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.

Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate solvable approximate equations for the variables of greater interest, and use data and statistical methods to account for the impact of the other variables. In the present paper we consider time-dependent problems and introduce a fully discrete solution method, which simplifies both the analysis of the data and the numerical algorithms. The resulting time series are identified by a NARMAX (nonlinear autoregression moving average with exogenous input) representation familiar from engineering practice. The connections with the MoriZwanzig formalism of statistical physics are discussed, as well as an application to the Lorenz 96 system.T here are many time-dependent problems in science, where the equations of motion are too complex for full solution, either because the equations are not certain or because the computational cost is too high, but one is interested only in the dynamics of a subset of the variables. Such problems appear, for example, in weather and climate modeling, e.g., refs. 1, 2; in economics, e.g., ref. 3; in statistical mechanics, e.g., refs. 4, 5; and in the mechanics of turbulent flow, e.g., refs. 6, 7. In these settings it is natural to look for simpler equations that involve only the variables of interest, and then use data to assess the effect of the remaining variables on the variables of interest in some approximate way. In the present paper we focus on stochastic methods for doing this, with data coming either from experimental observations or from prior numerical calculations.Consider a set of differential equations of the formwhere t is the time, x = ðx 1 , x 2 , . . . , x m Þ is the vector of resolved variables, and y = ðy 1 , y 2 , . . . , y ℓ Þ is the vector of unresolved variables, where ℓ is not necessarily finite, with initial data xð0Þ = α, yð0Þ = β. Consider a situation where this system is too complicated to solve, but where data are available, usually as sequences of measured values of x, assumed here to be observed with negligible observation errors. One can write Rðx, yÞ in the form Rðx, yÞ = R 0 ðxÞ + zðx, yÞ,where R 0 approximates Rðx, yÞ in some sense and is such that one is able to solve the equationThe remainder zðx, yÞ = Rðx, yÞ − R 0 ðxÞ is the contribution of the unresolved variables to the equation for x and must be taken into account. It has been variously called "unresolved tendency" (8, 9), "model error" (10, 11), or "model noise" (12); we call it simply the "noise." In the present paper we do not discuss general methods for constructing R 0 ; they depend on practical considerations which differ from case to case. A usual approach to the problem of computing x is to identify z from data (8, 13, 14), i.e., find a concise approximate representationẑ of z that can be evaluated on the computer, and then solve the equationWhenẑ is a...

No abstract

The problem of constructing data-based, predictive, reduced models for the Kuramoto-Sivashinsky equation is considered, under circumstances where one has observation data only for a small subset of the dynamical variables. Accurate prediction is achieved by developing a discrete-time stochastic reduced system, based on a NARMAX (Nonlinear Autoregressive Moving Average with eXogenous input) representation. The practical issue, with the NARMAX representation as with any other, is to identify an efficient structure, i.e., one with a small number of terms and coefficients. This is accomplished here by estimating coefficients for an approximate inertial form. The broader significance of the results is discussed.

No 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 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.

No abstract

No abstract