## 4 Outlook and Limitations  
In this paper, we focus on the application of multi-fidelity (MF) data in materials science, including methods for acquiring MF data, datasets in previous works, and machine learning algorithms. Additionally, we applied different machine learning methods to handle simulated MF data with different types of noise (additive, multiplicative, and mixed) as validation. Our results indicate that it is helpful to introduce a larger amount of lower fidelity data to work together with expensive higher fidelity data.  
In an era where materials science is increasingly becoming data driven, the transformation from a single-fidelity-data driven approach to a multi-fidelity-data driven paradigm seems inevitable. As a part of this ongoing evolution, we propose the following outlook and limitations for the future research in the area of multi-fidelity data driven methods:  
1. Classic methods, such as the Gaussian process regression as demonstrated in this study, will continue to play a pivotal role. For this paper, while classic methods such as Gaussian process regression prove effective, their computational demands might limit their performance on larger datasets or higher-dimensional problems. For researchers, choosing the appropriate model and having a clear understanding of the algorithm and its principles remain indispensable.
2. Alongside hardware advancements, newer methods are surfacing that offer innovative ways to address challenges. Techniques such as large language model (LLM)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Average MSE**} \\ \cline{2-4}  & **Additive Noise** & **Multiplicative Noise** & **Mixed Noise** \\ \hline HFDM & 2.545 & 2.531 & 2.389 \\ \hline AC & 0.065 & 0.753 & 0.501 \\ \hline MC & 0.608 & 0.609 & 1.535 \\ \hline CC & **0.057** & **0.436** & **0.425** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Average MSE of different methods on datasets with different noises, the lowest values are highlighted in bold.  
warrant close attention, as they could redefine how we handle multi-fidelity problems. However, concurrently, utilizing LLM or similar complex technologies may introduce challenges associated with interpretability, computational resources, and data requirements.
* Active learning strategies, especially when integrated with both experimental and theoretical multi-fidelity data, hold the potential for groundbreaking contributions. Machine learning models can determine not only material structures or features but also which levels of data fidelity are most informative for exploration. However, a pertinent issue is that active learning strategies necessitate high-quality and representative initial datasets. If the initial data lack quality or do not aptly represent the studied domain, it may impact the effectiveness of learning. Acquiring suitable multi-fidelity datasets is, thus, a critical concern for researchers.  
Validation, Z.W.; investigation, Z.W. and H.C.; resources, X.L., T.Y. and Y.H.; visualization, Z.W.; supervision, X.L. and T.Y.; funding acquisition, X.L., T.Y. and Y.H. All authors have read and agreed to the published version of the manuscript.  
The authors extend their sincere appreciation to the National Natural Science Foundation of China for supporting this work through the funding of project number 22203008, 22272009, 22002008.  
Not applicable.  
We generated the data used in this study, along with the corresponding code and results, which are available on our GitHub repository at [[https://github.com/1152041831](https://github.com/1152041831)]([https://github.com/1152041831](https://github.com/1152041831)) /MFSM_experiment (accessed on 3 December 2023).  
The authors declare no conflict of interest.