  Flocking refers to collective behavior of a large number of interacting
entities, where the interactions between discrete individuals produce
collective motion on the large scale. We employ an agent-based model to
describe the microscopic dynamics of each individual in a flock, and use a
fractional PDE to model the evolution of macroscopic quantities of interest.
The macroscopic models with phenomenological interaction functions are derived
by applying the continuum hypothesis to the microscopic model. Instead of
specifying the fPDEs with an ad hoc fractional order for nonlocal flocking
dynamics, we learn the effective nonlocal influence function in fPDEs directly
from particle trajectories generated by the agent-based simulations. We
demonstrate how the learning framework is used to connect the discrete
agent-based model to the continuum fPDEs in 1D and 2D nonlocal flocking
dynamics. In particular, a Cucker-Smale particle model is employed to describe
the microscale dynamics of each individual, while Euler equations with nonlocal
interaction terms are used to compute the evolution of macroscale quantities.
The trajectories generated by the particle simulations mimic the field data of
tracking logs that can be obtained experimentally. They can be used to learn
the fractional order of the influence function using a Gaussian process
regression model implemented with the Bayesian optimization. We show that the
numerical solution of the learned Euler equations solved by the finite volume
scheme can yield correct density distributions consistent with the collective
behavior of the agent-based system. The proposed method offers new insights on
how to scale the discrete agent-based models to the continuum-based PDE models,
and could serve as a paradigm on extracting effective governing equations for
nonlocal flocking dynamics directly from particle trajectories.
