Gradient-Flow SDEs Have Unique Transient Population Dynamics

Vincent Guan, Joseph Janssen, Nicolas Lanzetti, Antonio Terpin, Geoffrey Schiebinger, Elina Robeva

TL;DR: We prove that outside of equilibrium, both the drift and diffusion of a gradient-flow SDE are identifiable from its marginals, and we develop the first Schrodinger Bridge-based method that can infer the full SDE solely from marginals.

Abstract: Identifying the drift and diffusion of an SDE from its population dynamics is a notoriously challenging task. Researchers in machine learning and single-cell biology have only been able to prove a partial identifiability result: for potential-driven SDEs, the gradient-flow drift can be identified from temporal marginals if the Brownian diffusivity is already known. Existing methods therefore assume that the diffusivity is known a priori, despite it being unknown in practice. We dispel the need for this assumption by providing a complete characterization of identifiability: the gradient-flow drift and Brownian diffusivity are jointly identifiable from temporal marginals if and only if the process is observed outside of equilibrium. Given this fundamental result, we propose nn-APPEX, the first Schrödinger Bridge–based inference method that can simultaneously learn the drift and diffusion of a gradient-flow SDE solely from observed marginals. Extensive experiments show that nn-APPEX's ability to adjust its diffusion estimate enables accurate inference, while previous Schrödinger Bridge methods obtain biased drift estimates due to their assumed, and likely incorrect, diffusion.

Code Dataset Promise: Yes

Code Dataset Url: https://github.com/guanton/identifying-gradient-flow-sdes

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Submission Number: 1072