  We introduce a nonparametric algorithm to learn interaction kernels of
mean-field equations for 1st-order systems of interacting particles. The data
consist of discrete space-time observations of the solution. By least squares
with regularization, the algorithm learns the kernel on data-adaptive
hypothesis spaces efficiently. A key ingredient is a probabilistic error
functional derived from the likelihood of the mean-field equation's diffusion
process. The estimator converges, in a reproducing kernel Hilbert space and an
L2 space under an identifiability condition, at a rate optimal in the sense
that it equals the numerical integrator's order. We demonstrate our algorithm
on three typical examples: the opinion dynamics with a piecewise linear kernel,
the granular media model with a quadratic kernel, and the aggregation-diffusion
with a repulsive-attractive kernel.
