Page 84 - Read Online

Page 6 of 20 Gesch et al. J. Mater. Inf. 2025, 5, 42 https://dx.doi.org/10.20517/jmi.2025.23

time steps is set to 30,000,000, and no-flux boundary conditions are applied to all sides of the domain,
ensuring that microstructures stick to the edges of the simulation box and do not pass through. All
microstructure evolutionary trajectories are evenly divided into the 100 intervals, ensuring that each
simulation has a 256 × 256 × 100 data points. As it does not have microstructures, it is not included in the
dataset, but a snapshot of the initial condition is also saved for each trajectory. Given the five coupled fields
in each simulation, the total size of a microstructure evolution for 200 simulations is 256 × 256 × 100 × 5 ×
200 (26 billion data points) generated in this work. Each phase-field simulation takes about 377 minutes by
using 16 CPUs on HPC facilities at the Alabama Supercomputer Center to predict the microstructure
evolution of Ostwald ripening over the 100 time frames. All microstructure databases have been uploaded to
[59]
the materials data facility (MDF) [57,58] and the database is already available online .

Dimensionality reduction of microstructural evolution
A key component of our accelerated framework is the manifold hypothesis, which suggests that high-
[60]
dimensional data lies near a low-dimensional manifold inside high-dimensional space . Similarly, many
studies have shown that high-dimensional microstructure images can be effectively transformed into a
lower-dimensional space while preserving key features to represent their evolutionary behaviors. Therefore,
we applied dimensionality reduction to transform the 2D microstructure images of Ostwald ripening into a
reduced space. A recent study by Desai et al. has demonstrated that the autoencoder is a very powerful
technique in transforming microstructure images into low-dimensional space and provides smooth
[61]
trajectories in the latent space . As such, we primarily adopted this approach to process the 2D images of
Ostwald ripening for all five coupled fields [Figure 1B].

Autoencoder
An autoencoder is an unsupervised learning technique that is designed to encode high-dimensional data
into a compressed and meaningful representation [62,63] . Briefly speaking, an autoencoder has two major
components: an encoder that maps the input data into a lower-dimensional space, and a decoder that
reconstructs the original data from this compressed representation [Figure 1B]. The code typically has a
space Z, which is smaller than the original space of input data but can still retain useful features in its
compressed space. This capability makes autoencoders a powerful tool for transforming 2D microstructure
image data, X (x, t), into a low-dimensional representation, Z (x, t) [64,65] . The microstructure dataset used to
i
train the autoencoders consists of 80,000 2D images, which is 80% of all microstructure images generated
for Ostwald ripening. The microstructure images from the early time frames (1 to 80) were preferentially
selected as training data due to their larger diversity, while those from later time frames tend to exhibit more
uniform and constant features because most of the systems have reached a fully stabilized state after time
frame 80 and were thus included less often. This selection strategy also ensured that each microstructure
trajectory has an equal number of microstructure images (80) included in the entire dataset. The training
routine further creates a split between training data and validation data, with 80% used for training and 20%
used for testing. The autoencoders were trained to minimize the reconstruction error using the binary
[66]
cross-entropy (BCE) as the loss function . We selected BCE as the performance metric for evaluating the
reconstruction performance of the autoencoder due to two main reasons: (i) its focus on the probabilistic
difference between the predicted and actual outputs when there are discrete options, such as in phase
selection during reconstruction; and (ii) faster convergence speed during the training process. The training
was conducted for a total of 1,400 epochs, and the autoencoder reached convergence after approximately
700 epochs, based on the training and validation loss shown in Supplementary Figure 2. The reconstruction
error depends on the size of the latent dimension Z, i.e., the degree of data compression desired. By testing
the performance of autoencoders, we found that the 2D microstructure images with 256 × 256 pixels can be
effectively transformed a 16-value representation which can be viewed as a 4 × 4 matrix. The well-trained
autoencoders enable the complete encoding and reconstruction of any microstructure frame of the same