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Abstract

Stochastic differential equations (SDEs) are a fundamental tool for modelling dynamic processes, including gene regulatory networks (GRNs), contaminant transport, financial markets, and image generation. However, learning the underlying SDE from data is a challenging task, especially if individual trajectories are not observable. Motivated by research in single-cell datasets, we present the first comprehensive approach for jointly identifying the drift and diffusion of an SDE from its temporal marginals. First, a detailed explanation of the trajectory inference problem is given, and its connection to entropic optimal transport formulated as a convex variational problem is established. Assuming linear drift and additive diffusion, it is then proved that these parameters are identifiable from marginals if and only if the initial distribution lacks any generalized rotational symmetries. Further, it is shown that the causal graph of any SDE with additive diffusion can be recovered from the SDE parameters, thereby linking the problem to causal discovery. To complement this theory, entropy-regularized optimal transport is adapted to handle anisotropic diffusion, and APPEX (Alternating Projection Parameter Estimation from X0) is introduced as an iterative algorithm designed to estimate the drift, diffusion, and trajectories solely from temporal marginals. It is demonstrated that APPEX iteratively decreases Kullback–Leibler divergence to the true solution, and its effectiveness is illustrated on simulated data from linear additive noise SDEs.