  Interacting particle system (IPS) models have proven to be highly successful
for describing the spatial movement of organisms. However, it has proven
challenging to infer the interaction rules directly from data. In the field of
equation discovery, the Weak form Sparse Identification of Nonlinear Dynamics
(WSINDy) methodology has been shown to be very computationally efficient for
identifying the governing equations of complex systems, even in the presence of
substantial noise. Motivated by the success of IPS models to describe the
spatial movement of organisms, we develop WSINDy for second order IPSs to model
the movement of communities of cells. Specifically, our approach learns the
directional interaction rules that govern the dynamics of a heterogeneous
population of migrating cells. Rather than aggregating cellular trajectory data
into a single best-fit model, we learn the models for each individual cell.
These models can then be efficiently classified according to the active classes
of interactions present in the model. From these classifications, aggregated
models are constructed hierarchically to simultaneously identify different
species of cells present in the population and determine best-fit models for
each species. We demonstrate the efficiency and proficiency of the method on
several test scenarios, motivated by common cell migration experiments.
