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Gesch et al. J. Mater. Inf. 2025, 5, 42 https://dx.doi.org/10.20517/jmi.2025.23 Page 13 of 20

As the training time frames decrease to 50, it becomes more likely for differences to appear between the
LSTM-predicted 2D microstructures and the true images [Figure 5C], especially at later time steps such as
t = t and t = t . Furthermore, when using only 30 training time frames, the LSTM-predicted
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microstructures start to deviate significantly from the true evolution as early as t = t , and frequently
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diverges completely from the ground-true phase-field simulations [Figure 5D]. This large difference can be
clearly seen in the pointwise error plots in Figure 5D. Based on the above microstructural analyses, we can
conclude that the optimal training time frames for LSTM models lie between 40 and 60, which is once again
consistent with our prior analysis about feature values [Figure 4].

SVM prediction of coupled phase fields in latent space
Although LSTM models demonstrate exceptional performance in predicting the dynamic features of
multiple fields, they are not transferable across different fields. More specifically, the LSTM model must be
trained separately for each field by using their individual database. These processes significantly limit the
efficiency of LSTM models for predicting the microstructural evolution for more complex systems,
especially involving a large number of coupled fields. As such, we investigate the correlations between
different fields and explore the potential of using machine learning techniques to predict the
microstructural evolutions among these coupled fields.

Here, we employ SVM models to predict latent features of one field by using the reduced features of other
coupled fields as input parameters. For instance, Figure 6A shows a parity plot of the SVM-predicted latent
features of c field by using the other four η fields as the input parameters. The perfect linear correlation
between the SVM prediction and actual feature values, along with the low MSE values, suggests the high
accuracy and effectiveness of the SVM predictions. Using this SVM model, we can easily predict the
reduced microstructure of any c field. For example, the inset of Figure 6A compares the 4 × 4 latent features
of a representative c field obtained from the autoencoder with those predicted by the SVM model. The
excellent agreement between them further highlights the promising performance of the SVM models in
predicting the latent microstructure evolution of the c field.

Similarly, SVM models can be used to predict the latent representation of any of the η fields by using the
remaining fields as the input parameters. Figure 6B-E shows the parity plots which compare the SVM
prediction with the actual feature values for each η field. The perfect linear correlations observed in these
plots further verify the high accuracy of the SVM models in predicting η parameters. By comparing the MSE
values, Figure 6 indicates that the c field has a lower MSE value than the four η parameter fields. The lower
MSE in the c field is likely due to the non-redundant latent features contributed by the four individual η
parameters. In contrast, the latent representation of the c field may already embed information from the
reduced η fields, making the SVM predicting of these individual η fields less accurate.

To further assess the performance of SVM in predicting microstructural evolution across different fields, we
use well-trained autoencoders to reconstruct the SVM-predicted latent space of each field and compare the
results with reconstructed images using the ideal latent features. As shown in Figure 7, the reconstructed
microstructures using SVM-predicted microstructure exhibit perfect agreement with actual c and first two η
parameter fields (η and η ). Yet, the prediction performance is less accurate for the microstructures of η and
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η fields. This can be seen in the relatively large pointwise errors between autoencoder-reconstructed image
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and original microstructure images in the rightmost column in Figure 7. The slightly lower performance in
predicting the η and η fields is indeed due to the autoencoder reconstruction process rather than the SVM
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prediction, as pointwise error plots shown in the second to last column in Figure 7 compare the
reconstruction from the SVM-predicted latent representation to the reconstruction of the original latent