Inferring Mean-field Models from Microscale Observations: Strengths of a Weak Form

Abstract: Complex macroscale behavior, from cell migration to turbulence, can often be reduced to a sparse system of equations describing an underlying microscale process. An opposite pursuit is to find an effective macroscale description from microscale observations. In general, the inference of governing equations from experimental data, whether at the microscale or macroscale, has been a holy grail of scientific research for decades. In this talk I will describe an approach based on weak formulations of differential equations for inferring equations from data at a range of scales, not only those inherent to the observed data. By viewing the dynamics through the guise of test functions, so-called “weak-form inference methods” are designed to treat non-smooth dynamics and data corrupted by noise. I will show that these properties lead to a more general consequence, that the weak form can be used to reveal effective macroscale descriptions from microscale observations, such as mean-field and coarse-grained models, and in some cases identify microscale rules through their embedding in macroscale equations. After providing a broad overview of weak-form inference methods, with emphasis on the WSINDy algorithm (Weak-form Sparse Identification of Nonlinear Dynamics), I will describe recent results attempting to infer interaction rules for migrating cell populations, and in finding moment closure models and homogenized equations from particle simulations at the boundary between kinetic and hydrodynamic phenomena.