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

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 meanfield limits, transport methods, and a Gamma-convergence argument. A crucial condition for the learnability is a certain coercivity property of the least square functional, defined by the majorization of 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.