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

TL;DR

This paper applies machine learning to improve free-energy functionals in density functional theory for a Lennard-Jones system, enhancing the accuracy of thermodynamic and structural predictions from simulation data.

Contribution

It introduces a machine learning approach to refine the excess Helmholtz free energy functional within DFT for a 3D Lennard-Jones system, based on simulation data.

Findings

Accurately predicts 3D bulk equations of state

Successfully derives radial distribution functions

Struggles with reliable Ornstein-Zernike correlation functions

Abstract

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.