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Page 14 of 20 Zhu et al. J. Mater. Inf. 2025, 5, 8 https://dx.doi.org/10.20517/jmi.2024.76

values by DFT. Dots of different colors represent various crystal structure types, with the gray diagonal line indicating perfect agreement
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between predicted values and calculated values by DFT. IGNN: Intermetallics graph neural network; R : the coefficient of determination;
RMSE: root mean square error; DFT: density functional theory.

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In Hg decreased from 6.16 to 3.25 g/cm , and the error for PuPt reduced from 2.83 to 0.82 g/cm .
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To further assess the generalization capability of the IGNN model in handling the density of complex
intermetallic compounds, an independent density test set for ternary and quaternary compounds is applied.
The extended test set is more complex because the structures of ternary and quaternary compounds involve
more types of atoms and more complex crystal structures, providing a more comprehensive examination of
the model’s robustness across different compound types and structures. Figure 5A shows the performance
of IGNN on the ternary compound test set, with an R value of 0.9852 and an RMSE of 0.3313 g/cm ,
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indicating a high prediction accuracy for the ternary system. The IGNN prediction results on the quaternary
compound test set are shown in Figure 5B, with an R of 0.9694 and an RMSE of 0.3340 g/cm . Compared to
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ternary compounds, the prediction accuracy for quaternary compounds is slightly lower, likely due to the
higher structural complexity of quaternary compounds. From the scatter plot distribution, the IGNN
model’s prediction results on the ternary and quaternary compound test sets are highly consistent with
calculated values by DFT, with most data points concentrated near the ideal diagonal, indicating minimal
deviation between predicted values and calculated values by DFT. The data points for each crystal structure
are distributed relatively evenly, without significant deviation for any particular crystal structure. This
suggests that the IGNN model demonstrates strong stability and generalization ability when handling
diverse crystal structures.

Classification prediction and visualization
In studying the crystal structure and density prediction of intermetallic compounds, it is necessary not only
to perform accurate regression predictions but also to conduct density classification analysis to explore the
influence of different crystal structures on model classification. To gain a better understanding of the IGNN
model’s performance in classification tasks, t-SNE is employed to visualize classification results across
different layers of the IGNN model . Additionally, a comparative analysis of the classification performance
[38]
of IGNN and various traditional machine learning models is conducted.

t-SNE is a widely used dimensionality reduction technique that projects high-dimensional data into a two-
dimensional space, facilitating visual assessments of a model’s ability to differentiate between categories. In
this study, t-SNE is employed to visualize the feature representations at each layer of the IGNN model
[Figure 6]. Figure 6A presents the t-SNE visualization of the input layer. In the input layer, data points from
various crystal systems are not yet clearly separated, suggesting that the model has not deeply separated data
features at this initial stage. The t-SNE visualization of the normalization layer is shown in Figure 6B.
Following normalization, the data distribution becomes more concentrated, with emerging boundaries
between different crystal systems, suggesting initial feature refinement. The t-SNE visualization of the global
pooling layer is shown in Figure 6C. The data points from different crystal systems demonstrate a clear
clustering tendency, reflecting the model’s capacity to aggregate node features and capture significant
structural differences among the crystal systems. Figure 6D displays the t-SNE visualization of the output
layer, where data points from different crystal systems are distinctly separated, indicating a strong
classification outcome. The features extracted at the output layer of the IGNN model possess high
discriminative power, creating clear boundaries between data from different crystal structures in two-
dimensional space. Through the t-SNE visualization analysis of each layer in the IGNN model, a progressive
enhancement in the model’s ability to distinguish among different crystal structures is observed as the layers