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method. We showed that while for Nb alloys nonlinearity is unimportant, it is critical to Nb-Nb Si alloys.
5
3
We find that inter-feature coupling terms are unimportant or non-recoverable, demonstrating the utility of
more robust and interpretable additive models for the decoupled feature space. The method allows for
estimation of feature importance, although one should not exaggerate the general physical meaning of
feature importances during the interpretation of ML models. The relative importance of features can be
quite sensitive to detailed local configurations and feature distributions rather than generic to a class of
physically similar systems. Overinterpretation should be avoided when one correlates the feature’s
importance tightly with its physical significance, as commonly found in the literature.

We hope that this study will be helpful to researchers in designing optimal ML approaches, including
dataset augmentation, algorithm optimization, and feature analysis, for ML of materials properties under
limited data in solving doping problems for alloys or semiconductors. For example, if it is understood that
combining data is not advantageous as different subsets may not increase the density of sampling and have
different optimal hyperparameters, complicating rather than facilitating the ML task, this knowledge can
then be used to select appropriate methods for such data, such as methods taking into account data
hierarchy [25,53] . Once the kind of dependence of the target on the features (linear vs non-linear or coupled vs.
uncoupled) is understood, it can also be used to select more appropriate methods (e.g., simple linear
[20]
regressions or polynomial models instead of complex ML schemes ). Moreover, this work suggests that
data-driven feature learning becomes increasingly important rather than the optimization of algorithm and
parameters alone due to the feature dependent prediction accuracy.

The prediction of energy changes for substitutional elements in alloys serves as a fundamental theoretical
approach to guide the design and optimization of alloy compositions. By accurately forecasting energy
changes due to substitution, the CE-based ML approach makes it possible to identify stable alloy phases and
preferred occupancy for understanding and evaluating alloying effects, inform the selection of appropriate
alloying elements, and mitigate the necessity for extensive empirical experimentation.

DECLARATIONS
Authors’ contributions
Made substantial contributions to conception and design of the study and performed data analysis and
interpretation: Manzhos, S.; Liu, Y.
Performed data acquisition and provided administrative, technical, and material support: Tang, Y.; Xiao, B.;
Liu, Y.
Wrote the manuscript: Tang, Y.; Manzhos, S.; Liu, Y.
Review and editing: Tang, Y.; Manzhos, S.; Liu, Y.; Ihara, M.

Availability of data and materials
The data and code supporting the findings of this study are available at the following URL: https://github.
com/Don-sugar/ML_script.

Financial support and sponsorship
Liu, Y.; Tang, Y. and Xiao. B. thank the financial support of the National Natural Science Foundation of
China (Nos. 52373227, 52201016, and 91641128) and the National Key R&D Program of China (Nos.
2017YFB0701502 and 2017YFB0702901). This work was also supported by the Shanghai Technical Service
Center for Advanced Ceramics Structure Design and Precision Manufacturing (No. 20DZ2294000), and the
Shanghai Technical Service Center of Science and Engineering Computing, Shanghai University. The
authors acknowledge the Beijing Super Cloud Computing Center, Hefei Advanced Computing Center, and
Shanghai University for providing HPC resources. Manzhos, S. and Ihara, M. thank JST Mirai Program,