Analytical classical density functionals from an equation learning network
This work provides a method to improve density functional approximations for statistical physics, but it is incremental as it builds on prior equation learning networks with expanded functional spaces.
The study tackled the problem of deriving analytic classical free energy functionals for model fluids using machine learning, resulting in good approximations for hard rod and Lennard-Jones fluids that matched simulated density profiles both inside and outside the training region.
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.