  Large collections of coupled, heterogeneous agents can manifest complex
dynamical behavior presenting difficulties for simulation and analysis.
However, if the collective dynamics lie on a low-dimensional manifold then the
original agent-based model may be approximated with a simplified surrogate
model on and near the low-dimensional space where the dynamics live. This is
typically accomplished by deriving coarse variables that summarize the
collective dynamics, these may take the form of either a collection of scalars
or continuous fields (e.g. densities), which are then used as part of a reduced
model. Analytically identifying such simplified models is challenging and has
traditionally been accomplished through the use of mean-field reductions or an
Ott-Antonsen ansatz, but is often impossible.
  Here we present a data-driven coarse-graining methodology for discovering
such reduced models. We consider two types of reduced models: globally-based
models which use global information and predict dynamics using information from
the whole ensemble, and locally-based models that use local information, that
is, information from just a subset of agents close (close in heterogeneity
space, not physical space) to an agent, to predict the dynamics of an agent.
For both approaches we are able to learn laws governing the behavior of the
reduced system on the low-dimensional manifold directly from time series of
states from the agent-based system. A nontrivial conclusion is that the
dynamics can be equally well reproduced by an all-to-all coupled as well as by
a locally coupled model of the same agents.
