Learning Interaction Dynamics from Particle Trajectories and Density Evolution

We propose a family of parametric interaction functions in the general Cucker-Smale model such that the mean-field macroscopic system of equations can be iteratively solved in an optimization scheme aiming to learn the interaction dynamics of the microscopic model from observations of macroscopic quantities. We treat the interaction functions as Green's functions for a semi-linear Poisson differential operator, which allows the transformation of the non-local interaction terms of the macroscopic model into a system of PDEs. The resulting system of hydrodynamic equations is efficiently solved as part of an iterative learning algorithm that estimates the interaction function from particle density evolution data. Finally, we utilize the proposed interaction function model to formulate an efficient learning algorithm based on observations from particle trajectories, and discuss the trade-offs associated with each approach.