Learning mean-field equations from particle data using WSINDy.

We develop a weak-form sparse identification method for interacting particle systems (IPS) with the primary goals of reducing computational complexity for large particle number and offering robustness to either intrinsic or extrinsic noise. In particular, we use concepts from mean-field theory of IPS in combination with the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) to provide a fast and reliable system identification scheme for recovering the governing stochastic differential equations for an IPS when the number of particles per experiment is on the order of several thousands and the number of experiments is less than 100. This is in contrast to existing work showing that system identification for less than 100 and on the order of several thousand is feasible using strong-form methods. We prove that under some standard regularity assumptions the scheme converges with rate in the ordinary least squares setting and we demonstrate the convergence rate numerically on several systems in one and two spatial dimensions. Our examples include a canonical problem from homogenization theory (as a first step towards learning coarse-grained models), the dynamics of an attractive-repulsive swarm, and the IPS description of the parabolic-elliptic Keller-Segel model for chemotaxis. Code is available at https://github.com/MathBioCU/WSINDy_IPS.