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Estimating time-dependent entropy production from non-equilibrium trajectories

The rate of entropy production provides a useful quantitative measure of a non-equilibrium system and estimating it directly from time-series data from experiments is highly desirable. Several approaches have been considered for stationary dynamics, some of which are based on a variational characterization of the entropy production rate. However, the issue of obtaining it in the case of non-stationary dynamics remains largely unexplored. Here, we solve this open problem by demonstrating that the variational approaches can be generalized to give the exact value of the entropy production rate even for non-stationary dynamics. On the basis of this result, we develop an efficient algorithm that estimates the entropy production rate continuously in time by using machine learning techniques, and validate our numerical estimates using analytically tractable Langevin models in experimentally relevant parameter regimes. Our method is of great practical significance since all it requires is time-series data for the system of interest without requiring prior knowledge of the system parameters.