Why neural functionals suit statistical mechanics

TL;DR

This paper discusses recent advances in using neural functionals, inspired by density functional theory, to improve the statistical mechanical modeling of many-body systems, with a focus on validation and pedagogical demonstration.

Contribution

It introduces a neural functional approach that enables quality control and consistency in machine learning-based statistical mechanics, demonstrated through a one-dimensional hard core particle system.

Findings

Neural functionals provide a functional representation of correlations and thermodynamics.

The approach allows for rigorous quality control of AI methods in statistical mechanics.

A pedagogical tutorial demonstrates the neural functional concepts and related computational techniques.

Abstract

We describe recent progress in the statistical mechanical description of many-body systems via machine learning combined with concepts from density functional theory and many-body simulations. We argue that the neural functional theory by Samm\"uller et al. [Proc. Nat. Acad. Sci. 120, e2312484120 (2023)] gives a functional representation of direct correlations and of thermodynamics that allows for thorough quality control and consistency checking of the involved methods of artificial intelligence. Addressing a prototypical system we here present a pedagogical application to hard core particle in one spatial dimension, where Percus' exact solution for the free energy functional provides an unambiguous reference. A corresponding standalone numerical tutorial that demonstrates the neural functional concepts together with the underlying fundamentals of Monte Carlo simulations, classical density functional theory, machine learning, and differential programming is available online at https://github.com/sfalmo/NeuralDFT-Tutorial.