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Gesch et al. J. Mater. Inf. 2025, 5, 42 https://dx.doi.org/10.20517/jmi.2025.23 Page 3 of 20

dependent machine learning models, such as recurrent neural networks (RNNs), to predict the evolution of
microstructures in future time sequence [45-50] . PSMDs directly approximate the microstructural dynamics
from their original two-dimensional (2D) or three-dimensional (3D) space using the physics-informed
neural networks, such as neural operator learning. These PSDMs generally incorporate a temporal-
conditioning mechanism, which can further improve the accuracy and efficiency of time-to-solution
predictions for microstructural evolution. For instance, a deep-learning-based neural network with a U-Net
architecture has been developed to predict the microstructural evolution for a range of phase-field
[51]
problems, including spinodal decomposition, dendrite growth, and thin film deposition . Despite their
outstanding performance, most existing accelerated frameworks, including both LDMs and PSDMs have
been developed for relatively simple phase-field problems with a single evolution field, such as spinodal
decomposition [45-49] . Accelerated frameworks for more complex, coupled microstructural evolution
problems are still scarcely developed, thereby motivating this study.

[52]
Using the classical benchmark problem Ostwald ripening as a modeling system , we in this work
developed a highly integrated framework to accelerate the microstructural evolution of coupled phase fields.
This accelerated framework integrates high-throughput phase-field modeling, dimensionality reduction,
and long short-term memory (LSTM) networks to predict the microstructural evolution of Ostwald
ripening in latent representations. Specifically, autoencoders are first used to transform the 2D
microstructure images into a low-dimensional space for training LSTM models. Once trained, the LSTM
models are employed to predict the temporal evolution of microstructures in the latent space, and then the
autoencoder is used to reconstruct these predicted microstructure sequences back into the original 2D
space. Finally, supervised learning techniques are applied to predict the correlations between coupled fields
within the latent space, thereby further enhancing the overall efficiency of this accelerated phase-field
framework.

MATERIALS AND METHODS
Figure 1 illustrates a simplified workflow of our accelerated framework for Ostwald ripening . The high-
[52]
throughput phase-field simulations are first performed with various input parameters to generate a large
database of microstructural evolution over time [Figure 1A]. Using the large number of microstructure
images, we employ an autoencoder-based dimensionality reduction technique to reduce the data to the low-
dimensional space. Specifically, each parameter field in Ostwald ripening can be defined by a continuous
function X (x, t), where x and t represent a 2D spatial domain and time, respectively. After dimensionality
reduction, the microstructure of each parameter field can be given by a multivariate time-series x = (x , x ,
2
1
..., x ), where x is a vector in latent space and represents the reduced microstructure at time t = t i
N
i
[Figure 1B]. Next, we use one of the most popular RNN models, LSTM, to learn the nature of
microstructure evolution of Ostwald ripening in the latent space [Figure 1C]. This is realized by establishing
a functional relationship among the low-dimensional representation x, the current time step t and prior
i
i
lagged values (t , t ).
i-1
i-n
(1)
The LSTM-based microstructure-learning engine can efficiently learn the microstructural evolution as a
function of any model input and predict future time frames without the need to perform computationally
expensive phase-field simulations. Finally, the autoencoder will be used to transform the LSTM-predicted
microstructure in latent space back to full-scale 2D images [Figure 1D].