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Page 12 of 19 Tang et al. J. Mater. Inf. 2025, 5, 38 https://dx.doi.org/10.20517/jmi.2025.05

Table 3. RMSE in substitution energies (in eV/cell) of alloying elements predicted by GPR-NN and optimal hyperparameters (kernel
length parameter l and noise parameter logσ) for different datasets

RMSE (GPR-NN) Hyperparameters
Data set
Training Test l logσ
Nb (210) 0.17 ± 0.01 0.21 ± 0.05 20 -6
Nb -Nb Si (588) 0.18 ± 0.01 0.25 ± 0.03 3 -5
I 5 3
Nb -Nb Si (1,764) 0.23 ± 0.01 0.29 ± 0.02 0.5 -5
II 5 3
Si -Nb Si (784) 0.27 ± 0.01 0.32 ± 0.04 0.5 -3
I
3
5
Si -Nb Si (392) 0.11 ± 0.003 0.14 ± 0.02 1.5 -4
II 5 3
Combined Nb Si (3,528) 0.43 ± 0.01 0.52 ± 0.03 0.1 -2.5
5 3
The spread of values indicated by “±” is for 1 standard deviation over 100 runs differing by random train-test splits. RMSE: Root mean square error;
GPR-NN: Gaussian process regression-neural network.
Table 4. RMSE of substitution energies (in eV/cell) of alloying elements when using different numbers of terms N in the coupled
model of Equation (6)
N = 100 N = 200 N = 500
Data set
Training Test Training Test Training Test
Nb 0.17 ± 0.01 0.21 ± 0.05 0.16 ± 0.02 0.22 ± 0.05 0.16 ± 0.01 0.22 ± 0.06
Nb -Nb Si 0.13 ± 0.01 0.26 ± 0.05 0.12 ± 0.01 0.25 ± 0.05 0.12 ± 0.01 0.25 ± 0.05
I 5 3
Nb -Nb Si 3 0.11 ± 0.01 0.30 ± 0.07 0.11 ± 0.01 0.35 ± 0.11 0.11 ± 0.01 0.35 ± 0.13
5
II
Si -Nb Si 0.24 ± 0.01 0.31 ± 0.04 0.24 ± 0.01 0.31 ± 0.04 0.24 ± 0.01 0.31 ± 0.04
I 5 3
Si -Nb Si 0.09 ± 0.003 0.13 ± 0.01 0.09 ± 0.003 0.13 ± 0.02 0.09 ± 0.003 0.13 ± 0.01
II 5 3
Combined Nb Si 3 0.36 ± 0.004 0.52 ± 0.05 0.37 ± 0.003 0.53 ± 0.04 0.38 ± 0.003 0.54 ± 0.03
5
The spread of values indicated by “±” is for 1 standard deviation over 100 runs differing by random train-test splits. RMSE: Root mean square error.

Figure 6. Component functions of the additive model for Nb alloys.

calculate various configurations to cover the full feature space more uniformly and adopt the CE feature
models incorporating both compositional and structural information. It also helps explain why the model
on the combined dataset performed worse than the individual models on the data subsets since these
features are orthogonal independently each other.