Why neural functionals suit statistical mechanics

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 Sammüller et al. [Proc. Natl. 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.

Noether invariance theory for the equilibrium force structure of soft matter

“…Relating to forcesampling methods that reduce the statistical variance inherent in sampling results [68][69][70][71][72][73][74] is another interesting point for future work, as can be relating to force-based density functional theory [29,48]. The recent neural functional theory [75][76][77] is based on the powerful concept of using neural networks to represent functional relationships which encapsulate the correlation behaviour of complex systems. Noether sum rules have been shown to provide valuable consistency checks for these neural functionals and they give much inspiration for further theoretical developments in the spirit of physics-informed machine learning [78][79][80][81][82][83].…”