Learning hydrodynamic equations from microscopic Langevin simulations of self-propelled particles dynamics

In nonequilibrium systems, a collective movement of microscopic active particles often displays several common emerging properties, such as swarming, motility-induced phase separation, disorder-order transitions, anomalous density fluctuation, spatiotemporal patterning, and unusual rheological properties. However, those universal aspects of collective behaviors are hardly captured from microscopic particle-based simulation methods. The macroscopic properties obtained from nonlinear hydrodynamic equations are useful for understanding those aspects. Therefore, we start from the numerical Langevin simulations of the microscopic particle dynamics and present a data-driven strategy for the collection of self-propelled particles to develop the hydrodynamics equations. In our method, microscopic particle data is our input. Hence, the hydrodynamics fields are obtained by coarse-graining from the discrete description of particle dynamics. For partial differential equation (PDE) learning, the spectral representation gives the efficient and accurate computation of spatial and temporal derivatives of density and polarization density fields. Using sparse regression on the fields, we generate hydrodynamic equations.

The estimated PDEs from microscopic models are beneficial to understand the universal features of the system in comparison to standard supervised learning. Hence, the macroscopic features will be shared both by microscopic models and hydrodynamic equations.