \(k\)-means clustering algorithm is used to select \(n_{complete}\) samples from all training samples for subsequent high-fidelity CFD simulations. This divides the training samples into two sets, one consisting of \(n_{complete}\) samples, denoted as the complete sample set, and the other consisting of the remaining (\(n_{train}\) - \(n_{complete}\)) samples, denoted as the incomplete sample set.
4. High-fidelity responses are acquired via CFD simulations across the complete sample set. At this point, for the complete samples, there are both high- and low-fidelity responses. For the incomplete samples, only low-fidelity responses are available.
5. The Gappy-POD method is employed, based on the high- and low-fidelity responses of the complete samples and the low-fidelity responses of the incomplete samples, to predict the high-fidelity responses and the projection error of low-fidelity responses for incomplete samples. Traversing through incomplete samples, if the maximum projection error is less than the preset value, it can be considered that the Gappy-POD method based on the existing complete sample set can well predict the high-fidelity responses of incomplete samples; therefore, the multi-fidelity modeling process is finished. Otherwise, it is considered that the prediction ability is insufficient, and additional complete sample data are selected through the sixth step.  
Figure 1: The framework of the adaptive multi-fidelity model.  
* Traversing through the incomplete samples, the one with the maximum projection error is selected as the next additional sample. Its high-fidelity responses are obtained through high-fidelity CFD calculations, and added to the complete sample set together with the previously obtained low-fidelity responses.  
The fifth and sixth steps are repeated until the maximum projection error is less than the present value.  
At this point, high-fidelity responses for all training samples are available, obtained by either the prediction model or CFD simulations.  
* Uncertainty propagation model  
After obtaining the high-fidelity responses of all training samples, we can establish a predictive model that maps the uncertain model parameters to multi-dimensional flow field responses. Through Monte Carlo sampling on this surrogate, we quantify input uncertainty propagation.  
The procedure is consistent with the method of combining POD and surrogate modeling proposed in the literature [17; 18; 19; 20; 21]. The dominant orthogonal basis functions are obtained by performing POD [29; 30] on the high-fidelity responses of all training samples. The reduced-dimensional response for each training sample is obtained by the projection method. The surrogate model (the Kriging model [31; 32] used in this paper) between uncertain model parameters and the reduced-dimensional response is then constructed. When given new model parameters, the corresponding reduced-dimensional response can be predicted, and the complete response is recovered via POD.