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Many complex systems operating far from the equilibrium exhibit stochasticdynamics that can be described by a Langevin equation. Inferring Langevinequations from data can reveal how transient dynamics of such systems give riseto their function. However, dynamics are often inaccessible directly and can beonly gleaned through a stochastic observation process, which makes theinference challenging. Here we present a non-parametric framework for inferringthe Langevin equation, which explicitly models the stochastic observationprocess and non-stationary latent dynamics. The framework accounts for thenon-equilibrium initial and final states of the observed system and for thepossibility that the system's dynamics define the duration of observations.Omitting any of these non-stationary components results in incorrect inference,in which erroneous features arise in the dynamics due to non-stationary datadistribution. We illustrate the framework using models of neural dynamicsunderlying decision making in the brain.

Learning non-stationary Langevin dynamics from stochastic observations of latent trajectories