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Density functional theory (DFT) [1] provides an alternative representation of quantum many-body states to the wavefunction formalism. Its central theorems establish that the ground state of a many-electron system is uniquely determined by the electron number density distribution
() in real space. Based on this principle, the Kohn–Sham equations [2], which form the foundation of modern first-principles electronic-structure calculations, are derived.

A central challenge in DFT is the construction of accurate models for the exchange–correlation energy xc as a functional of (). This term represents many-body effects that cannot be accounted for by the classical electrostatic interaction alone. Developing reliable approximations for xc has long been a grand challenge in quantum mechanics and is essential for extending the predictive power and applicability of first-principles calculations. However, analytical development of exchange–correlation approximations is intrinsically difficult. One seeks formulas that relate () and the corresponding xc, which are both outputs of the Schrödinger equation. The task can therefore be viewed as approximating the sequential inverse-and-forward solution of the Schrödinger equation.

Recently, the rapid progress of machine-learning (ML) techniques has enabled new approaches to the construction of exchange–correlation and other density functionals. These approaches can in essence be viewed as a sophisticated extension of traditional parameter fitting, but they have also yielded new practical tools and insights into the structure of DFT, as demonstrated by studies [3].

In this talk, I provide a brief introduction to the fundamentals of DFT and review recent developments in ML-based density functional construction. Some recent results will also be presented.

[1] P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)
[2] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)
[3] RA, M. Sogal and K. Burke, arXiv:2503.01709

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