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

Phase-Field DeepONet: Physics-informed deep operator neural network for  fast simulations of pattern formation governed by gradient flows of  free-energy functionals