Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories

Understand

We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances.

• We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories.
• We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data.
• We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of $1$-dimensional non-parametric regression.