Machine-learning free-energy functionals using density profiles from simulations

The formally exact framework of equilibrium Density Functional Theory (DFT) is capable of simultaneously and consistently describing thermodynamic and structural properties of interacting many-body systems in arbitrary external potentials. In practice, however, DFT hinges on approximate (free-)energy functionals from which density profiles (and hence the thermodynamic potential) follow via an Euler-Lagrange equation. Here, we explore a relatively simple Machine-Learning (ML) approach to improve the standard mean-field approximation of the excess Helmholtz free-energy functional of a 3D Lennard-Jones system at a supercritical temperature. The learning set consists of density profiles from grand-canonical Monte Carlo simulations of this system at varying chemical potentials and external potentials in a planar geometry only. Using the DFT formalism, we nevertheless can extract not only very accurate 3D bulk equations of state but also radial distribution functions using the Percus test-particle method. Unfortunately, our ML approach did not provide very reliable Ornstein-Zernike direct correlation functions for small distances.

Automated summary

The density functional theory provides an accurate framework to study thermodynamic and structural properties of interacting many-body systems, but relies on approximate energy functionals in practice. A machine-learning approach was explored to improve the standard mean-field approximation in this study, with a focus on a 3D Lennard-Jones system at a supercritical temperature. The approach showed promise in extracting accurate bulk equations of state but faced challenges in providing reliable direct correlation functions.