Inferring interaction rules from observations of evolutive systems I: The variational approach

Abstract

In this paper we are concerned with the learnability of nonlocal interaction kernels for first order systems modeling certain social interactions, from observations of realizations of their dynamics. This paper is the first of a series on learnability of nonlocal interaction kernels and presents a variational approach to the problem. In particular, we assume here that the kernel to be learned is bounded and locally Lipschitz continuous and that the initial conditions of the systems are drawn identically and independently at random according to a given initial probability distribution. Then the minimization over a rather arbitrary sequence of (finite dimensional) subspaces of a least square functional measuring the discrepancy from observed trajectories produces uniform approximations to the kernel on compact sets. The convergence result is obtained by combining mean-field limits, transport methods, and a Γ-convergence argument. A crucial condition for the learnability is a certain coercivity property of the least square functional, majoring an L 2 -norm discrepancy to the kernel with respect to a probability measure, depending on the given initial probability distribution by suitable push forwards and transport maps. We illustrate the convergence result by means of several numerical experiments.

Nonparametric inference of interaction laws in systems of agents from trajectory data

“…3 and similar experiments for the other systems ) do suggest that the estimator does improve as L increases, at least to a point, limited by the information contained in a single trajectory. Comparing to [6], where the mean field limit N → ∞, M = 1, is studied, we see the rates in [6] are no better than N −1/d , i.e. they are cursed by dimension.…”

“…We illustrate the learning procedure on a particle system with N = 7 particles in R 2 , interacting according to (1) with φ(r) = Φ LJ (r)/r, where Φ LJ (r) := 4 (σ/r) 12 − (σ/r) 6 is the Lennard-Jones potential, consisting of a strong near-field repulsion and a long-range attraction. The system converges quickly to equilibrium configurations, which often consist of ordered, crystal-like structures.…”

Learning interacting particle systems: Diffusion parameter estimation for aggregation equations

“…The problem is also related to uncertainty quantification, which is important as it enables building of more realistic models and making better predictions of their behavior in the future. In the modeling of self-organized systems, different ways to qualify uncertainties have been studied (see for example [2,8,15,19,37,49]).…”

“…where 1 ≪ K ≪ N , which means we only have partial observations. Our main result quantifies the estimation error of the proposed estimator (8), which is summarized as below…”

“…at time t n . For any α > 0, there exists some constant N 0 > 0 depending only on ν, α, T and ρ 0 L 1 ∩L ∞ (R d ) , such that for N ≥ N 0 , the estimator ν defined in (8) is an approximation of ν, and the following estimate holds…”