Macroscopic dynamical descriptions of complex physical systems are crucial for understanding and controlling material behavior. With the growing availability of data and compute, machine learning has become a promising alternative to first-principles methods to build accurate macroscopic models from microscopic trajectory simulations. However, for spatially extended systems, direct simulations of sufficiently large microscopic systems that inform macroscopic behavior are prohibitive. In this work, we propose a framework that learns the macroscopic dynamics of large stochastic microscopic systems using only small-system simulations. Our framework employs a partial evolution scheme to generate training data pairs by evolving large-system snapshots within local patches. We subsequently derive the closure variables associated with the macroscopic observables and learn the macroscopic dynamics using a custom loss. Furthermore, we introduce a hierarchical upsampling scheme that enables the efficient generation of large-system snapshots from small-system snapshots. We empirically demonstrate the accuracy and robustness of our framework through a variety of stochastic spatially extended systems, including those described by stochastic partial differential equations, idealized lattice spin systems, and a more realistic NbMoTa alloy system.
Chen et al. (Fri,) studied this question.