  This paper delves into a nonparametric estimation approach for the
interaction function within diffusion-type particle system models. We introduce
two estimation methods based upon an empirical risk minimization. Our study
encompasses an analysis of the stochastic and approximation errors associated
with both procedures, along with an examination of certain minimax lower
bounds. In particular, we show that there is a natural metric under which the
corresponding minimax estimation error of the interaction function converges to
zero with parametric rate. This result is rather suprising given complexity of
the underlying estimation problem and rather large classes of interaction
functions for which the above parametric rate holds.
