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

TL;DR

This paper develops a nonparametric method to infer interaction kernels in stochastic particle systems from multiple trajectories, achieving near-optimal convergence rates and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a data-adaptive regularized maximum likelihood estimator for interaction kernels, proving its consistency and near-optimal learning rate, independent of high-dimensional state spaces.

Findings

Estimator converges at a near-optimal rate of 1-dimensional non-parametric regression.

Discretization errors are of order 1/2, affecting convergence when large.

Numerical tests validate the efficiency and accuracy of the proposed algorithm.

Abstract

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. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order $1/2$ in terms of the time gaps between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel algorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.