Inferring Interaction Rules From Observations of Evolutive Systems I: The Variational Approach

TL;DR

This paper introduces a variational method for learning nonlocal interaction kernels in social dynamics models from observed data, ensuring uniform approximation and convergence under certain conditions.

Contribution

It presents a novel variational approach for inferring interaction kernels, combining mean-field limits and $ ext{Gamma}$-convergence for the first time.

Findings

Uniform approximation of kernels achieved on compact sets.

Convergence proven under coercivity and Lipschitz conditions.

Numerical experiments validate the theoretical results.

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 $\Gamma$-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.