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Since a prior study found that deep LSTM architectures often failed to converge during the training
[46]
process , we used 2 hidden layers and trained each LSTM model for 1,000 epochs for each field. The
training and validation loss, as shown in Supplementary Figure 3, indicates that all LSTM models can
achieve convergence after around 500 epochs. The mean square error (MSE) loss function was used as the
performance metric to assess the training process of LSTM models . All LSTM models were trained and
[69]
tested using the PyTorch library with Adam optimizer.

Predicting the correlation between coupled phase fields
Support vector machine (SVM) is a supervised learning technique that maps each input to a specific vector
in the output space . The output vector is scaled based on the magnitude of the input, and because each
[70]
input corresponds to a vector with a unique direction, combining these vectors allows the SVM model to
reach any point in the output domain. This capability makes SVM particularly effective for capturing the
correlation of the microstructure evolution among different fields in latent space.

We trained SVM models to capture the relationships among the autoencoder-reduced representations of
the five coupled fields involved in Ostwald ripening. The input to each SVM model is the reduced
microstructure sequences of four phases, while the output corresponds to the microstructures of the
remaining one field in the latent space. Accordingly, five distinct SVM models were developed for each field
in this work. We used 80% of the microstructure sequences for training and the remaining 20% for testing.
The performance of the SVM models was evaluated using MSE values as the scoring metric.

RESULTS AND DISCUSSION
Microstructural evolution of Ostwald ripening
Ostwald ripening is a classical phase-field benchmark problem in which multiple phases are coupled with
each other and evolve spontaneously . This phenomenon is commonly observed in real material systems,
[52]
[71]
such as the growth and coarsening of γ′ precipitates in a γ matrix in nickel-based superalloys . By varying
the dominant input parameters (e.g., M, c , c , and ϵ), a wide range of microstructures can be obtained for
a
0
the Ostwald ripening problem. Figure 2 illustrates the representative microstructural evolution for both
solute and four structural order parameter fields at different time steps. At the initial stage, the conservative
c fields (left column in each panel) always consist of small particles, which gradually coalesce into larger
particles over time. The four η fields exhibit similar evolutionary behaviors, where the initial small particles
merge and grow larger. By comparing the microstructure morphology between the c and the four η fields in
Figure 2, it is interesting to see that the η fields consist of particles that are isolated from the c field.
Moreover, the sum of all particles in the four η fields is always identical to the c field at all time steps. As
time evolves, some isolated particles in the η fields rapidly coalesce and form larger particles, while other
small particles may disappear, such as η fields as shown in Figure 2A.
2-4

Transforming microstructural evolution in latent space
Based on the diverse microstructural evolution paths of Ostwald ripening, all 2D original images were
transformed into the low-dimensional space using autoencoder technique. Mathematically, autoencoder is
used to achieve a nonlinear embedding of high-dimensional microstructure data X (x, t) into a low-
dimensional representation, Z (x, t). Figure 3 shows the autoencoder-reduced features for a representative
i
microstructural evolution of five fields in Ostwald ripening at three different time frames: t = t , t = t , and t
10
1
= t . The results demonstrate that the autoencoder effectively transforms the 2D images from 256 × 256
100
pixels into 4 × 4 reduced matrices for all five fields at different time frames. By using these latent
microstructures, the well-trained autoencoder can easily transform them back into the original space, and
the agreement with phase-field-simulated images demonstrated the promising performance of autoencoder
in reconstructing 2D images for the Ostwald ripening problem Figure 3.