# Analytical classical density functionals from an equation learning network

arXiv: https://arxiv.org/abs/1910.12752 · HTML: https://mlsanity.com/paper/1910.12752/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2019.

## Built on by (most cited first, 7 of 7)

- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Evolving symbolic density functionals](https://mlsanity.com/paper/2203.02540/) (2022)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Informed Equation Learning](https://mlsanity.com/paper/2105.06331/) (2021)
- [Physics-informed Bayesian inference of external potentials in classical density-functional theory](https://mlsanity.com/paper/2309.07065/) (2023)
- [Perfecting Liquid-State Theories with Machine Intelligence](https://mlsanity.com/paper/2311.05167/) (2023)
- [Neural force functional for non-equilibrium many-body colloidal systems](https://mlsanity.com/paper/2406.03606/) (2024)

## Abstract (arXiv)

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.