  Machine learning, with its remarkable ability for retrieving information and
identifying patterns from data, has emerged as a powerful tool for discovering
governing equations. It has been increasingly informed by physics, and more
recently by thermodynamics, to further uncover the thermodynamic structure
underlying the evolution equations, i.e., the thermodynamic potentials driving
the system and the operators governing the kinetics. However, despite its great
success, the inverse problem of thermodynamic model discovery from macroscopic
data is in many cases non-unique, meaning that multiple pairs of potentials and
operators can give rise to the same macroscopic dynamics, which significantly
hinders the physical interpretability of the learned models. In this work, we
propose a machine learning framework, named as Statistical-Physics-Informed
Neural Networks (Stat-PINNs), which further encodes knowledge from statistical
mechanics and resolves this non-uniqueness issue for the first time. The
framework is here developed for purely dissipative isothermal systems. It only
uses data from short-time particle simulations to learn the thermodynamic
structure, which can be used to predict long-time macroscopic evolutions. We
demonstrate the approach for particle systems with Arrhenius-type interactions,
common to a wide range of phenomena, such as defect diffusion in solids,
surface absorption and chemical reactions. Stat-PINNs can successfully recover
the known analytic solution for the case with long-range interaction and
discover the hitherto unknown potential and operator governing the short-range
interaction cases. We compare our results with an analogous approach that
solely excludes statistical mechanics, and observe that, in addition to
recovering the unique thermodynamic structure, statistical mechanics relations
can increase the robustness and predictability of the learning strategy.
