# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
###### Abstract  
To quantify the uncertainties in multi-dimensional flow field correlated responses caused by uncertain model parameters, this paper presents an adaptive multi-fidelity model based on gappy proper orthogonal decomposition (Gappy-POD), which integrates the two conventional approaches for enhancing the efficiency of surrogate modeling, namely, multi-fidelity modeling and adaptive sampling algorithms. The challenges surrounding the selection of initial high-fidelity samples and the subsequent incremental augmentation of these samples are addressed. The \(k\)-means clustering algorithm is employed to identify locations within the parameter space for conducting high-fidelity simulations, leveraging insights gained from low-fidelity responses. An adaptive sampling criterion, leveraging the low-fidelity projection error derived from the Gappy-POD method, is implemented to progressively augment high-fidelity samples. The results demonstrate that the adaptive model consistently outperforms random sampling methods, highlighting its superiority in terms of accuracy and reliability, providing an efficient and reliable prediction model for uncertainty quantification. a 2025 15 3359 10.122015 15 3359 10.122015 15 3359 10.122015 15 3359 10.122015 15 3359 10.122015 15 3359 10.122015 15 3359 10.122015 15 3359 10.122015 15.1  
Multi-Fidelity Integration Framework: This strategy synergizes heterogeneous simulation models by combining abundant low-fidelity data (computationally economical) with selective high-fidelity samples (accuracy-assumed). Established implementations include co-Kriging architectures [9; 10; 11] and hierarchical neural networks [12; 13], which have demonstrated enhanced cost-accuracy equilibria in complex engineering optimizations. Sequential Sampling Methodology: Diverging from conventional static designs that predefine sample sets, this approach dynamically allocates computational resources through iterative sample selection. Guided by information entropy criteria or error minimization principles (e.g., MSE, IMSE [14; 15; 16]), it strategically identifies critical regions in parameter spaces for supplemental sampling. In CFD, quantities of interest extend beyond scalar outputs to include multi-dimensional, correlated flow field responses with temporal/spatial variations, such as spatially distributed wall pressure coefficients or time-dependent aerodynamic loads. These field variables exhibit inherent cross-correlations across discrete spatial nodes or time steps, with output dimensionality frequently reaching O(10\({}^{2}\))-O(10\({}^{4}\)) magnitudes in practical applications. When modeling multi-dimensional correlated responses, although surrogate models such as multi-output Kriging models [17] and multi-output neural networks exist, these models require the estimation of a large number of parameters, and the training time for the models themselves is relatively long. Consequently, scholars have developed surrogate model methods that combine with flow field reduction techniques. [PERSON] and [PERSON] [18] developed a non-intrusive reduced basis framework synergizing proper orthogonal decomposition (POD) with Gaussian process regression, specifically targeting non-linear structural response prediction. [PERSON] et al. [19] proposed the construction of active subspaces for POD modal coefficients and built a surrogate model with active variables. [PERSON] et al. [20; 21] proposed the non-intrusive POD-based approach enhanced by multivariate interpolation for the parametric analyses of aero-icing problems. However, within the framework of surrogate modeling combined with the reduced order method, enhancing modeling efficiency through the utilization of the aforementioned multi-fidelity modeling and adaptive sampling remains a significant challenge. Existing multi-fidelity models and adaptive sampling algorithms, while effective in single-output scenarios, often fall short when dealing with the multi-dimensional correlated flow field responses encountered in CFD simulations.