Neural network approximation of regularized density functionals
Mihály A. Csirik, Andre Laestadius, Mathias Oster
TL;DR
The paper tackles constructing principled, convergent approximations to the universal density functional of density-functional theory by applying Moreau--Yosida regularization to render the functionals continuous and differentiable on non-reflexive spaces, then approximating the regularized functionals with neural networks that preserve positivity and convexity. It develops a constrained universal-approximation framework for neural nets on separable Banach (Fréchet) spaces, extends Moreau--Yosida regularization to non-reflexive settings, and provides an explicit error bound for the resulting approximate ground-state energy. The core contributions include a generalized constrained universal-approximation theorem, a non-reflexive regularization scheme with proximal mappings and energy-shift properties, and a rigorous a priori error estimate for NN-based approximations of $E_N(v)$ on suitable compact density sets. The work offers a principled, first-principles route to differentiable, convex surrogates for density functionals that are directly usable in Kohn--Sham calculations, potentially enabling more robust and convergent electronic-structure computations.
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.