Collective behaviors (clustering, flocking, milling, etc.) are among the most interesting and challenging phenomena to understanding from the mathematical point of view. We offer a non-parametric learning approach to discover the governing structure, i.e. the interaction functions between agents, of collective dynamics from observation of the trajectory data. Our learning approach can aid in validating and improving the modeling of collective dynamics.
Having established the convergent properties of our learning approach, we investigate the steady state behavior of the learned dynamics evolved using the estimators inferred from observation data. We then apply our extended learning approach to study the celestial motion of the Solar system using the NASA JPL's modern Ephemeris. We are able to reproduce trajectory data with a precession rate of 544'' per Earth-century for Mercury's orbit. Compared to Newton's theoretical 532'' rate and the observed 575'' rate, we are able to learn portion of the general relativity effect directly from the data. Convergence properties of the extended learning approaches on second-order models are analyzed. Learning collective dynamics on non-Euclidean manifolds has been developed and discussed.