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 1dimensional 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.

Learning particle swarming models from data with Gaussian processes

“…The positivity of c H K is related to the positivity of relevant integral operators. Leveraging the recent advancement in [31,32], our ongoing work shows that, in the case of L = 1, the coercivity constant c H K ≥ N −1 N 2 for general systems and H K . Furthermore, it can be a positive constant independent of N if µ 0 is exchangeable Gaussian.…”

“…The learning theory developed in this paper is related to but significantly departs from the ones for the previous approach [29,[31][32][33]. All of these works investigate the performance of estimators as M → ∞.…”

Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles

“…Note that s ρ T depends on the initial distribution up¨, 0q and the true interaction kernel φ. We point out that the measure s ρ T is different from the empirical measure of pairwise distances in particle systems [22,21,19], because s X t and s X 1 t are independent copies and are no longer interacting particles. However, in view of inference, the high probability region of s ρ T is where ˇˇs X t ´s X 1 t ˇˇexplores the interaction kernel the most, as such, the natural function space of inference is L 2 ps ρ T q.…”

“…When these basis functions are orthonormal in L 2 ps ρ T q, we need the following coercivity condition. It extends of the coercivity condition for Nparticle systems defined in [19,21,22]. Definition 3.3 (Coercivity condition).…”

“…Thus, it is crucial to develop new methods beyond parametric estimation (see e.g., [9]). Towards this direction, recent efforts [2,22,20,21,31] estimate the kernel by nonparametric regression for systems with finitely many particles from multiple trajectories. For large systems, data of trajectories of all particles are often unavailable, instead, it is practical to consider data consisting of a macroscopic concentration density of the particles, i.e., the solution of the mean-field equation.…”