Page 80 - Read Online

Page 2 of 20 Gesch et al. J. Mater. Inf. 2025, 5, 42 https://dx.doi.org/10.20517/jmi.2025.23

INTRODUCTION
Phase-field simulation is a highly powerful computational tool for modeling the dynamic behaviors of
[1-4]
microstructures across a wide range of material systems . It employs a variety of approaches, from
continuum theory to classical thermodynamics, to explore the evolution of material interfaces in both
spatial and temporal resolutions. Unlike atomistic simulations, such as molecular dynamics and first-
principles, phase-field simulations do not rely on atomistic details of the materials, enabling them to study
materials dynamics over large length and time scales. For instance, these simulations are always used to
[5,6]
model the microstructural evolutions of materials in various scenarios such as spinodal decomposition ,
Ostwald ripening , grain growth [8-10] , and dendrite growth [11-13] . Additionally, phase-field simulations have
[7]
been broadly applied to practical engineering problems in solid mechanics [14-16] , fluid mechanics [17-19] ,
fracture mechanics [20-22] , and other disciplines [23-25] .

A major drawback of high-fidelity phase-field simulations is their high computational cost, which stems
from the need to solve a system of partial differential equations (PDEs) for various continuous variables that
evolve both in space and in time . In addition, phase-field simulations typically require refined spatio-
[1]
temporal discretization and an accurate description of the free energy of the entire system, making them
cumbersome to implement for a specific engineering application. To effectively solve the convoluted PDEs,
direct numerical solvers, such as finite-elements [26-28] , finite-difference [29-31] , finite-volume , and spectral
[32]
methods [33-35] , are widely adopted for phase-field simulations. Despite their improved effectiveness, these
numerical schemes still require a large number of degrees of freedom and substantial computational
resources to obtain accurate spatial and temporal solutions [33,36,37] . This limits the performance of phase-field
simulation, particularly when dealing with large or complex problems. One approach to accelerate the
solution of PDEs is to leverage the high-performance computing (HPC) architectures. For instance,
researchers have found that graphics processing unit (GPU)-based algorithm can speed up phase-field
simulation by 50 times compared to central processing unit (CPU) implementations . Yet, the high
[38]
demand and long waiting times for HPC resources may limit their effectiveness in speeding up phase-field
simulations. As a result, developing alternative strategies to accelerate phase-field simulations has become
an urgent research task.

As an emerging alternative, machine learning techniques are rapidly gaining research interest in the phase-
field communities due to their universal function approximation capabilities . For instance, deep neural
[39]
networks (DNNs) have been adopted to learn the high-dimensional free energy functions and their
derivatives from atomistic simulations to solve PDEs for phase-field simulations [40-42] . Data-driven surrogate
models, which use a reduced-order statistical representation of the microstructures combined with
regression models to link phase-field input parameters (e.g., deposition rate and phase fraction), have been
employed to forecast the corresponding microstructure at a given time [43,44] . Although these examples
illustrate the huge potential to enhance the usability of integrating machine-learning models into phase-field
simulations, such integration must be systematically examined to ensure high credibility. For instance, the
accuracy and robustness of these integrated machine-learning models highly rely on the quality of the
microstructure database used to train them. Generating these databases is both costly and often challenging
to achieve, especially for real-world engineering problems.

Another strategy involves developing an artificial microstructure-learning engine to accelerate the
prediction of microstructural evolution into the unknown future time sequence. Based on the architecture
of these learning engines, two major types of accelerated phase-field frameworks have been developed to
date: latent dynamics models (LDMs) and pixel-space dynamics models (PSDMs) . LDMs typically learn
[39]
microstructures in the latent space using dimensionality reduction techniques, and then employ history-