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Inferring governing dynamics from noisy single-cell time-series typically suffers from numerical differentiation artifacts and fails to cleanly separate individual heterogeneity from measurement error. To resolve this, the authors combine Weak-form Estimation of Nonlinear Dynamics (WENDy) with an empirical Bayes random-effects model to simultaneously select candidate growth laws and infer single-cell parameter variations. Validated on synthetic and experimental single-cell size trajectories, the framework accurately reconstructs ground-truth nonlinear laws and latent parameter distributions under realistic noise, offering a principled path to bridge individual trajectories to population densities.
Weak-form dynamical inference can jointly uncover nonlinear governing equations and individual-level parameter heterogeneity across trajectory ensembles without succumbing to derivative-induced noise.
Single-cell time-lapse microscopy now provides high-resolution size trajectories for thousands of individual cells, yet principled methods for simultaneously learning the functional form of growth laws and the distribution of cell-to-cell physiological variability remain limited. We present a data-driven framework based on the recent Weak-form Estimation of Nonlinear Dynamics (WENDy) algorithm to (i) select the best supported growth law from a set of candidate laws, and (ii) recover individual-level growth parameters via empirical Bayes shrinkage within a random-effects model. We demonstrate the approach on both synthetic individual size data incorporating commonly found growth laws and recorded real size trajectories, showing that the method accurately recovers both growth-law structure and parameter heterogeneity at reasonable noise levels. We finally end with a discussion on how the results obtained by the method here can be upscaled to the population level.