  Coarse-grained descriptions of collective motion of flocking systems are
often derived for the macroscopic or the thermodynamic limit. However, many
real flocks are small sized (10 to 100 individuals), called the mesoscopic
scales, where stochasticity arising from the finite flock sizes is important.
Developing mesoscopic scale equations, typically in the form of stochastic
differential equations, can be challenging even for the simplest of the
collective motion models. Here, we take a novel data-driven equation learning
approach to construct the stochastic mesoscopic descriptions of a simple
self-propelled particle (SPP) model of collective motion. In our SPP model, a
focal individual can interact with k randomly chosen neighbours within an
interaction radius. We consider k = 1 (called stochastic pairwise
interactions), k = 2 (stochastic ternary interactions), and k equalling all
available neighbours within the interaction radius (equivalent to Vicsek-like
local averaging). The data-driven mesoscopic equations reveal that the
stochastic pairwise interaction model produces a novel form of collective
motion driven by a multiplicative noise term (hence termed, noise-induced
flocking). In contrast, for higher order interactions (k > 1), including
Vicsek-like averaging interactions, yield collective motion driven primarily by
the deterministic forces. We find that the relation between the parameters of
the mesoscopic equations describing the dynamics and the population size are
sensitive to the density and to the interaction radius, exhibiting deviations
from mean-field theoretical expectations. We provide semi-analytic arguments
potentially explaining these observed deviations. In summary, our study
emphasizes the importance of mesoscopic descriptions of flocking systems and
demonstrates the potential of the data-driven equation discovery methods for
complex systems studies.
