  Many complex systems operating far from the equilibrium exhibit stochastic
dynamics that can be described by a Langevin equation. Inferring Langevin
equations from data can reveal how transient dynamics of such systems give rise
to their function. However, dynamics are often inaccessible directly and can be
only gleaned through a stochastic observation process, which makes the
inference challenging. Here we present a non-parametric framework for inferring
the Langevin equation, which explicitly models the stochastic observation
process and non-stationary latent dynamics. The framework accounts for the
non-equilibrium initial and final states of the observed system and for the
possibility 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 data
distribution. We illustrate the framework using models of neural dynamics
underlying decision making in the brain.
