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We introduce a nonparametric algorithm to learn interaction kernels ofmean-field equations for 1st-order systems of interacting particles. The dataconsist of discrete space-time observations of the solution. By least squareswith regularization, the algorithm learns the kernel on data-adaptivehypothesis spaces efficiently. A key ingredient is a probabilistic errorfunctional derived from the likelihood of the mean-field equation's diffusionprocess. The estimator converges, in a reproducing kernel Hilbert space and anL2 space under an identifiability condition, at a rate optimal in the sensethat it equals the numerical integrator's order. We demonstrate our algorithmon three typical examples: the opinion dynamics with a piecewise linear kernel,the granular media model with a quadratic kernel, and the aggregation-diffusionwith a repulsive-attractive kernel.

Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles