  Thermodynamic uncertainty relations (TURs) are the inequalities which give
lower bounds on the entropy production rate using only the mean and the
variance of fluctuating currents. Since the TURs do not refer to the full
details of the stochastic dynamics, it would be promising to apply the TURs for
estimating the entropy production rate from a limited set of trajectory data
corresponding to the dynamics. Here we investigate a theoretical framework for
estimation of the entropy production rate using the TURs along with machine
learning techniques without prior knowledge of the parameters of the stochastic
dynamics. Specifically, we derive a TUR for the short-time region and prove
that it can provide the exact value, not only a lower bound, of the entropy
production rate for Langevin dynamics, if the observed current is optimally
chosen. This formulation naturally includes a generalization of the TURs with
the partial entropy production of subsystems under autonomous interaction,
which reveals the hierarchical structure of the estimation. We then construct
estimators on the basis of the short-time TUR and machine learning techniques
such as the gradient ascent. By performing numerical experiments, we
demonstrate that our learning protocol performs well even in nonlinear Langevin
dynamics. We also discuss the case of Markov jump processes, where the exact
estimation is shown to be impossible in general. Our result provides a platform
that can be applied to a broad class of stochastic dynamics out of equilibrium,
including biological systems.
