  Entropy production (EP) is a central measure in nonequilibrium
thermodynamics, as it can quantify the irreversibility of a process as well as
its energy dissipation in special cases. Using the time-reversal asymmetry in a
system's path probability distribution, many methods have been developed to
estimate EP from only trajectory data. However, estimating the EP of a system
with odd-parity variables, which prevails in nonequilibrium systems, has not
been covered. In this study, we develop a machine learning method for
estimating the EP in a stochastic system with odd-parity variables through
multiple neural networks. We demonstrate our method with two systems, an
underdamped bead-spring model and a one-particle odd-parity Markov jump
process.
