## 1 Introduction  
Multi-fidelity (MF) often appears in the field of science and engineering. It is often used to describe a variety of data or models with different precision, which are called MF data and MF models. Generally, in MF data, the quality of a large amount of data with a low acquisition cost is coarse (i.e., low-fidelity (LF) data), while there are also small amounts of data with very high accuracy (i.e., high-fidelity (HF) data). This is the so-called \"cost-accuracy trade-off\". In practical applications, multi-fidelity data may not be limited to just two levels. The levels of data could be five [1, 2] or even more. However, most of the recent works still largely focus on the performance of models trained on single-fidelity data [3, 4, 5]. These efforts either face issues from insufficient data or problems with poor data quality [6].  
To overcome these problems, many methods are designed to exploit the multi-fidelity data efficiently, such as iterative denoising [1], information fusion [7, 8], Bayesian optimization [9, 10], etc. Iterative denoising enhances the model's accuracy by iteratively training on MF data with specific permutations and applying denoising operations on specific data after each iteration. Information fusion algorithms fuse data information by learning the relationships between HF and LF data, effectively leveraging MF data. Bayesian optimization uses the correlation between HF and LF data to reduce the computational cost and improve the prediction efficiency when iteratively searching for the best value.  
The systematic exposition and validation of multi-fidelity data learning methods in other fields have been extensively reviewed [11, 12, 13]. However, within the realm of materialsscience, there remains a notable absence of an overview and validation of multi-fidelity data learning methods. This paper aims to address this gap by presenting an introduction and validation of multi-fidelity data learning methods, with a specific focus on their application in materials science. The paper highlights the widespread presence of multi-fidelity data in the materials domain, emphasizing that employing multi-fidelity data learning methods in the face of such data scenarios is undoubtedly a promising choice. Additionally, the article introduces several typical multi-fidelity datasets within the materials domain, enabling interested readers to swiftly access them.  
The partitioning of multi-fidelity data needs a combination between prior knowledge and quantitative evaluation methods. Through the consideration of the application fields and the methodologies for data acquisition, a categorization of the generation of multi-fidelity data are summarized in Table 1. MF data often appear in the fields of materials science, aerospace science, and mechanical engineering, as shown in Figure 1. In every field, there are often a variety of different calculation results for the same object, and these results may have different accuracy and quantity. We will engage in a section-by-section discussion.  
\begin{table}
\begin{tabular}{c c c c} \hline  & & functional & [14; 15; 16; 17; 18] citations (273, 60, 4, 43, 25) \\ \cline{3-4}  & \multirow{2}{*}{\
\begin{tabular}{c} **Different** \\ **Algorithms** \\ \end{tabular} } & \multicolumn{2}{c}{experiment} & [19] citations (18) \\ \cline{3-4}  & & functional and experiment & [1,2] citations (1, 71) \\ \cline{3-4}  & & physical models & [20] citations (14) \\ \cline{3-4}  & & functional and physical models & [9] citations (55) \\ \cline{3-4}  & & mesh size & [21] citations (145) \\ \cline{3-4}  & \multirow{2}{*}{
\begin{tabular}{c} **Different** \\ ** hyperparameters** \\ \end{tabular} } & \multicolumn{2}{c}{time step} & [22; 23] citations (21, 23) \\ \cline{3-4}  & & different hyper parameters & [24] citations (35) \\ \hline  & & CFD methodology & [25; 26; 27; 28; 29; 30;