Daniel Messenger, Department of Applied Mathematics, University of Colorado Boulder

Recent advances in Weak-Form Equation Learning with applications to Collective Cell Migration

Weak-form Equation Learning has the potential to efficiently solve expensive inference tasks in cell biology. We will demonstrate this using the Weak-Form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm in the discovery of nonlocal equations relevant to collective cell migration, and in a cell-sorting application given a heterogeneous populations of unlabeled cells. Until recently, the benefits of weak-form methodologies in equation learning (efficiency, accuracy, robustness to noise, low regularity requirements, etc.) have only been observed empirically. In this talk we will discuss our recent results concerning provable recovery guarantees for WSINDy, and in particular, how our theoretical findings have revealed a key limitation of equation learning methods. To correct for this limitation, originating in a misspecification of the underlying statistical model of the data inherent to all SINDy-based methods, we present the WENDy algorithm (weak-form estimation of nonlinear dynamics), which improves the inner-loop parameter estimates within WSINDy by iteratively modifying the goodness-of-fit measure. After reviewing WSINDy in the context of inference problems in cell biology, we will present recent work on theoretical recovery guarantees and the WENDy algorithm.