Bridging statistical mechanics and thermodynamics away from equilibrium: A data-driven approach for learning internal variables and their dynamics

Thermodynamics with internal variables is a common approach in continuum mechanics to model inelastic (i.e., non-equilibrium) material behavior. It consists of enlarging the space of the state variables by introducing internal variables to eliminate the memory effects that would otherwise arise in the constitutive response when driving the system away from equilibrium. While this approach is computationally and theoretically very attractive, it currently lacks a well-established statistical mechanics foundation. As a result, internal variables are typically chosen phenomenologically and lack a direct link to the underlying atomistic or particle description. This hinders the predictability of the ensuing continuum models as well as the inverse problem of material design. In this work, we propose a machine learning approach that directly tackles these underlying issues, by learning internal variables and the evolution equations of the system, consistently with the principles of statistical mechanics and thermodynamics. The proposed approach leverages the following machine learning techniques (i) the information bottleneck (IB) method to ensure that the learned internal variables are functions of the microstates and are capable of capturing the salient feature of the microscopic distribution; (ii) conditional normalizing flows to represent arbitrary probability distributions of the microscopic states as functions of the state variables (these will be distinct from the Boltzmann distribution away from equilibrium); and (iii) Variational Onsager Neural Networks (VONNs) to guarantee thermodynamic consistency of the learned evolution equations and that the state variables are sufficient to predict the future state of the system in the absence of memory effects. The resulting framework, called IB-VONNs, is here tested on two problems on colloidal systems, governed at the microscale by overdamped Langevin dynamics. The first one is a prototypical model for a colloidal particle in an optical trap, which can be solved analytically thanks to its simplicity, and it is thus ideal to verify the framework. The second problem is a one-dimensional phase-transforming system, whose macroscopic description still lacks a statistical mechanics foundation under general conditions. The results in both cases indicate that the proposed machine learning strategy can indeed bridge statistical mechanics and thermodynamics with internal variables away from equilibrium.