  One of the main questions regarding complex systems at large scales concerns
the effective interactions and driving forces that emerge from the detailed
microscopic properties. Coarse-grained models aim to describe complex systems
in terms of coarse-scale equations with a reduced number of degrees of freedom.
Recent developments in machine learning (ML) algorithms have significantly
empowered the discovery process of the governing equations directly from data.
However, it remains difficult to discover partial differential equations (PDEs)
with high-order derivatives. In this paper, we present new data-driven
architectures based on multi-layer perceptron (MLP), convolutional neural
network (CNN), and a combination of CNN and long short-term memory (CNN-LSTM)
structures for discovering the non-linear equations of motion for phase-field
models with non-conserved and conserved order parameters. The well-known
Allen--Cahn, Cahn--Hilliard, and the phase-field crystal (PFC) models were used
as the test cases. Two conceptually different types of implementations were
used: (a) guided by physical intuition (such as local dependence of the
derivatives) and (b) in the absence of any physical assumptions (black-box
model). We show that not only can we effectively learn the time derivatives of
the field in both scenarios, but we can also use the data-driven PDEs to
propagate the field in time and achieve results in good agreement with the
original PDEs.
