## 1 Introduction
### Multi-fidelity Data from Different Algorithms  
In the field of materials science, the existence of multi-fidelity data is primarily attributed to the diverse methodologies. These approaches often range from experimental procedures to theoretical calculations. Experimental methods are generally considered to be more time-consuming than their theoretical counterparts, but they offer a higher degree of accuracy. Here, for instance, some works solely focus on theoretical methods, and their key results are summarized in Table 2. It is essential to recognize that even within the realm of theoretical computation, fidelity varies significantly. For example, the commonly used empirical potentials are generally considered low-fidelity (LF) methods when compared to density functional theory (DFT), which is often regarded as high-fidelity (HF). However, it is important to note that even within DFT, there exists a ladder of accuracy. The Perdew-Burke-Ernzerhof (PBE) functional, for instance, can sometimes be categorized as high-fidelity depending on the context, while in other cases, it might be considered low-fidelity. There are often systematic errors [54; 55; 56] and random errors compared to true values when using DFT calculations to determine material properties. For example, it has been reported in the literature that DFT calculations tend to underestimate the band gap width by 30% to 100% [57]. Random errors typically arise from the initial trial charge density used in the DFT calculations. Different trial charge densities can lead to varying random errors. Additionally, different convergence criteria in self-consistency may also introduce random errors. Systematic errors, on the other hand, are often introduced during the calculation of the Hamiltonian in DFT. The Kohn-Sham equations, which are used to compute the exchange-correlation potential and solve for the electronic wave functions, typically employ different approximation methods, in which different approximation methods introduce different errors. All of these errors arise during the DFT calculation process. For the same calculation target, the different errors resulting from different functionals can be viewed as discrepancies between HF and LF. These discrepancies can be corrected using multi-fidelity data learning methods. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **LDA [58]** & **PBE [59]** & **PBE0 [60]** & **HSE [61; 62]** & **Potential [63]** \\ \hline
[9] & & HF & & LF \\ \hline
[14; 15] & & LF & HF & \\ \hline
[16] & LF & LF & HF & \\ \hline
[17] & & LF & HF & \\ \hline
[18] & & LF & HF & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Distribution of high/low fidelity (HF/LF) data acquisition methods in the materials field. Figure 1: Multi-fidelity data in different fields. In addition to different algorithm applications in materials science and aerospace science, different computational fluid dynamics (CFD) methodologies or different CFD tools (Q3D and MATRICS-V [33]) are commonly used to obtain MF data. Reynolds-Averaged Navier-Stokes (RANS) method is commonly used to obtain HF data [30; 31; 32], and the potential flow model to obtain LF data [26; 29]. In reference [25], RANS is used to obtain LF data, while large-eddy simulations are used to obtain HF data. In reference [27], URANS and Euler models are used to provide LF and HF data, respectively. In reference [28], the doublet-lattice method and CFD-based flutter calculation are used to obtain LF and HF data, respectively. In mechanical engineering, different mathematical models (e.g., different finite element models [39; 42] and surrogate models [12; 40; 41]) or different measuring tools (e.g., different strain sensors [43] and different wind-turbine-specific aero-servo-elastic computer simulators [44]) are commonly used to obtain MF data.