Neural network approximation of regularized density functionals

TL;DR

This paper introduces a new method combining Moreau--Yosida regularization and neural networks to approximate the universal density functional in density functional theory, ensuring mathematical properties and enabling direct use in calculations.

Contribution

The paper presents a novel procedure that makes the density functional continuous and differentiable, and approximates it with a neural network preserving key properties.

Findings

Neural network approximations maintain positivity and convexity.

Regularized functionals are differentiable and suitable for Kohn--Sham calculations.

Proposed method offers a first-principles, mathematically consistent approximation.

Abstract

Density functional theory is one of the most efficient and widely used computational methods of quantum mechanics, especially in fields such as solid state physics and quantum chemistry. From the theoretical perspecive, its central object is the universal density functional which contains all intrinsic information about the quantum system in question. Once the external potential is provided, in principle one can obtain the exact ground-state energy via a simple minimization. However, the universal density functional is a very complicated mathematical object and almost always it is replaced with its approximate variants. So far, no ``first principles'', mathematically consistent and convergent approximation procedure has been devised that has general applicability. In this paper, we propose such a procedure by first applying Moreau--Yosida regularization to make the exact functionals continuous (even differentiable) and then approximate the regularized functional by a neural network. The resulting neural network preserves the positivity and convexity of the exact functionals. More importantly, it is differentiable, so it can be directly used in a Kohn--Sham calculation.