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

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.

38 pages; 9 figures

References in corpus (11)

• Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm
• Nonparametric inference of interaction laws in systems of agents from trajectory data
• Uniform convergence to equilibrium for granular media
• Kinetic models of opinion formation in the presence of personal conviction
• A Comparison of Methods for Computing Autocorrelation Time
• Learning interaction kernels in heterogeneous systems of agents from multiple trajectories
• A stochastic version of Stein Variational Gradient Descent for efficient sampling
• $L^p$ Bernstein Inequalities and Inverse Theorems for RBF Approximation on ${\mathbb{R}}^d$
• Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data
• On the identifiability of interaction functions in systems of interacting particles
• Cluster Prediction for Opinion Dynamics from Partial Observations

Cited by in corpus (3)

• Random Feature Models for Learning Interacting Dynamical Systems
• Learning dynamics on invariant measures using PDE-constrained optimization
• Learning particle swarming models from data with Gaussian processes