Inferring the laws of interaction in agent-based systems from observational data is a fundamental challenge in a wide variety of disciplines. We propose a non-parametric statistical learning approach for distance-based interactions, with no reference or assumption on their analytical form, given data consisting of sampled trajectories of interacting agents. We demonstrate the effectiveness of our estimators both by providing theoretical guarantees that avoid the curse of dimensionality, and by testing them on a variety of prototypical systems used in various disciplines. These systems include homogeneous and heterogeneous agents systems, ranging from particle systems in fundamental physics to agent-based systems that model opinion dynamics under the social influence, preypredator dynamics, flocking and swarming, and phototaxis in cell dynamics.

Let Y = {f (i), Af (i), . . . , A l i f (i) : i ∈ Ω}, where A is a bounded operator on 2 (I). The problem under consideration is to find necessary and sufficient conditions on A, Ω, {li : i ∈ Ω} in order to recover any f ∈ 2 (I) from the measurements Y . This is the so called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states A l f . We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, the Müntz-Szász Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that Y can never be a Riesz basis when Ω is finite. We can also show that, when Ω is finite, Y = {f (i), Af (i), . . . , A l i f (i) : i ∈ Ω} is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H 2 (D).

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 one-dimensional nonparametric 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 spacings 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.

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