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5
10 ) is slightly higher than that achieved with 60 training frames, it comes at the cost of reduced predicting
accuracy for microstructure evolution in latent space. Thus, we estimate the optimal computational gain of
5
our accelerated framework to be approximately 3.35 × 10 for Ostwald ripening, which represents a
significant speedup compared to high-fidelity phase-field simulations.

Furthermore, integrating SVM models into our accelerated phase-field framework improves the overall
efficiency. For instance, once the framework is established for any four fields, SVM models can be employed
to predict the latent dynamic features of the remaining field. Subsequently, these predicted latent
representations can be reconstructed into the original 2D microstructure using the well-trained
autoencoder. In this way, there is no need to develop an LSTM model for the remaining field, thereby
significantly improving the overall computational efficiency of the accelerated phase-field framework.

Limitations and future work
The accelerated framework demonstrates the feasibility of expediting the phase-field simulations for
coupled microstructural evolution. Through the analysis of microstructure prediction performance, we
found that autoencoder-based LSTM models exhibit high accuracy and efficiency in forecasting the
microstructure evolution within the latent space. However, there are several limitations of this accelerated
framework that should be acknowledged. First, as shown in Figure 7, the autoencoder sometimes has
trouble recreating the 2D microstructures of the η fields from the latent space. This limitation is likely due to
the relatively simple microstructure patterns presented in the η fields compared to the c field, as well as the
limited size of microstructure database (200 simulations), which may be insufficient for training a robust
autoencoder-based neural network. The second limitation is that LSTM models are heavily dependent on
the number of time frames used for their training, as these models demonstrate lower performance when
trained with a smaller number of time steps [Figure 4]. Thus, enhancing the predictive performance of
LSTM models with fewer number of time frames is crucial for improving the overall efficiency of the
accelerated phase-field framework.

It is worthy to note that an interesting direction for accelerating Ostwald ripening is the development of
[51]
physics-informed models, such as PSDMs based on U-Net . This approach is motivated by recent studies
which demonstrate that the PSDMs can achieve very high accuracy in predicting the microstructural
evolution with low prediction error and reduced error accumulation during autoregressive predictions .
[39]
According to a review article by Dingreville et al., the accelerated framework developed for Ostwald
ripening falls in the category of LDM formulation, as the microstructure evolution is predicted within the
[39]
latent space . Since LDMs typically scale better with smaller dataset , we anticipate the accelerated
[39]
framework will outperform the PSDMs in modeling Owstwald ripening. Yet, as the size of database
increased or microstructural features become more complex, PSDMs are expected to outperform LDMs in
predictive performance for coupled microstructure evolution.

Although the accelerated framework developed in this study is for 2D Ostwald ripening problem, it is
anticipated that this framework can be easily extended to the 3D systems. This is because the
microstructural features in both 2D and 3D cases always share similar evolutionary behaviors, which
indicates that the key features in latent space demonstrate similar variations, making it easier for machine-
learning models to capture them. Additionally, our accelerated framework is applicable to the more
complex, coupled phase-field problems by incorporating various physical coupling interactions between
various fields, such as elastic strain, component diffusion heterogeneity, and inhomogeneous interface
energy. Once the new microstructural database is developed, the accelerated framework can be readily
retrained to capture the new microstructural features in latent representations and predict their evolution