A stochastic framework for evolving grain statistics using a neural network model for grain topology transformations

We develop a stochastic framework to evolve grain statistics during 2D isotropic grain growth. Grain microstructures observed in simulations and experiments often demonstrate the existence of asymptotic distributions for descriptors such as grain size and topology (number of sides). Moreover, such grain descriptors/states are used to characterize a microstructure as they influence the mechanical properties of a polycrystal. This motivates us to discover laws that govern the distributions of grain microstructure descriptors. In particular, we focus on the evolution of distributions for grain sizes and the number of sides. Since the surface tension is assumed to be isotropic, the von Neumann–Mullins law provides an evolution equation for grain areas in terms of their topological states—number of edges. However, since grains change their topology as they evolve by interacting with neighbors, we construct a topology transformation model (TTM) that predicts the probability of topology transformation of a grain in terms of its current state and the states of its neighbors. The construction of the TTM relies on a data-driven approach using a fully connected deep neural network. Topology transformations recorded in phase field simulations are used as training data. The resulting neural network model is used in a Monte Carlo framework to evolve grain microstructures in a statistical sense. The stochastic method is validated using the asymptotic and transient grain statistics predicted by large-scale phase field simulations. In addition, our results suggest that the TTM incorporates, in a statistical sense, the geometric constraint that grains fill the space.