Machine-learning-based data-driven discovery of nonlinear phase-field dynamics

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 algorithms have significantly empowered the discovery process of governing equations directly from data. However, it remains difficult to discover partial differential equations (PDEs) with high-order derivatives. In this paper, we present data-driven architectures based on a multilayer perceptron, a convolutional neural network (CNN), and a combination of a CNN and long short-term memory structures for discovering the nonlinear equations of motion for phase-field models with nonconserved and conserved order parameters. The well-known Allen-Cahn, Cahn-Hilliard, and phase-field crystal models were used as test cases. Two conceptually different types of implementations were used: (a) guided by physical intuition (such as the 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.