Self-organization can be used to explain crystal formation, aggregation of cells, social behaviors of insects, synchronization of heart beats, etc. It is challenging to understand these types of phenomena from the mathematical point of view. We offer a statistical/machine learning approach to understand these behaviors from observation; moreover, our learning approach can aid in validating and improving the modeling of self-organization. 

We develop a learning framework to derive physically meaningful dynamical models to explain the observation. We show the convergence property of our learning method in terms of the number of different initial conditions for first-order systems of homogeneous agents, and investigate its performance for various first- and second-order systems of heterogeneous agents. We also study the steady state properties of our learned models. We extend the learning approach to dynamical constrained on Riemannian manifolds, and we provide a convergence study for second order systems.
Finally, we apply our learning method on the NASA Jet Propulsion Laboratory's modern Ephemerides.