Machine-learning the DFT of a classical statistical-mechanical system*

We apply machine-learning (ML) to the construction of mean-field theories of classical statistical-mechanical systems. In the density functional formulation of classical statistical physics, the Helmholtz free energy generates a hierarchy of many-body direct correlation functions, and the one-body density is its first member. In equilibrium, the latter minimises the free energy functional of the system. Thus, knowing the free energy functional allows one to solve classical statistical mechanics. In this talk, we address the inverse problem of finding the free energy functional, given the particle data corresponding to the system in equilibrium. Introducing an adversarial ML methodology, we reformulate the learning problem as a two-player game, with the best fitting parameters obtained as the solution of a minimax problem. As proof of concept, we consider the Percus’ model of a 1D fluid, consisting of hard rods on a line, for which the exact functional is known. We emphasize the physics-informed aspect of ML, where the physical constraints, including the "physical intuition”, are combined with ML methods to obtain meaningful results.