  Recent advances in scientific machine learning have shed light on the
modeling of pattern-forming systems. However, simulations of real patterns
still incur significant computational costs, which could be alleviated by
leveraging large image datasets. Physics-informed machine learning and operator
learning are two new emerging and promising concepts for this application.
Here, we propose "Phase-Field DeepONet", a physics-informed operator neural
network framework that predicts the dynamic responses of systems governed by
gradient flows of free-energy functionals. Examples used to validate the
feasibility and accuracy of the method include the Allen-Cahn and Cahn-Hilliard
equations, as special cases of reactive phase-field models for nonequilibrium
thermodynamics of chemical mixtures. This is achieved by incorporating the
minimizing movement scheme into the framework, which optimizes and controls how
the total free energy of a system evolves, instead of solving the governing
equations directly. The trained operator neural networks can work as explicit
time-steppers that take the current state as the input and output the next
state. This could potentially facilitate fast real-time predictions of
pattern-forming dynamical systems, such as phase-separating Li-ion batteries,
emulsions, colloidal displays, or biological patterns.
