LEARNING INTERACTION KERNELS IN MEAN-FIELD EQUATIONS OF FIRST-ORDER SYSTEMS OF INTERACTING PARTICLES

We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for first-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 efficiently on data-adaptive hypothesis spaces. A key ingredient is a probabilistic error functional derived from the likelihood ratio of the mean-field equation's diffusion process. The estimator converges in a weighted L² space at a rate determined by the trade-off between the numerical error and approximation error. 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.

Duke Scholars

Author Quanjun Lang Mathematics