Estimating entropy production in a stochastic system with odd-parity variables

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.