# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 2 The Multi-Fidelity Modeling Framework  
In this paper, we develop an adaptive multi-fidelity modeling approach based on the Gappy-POD algorithm for predicting multi-dimensional corrected flow field responses. The Gappy-POD method facilitates high-fidelity responses from low-fidelity ones, thereby circumventing computationally expensive high-fidelity CFD simulations. Assuming that all training samples are divided into two parts, one part constitutes the complete sample set, where both high- and low-fidelity responses are obtained through CFD simulations. The remaining samples are referred to as incomplete samples, with only low-fidelity responses obtained via CFD simulations, and their high-fidelity responses are unknown and predicted by the surrogate model. For the \(i\)-th sample in the complete sample set, we define its input vector as \(\theta_{i}\), its low-fidelity response vector as \(s_{i}^{L}=\left(s_{1,i}^{L},s_{2,i}^{L},\cdots,s_{n_{L},i}^{L}\right)^{T}\) with a dimension of \(n_{L}\), and its high-fidelity response vector as \(s_{i}^{H}=\left(s_{1,i}^{H},s_{2,i}^{L},\cdots,s_{n_{H},i}^{H}\right)^{T}\) with a dimension of \(n_{H}\). The high- and low-fidelity vectors are combined into a multi-fidelity snapshot denoted as \(s_{i}=\left(\frac{s_{i}^{L}}{s_{i}^{H}}\right)\) with a dimension of \(n_{L}+n_{H}\). The conventional POD methodology is applied to multi-fidelity snapshots \(\left\{s_{i}\right\}_{i=1}^{m}\). Based on the generalized energy criterion, truncation is performed to obtain an orthogonal space \(\Phi=[\varphi_{1},\varphi_{2},\cdots,\varphi_{n}]\), which is composed of \(n\) orthogonal basis functions. Therefore, any multi-fidelity sample snapshot can be represented as follows:  
Employing a generalized energy threshold criterion, we perform modal truncation to construct a reduced-order orthogonal subspace \(\Phi=[\varphi_{1},\varphi_{2},\cdots,\varphi_{n}]\). Consequently, any multi-fidelity snapshot can be expressed as follows:  
\[s\approx\overline{s}+\sum\limits_{j=1}^{n}\alpha_{j}\varphi_{j} \tag{1}\]  
where \(\overline{s}=\frac{1}{m}\sum\limits_{i=1}^{m}s_{i}\). The basis function coefficients vector \(\overset{\rightarrow}{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})^{T}\) is obtained by the projection method. Each orthogonal basis function \(\varphi_{i}(1\leq i\leq n)\) is a \((n_{L}+n_{H})\)-dimensional vector. The first \(n_{L}\) elements of each orthogonal basis function,denoted as \(\varphi_{i}^{L}\), represent features of low-fidelity responses, while the last \(n_{H}\) elements, denoted as \(\varphi_{i}^{H}\), represent features of high-fidelity responses. Therefore, the orthogonal space can be expressed as follows:  
\[\Phi=\left[\frac{\varphi_{1}^{L}}{\varphi_{1}^{H}},\frac{\varphi_{2}^{L}}{ \varphi_{2}^{H}},\cdots\frac{\varphi_{n}^{L}}{\varphi_{n}^{H}}\right]=\left[ \frac{\Phi^{L}}{\Phi^{H}}\right] \tag{2}\]  
For the \(j\)-th sample in the incomplete sample set, we denote its input parameter vector as \(\theta_{j}\) and its multi-fidelity response vector as \(s_{j}=\left(\frac{s_{j}^{L}}{s_{j}^{H}}\right)\). The low-fidelity part \(s_{j}^{L}=\left(s_{1,j^{\prime}}^{L}s_{2,j^{\prime}}^{L},\cdots,s_{n_{L,j}}^{L }\right)^{T}\) is obtained with CFD simulation, while its high-fidelity part \(s_{j}^{H}\) is unknown.