Gradient-flow SDEs have unique transient population dynamics

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 Schr\"odinger Bridge-based inference method that can simultaneously learn the drift and diffusion of gradient-flow SDEs solely from observed marginals. Extensive numerical experiments show that nn-APPEX's ability to adjust its diffusion estimate enables accurate inference, while previous Schr\"odinger Bridge methods obtain biased drift estimates due to their assumed, and likely incorrect, diffusion.