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

• Main Authors: Peter Cats, Sander Kuipers, Sacha de Wind, Robin van Damme, Gabriele M. Coli, Marjolein Dijkstra, René van Roij
• Format: Article
• Language: English
• Published: AIP Publishing LLC 2021-03-01
• Series: APL Materials
• Online Access: http://dx.doi.org/10.1063/5.0042558

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.