Learning hydrodynamic equations for active matter from particle simulations and experiments*

Recent advances in particle-based simulation methods and high-resolution imaging techniques have enabled the precise characterization of collective dynamics in various biological and engineered active fluids. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying PDEs from continuum simulations. By contrast, learning macroscopic hydrodynamic equations and closure relations from microscopic particle simulations remains a major challenge. Here, we present a framework that leverages sparse regression learning algorithms to discover PDE models from coarse-grained microscopic data, while incorporating the relevant physical symmetries. We illustrate the practical potential through an application to a polar active particle model with alignment interactions mimicking those of swimming sperm cells. We further verify the framework with applications to recent micro-roller experiments and to tracked trajectories of the collective motion of animals. Our scheme succeeds in learning hydrodynamic equations that reproduce the characteristic dynamics observed in these systems, demonstrating how one can learn continuum theories directly from large-scale microscopic simulations and observations of complex systems.