# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 3 The Strategies for Selecting High-Fidelity Samples  
In the Gappy-POD method, it is assumed that there are a relatively large number of low-fidelity samples. A portion of these samples needs to be selected for high-fidelity CFD simulations to obtain their high-fidelity responses, while the high-fidelity responses of the remaining samples will be predicted using Equation (4). The key challenge lies in determining which samples to select for high-fidelity CFD simulations. From the perspective of dynamic sampling, this can be divided into two problems: how to select initial high-fidelity samples and how to incrementally add high-fidelity samples based on model feedback. This paper addresses these two issues separately. (1) Selection of the initial high-fidelity samples  
To construct the multi-fidelity surrogate model, it is necessary to determine, through design-of-experiment, where to conduct high- and low-fidelity CFD simulations. During the initial modeling process, Latin hypercube sampling is used to obtain training samples for low-fidelity CFD simulations. Latin hypercube sampling (LHS) synthesizes the space-filling advantages of Monte Carlo methods with the stratification principles of experimental design, establishing itself as an optimal sparse sampling technique for high-dimensional parameter spaces. After obtaining the low-fidelity responses of all training samples, a subset of samples needs to be selected from all training samples for high-fidelity CFD simulations. The objective is to ensure that the selected samples comprehensively represent the distinct characteristics of the output responses while exhibiting a certain discrepancy between each other. Previously, [PERSON] and [PERSON] [26] developed an exchange algorithm employing the Morris-Mitchell criterion [27] for nested sample selection. However, this algorithm relies on the distance measured within the input parameter space, which may not adequately capture the similarity in output responses for non-linear CFD problems. Therefore, we shifted the focus to an approach based on the output responses. To achieve this, we utilize the \(k\)-means clustering algorithm [28] to partition the \(n_{\text{train}}\) training samples into \(n_{\text{complete}}\) clusters based on their low-fidelity responses. Subsequently, we select the sample closest to the centroid of each cluster as the representative of that cluster and incorporate it into the complete sample set. The \(k\)-means clustering algorithm assesses the similarity between different sample points by quantifying their Euclidean distance, and samples that are grouped into the same cluster exhibit higher similarity. The clustering algorithm is implemented using the scikit-learn library. (2) Adaptive refinement  
From the perspective of dynamic sampling, if the prediction capability of the model constructed with the initial complete sample set is inadequate, more training samples need to be incrementally added. This process involves three key steps: (a) selecting samples from the incomplete set based on specific criteria, (b) obtaining their high-fidelity responses through CFD simulations, and (c) incorporating them into the complete sample set.