Learning Mean-Field Equations from Particle Data Using WSINDy
This provides a fast and robust system identification method for large-scale particle systems, addressing a computational bottleneck in fields like physics and biology, though it is incremental as it builds on existing weak-form sparse identification techniques.
The authors tackled the problem of learning governing stochastic differential equations from interacting particle systems with large particle numbers and few experiments, achieving a convergence rate of O(N^{-1/2}) and demonstrating it on systems like swarms and chemotaxis models.
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 $N$ 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 $N$ is on the order of several thousand and the number of experiments $M$ is less than 100. This is in contrast to existing work showing that system identification for $N$ less than 100 and $M$ 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 $\mathcal{O}(N^{-1/2})$ 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.