Analytical classical density functionals from an equation learning network

TL;DR

This paper demonstrates how machine learning, specifically an equation learning network, can derive accurate analytic classical free energy functionals for one-dimensional fluids like hard rods and Lennard-Jones, improving over previous polynomial-based models.

Contribution

The study introduces a modified equation learning network that constructs flexible free energy density functionals from basis functions, significantly expanding the functional space compared to prior polynomial models.

Findings

Accurate approximation of the hard rod functional and correlation functions.

Good agreement of learned functionals with simulated density profiles.

Successful learning of both the full and attraction-related free energy functionals for Lennard-Jones fluids.

Abstract

We explore the feasibility of using machine learning methods to obtain an analytic form of the classical free energy functional for two model fluids, hard rods and Lennard--Jones, in one dimension . The Equation Learning Network proposed in Ref. 1 is suitably modified to construct free energy densities which are functions of a set of weighted densities and which are built from a small number of basis functions with flexible combination rules. This setup considerably enlarges the functional space used in the machine learning optimization as compared to previous work 2 where the functional is limited to a simple polynomial form. As a result, we find a good approximation for the exact hard rod functional and its direct correlation function. For the Lennard--Jones fluid, we let the network learn (i) the full excess free energy functional and (ii) the excess free energy functional related to interparticle attractions. Both functionals show a good agreement with simulated density profiles for thermodynamic parameters inside and outside the training region.