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Systems of interacting particles or agents have wide applications in manydisciplines such as Physics, Chemistry, Biology and Economics. These systemsare governed by interaction laws, which are often unknown: estimating them fromobservation data is a fundamental task that can provide meaningful insights andaccurate predictions of the behaviour of the agents. In this paper, we considerthe inverse problem of learning interaction laws given data from multipletrajectories, in a nonparametric fashion, when the interaction kernels dependon pairwise distances. We establish a condition for learnability of interactionkernels, and construct estimators that are guaranteed to converge in a suitable$L^2$ space, at the optimal min-max rate for 1-dimensional nonparametricregression. We propose an efficient learning algorithm based on least squares,which can be implemented in parallel for multiple trajectories and is thereforewell-suited for the high dimensional, big data regime. Numerical simulations ona variety examples, including opinion dynamics, predator-swarm dynamics andheterogeneous particle dynamics, suggest that the learnability condition issatisfied in models used in practice, and the rate of convergence of ourestimator is consistent with the theory. These simulations also suggest thatour estimators are robust to noise in the observations, and produce accuratepredictions of dynamics in relative large time intervals, even when they arelearned from data collected in short time intervals.

Learning interaction kernels in heterogeneous systems of agents from multiple trajectories