Electronic structure is usually an expensive iterative solve. Two recent programs skip the solve by learning an operator instead. One works at the mean-field density and the other at the correlated two-electron marginal, yet they turn out to be the same idea on two rungs of a ladder, held up by one principle.
Learning the map, not the number
A Kohn–Sham calculation spends its cost diagonalizing orbitals, over and over, until the density is self-consistent. Most machine learning for electronic structure tries to shortcut the answer, predicting the energy or the ground state, and then extrapolates badly. The better move, shared by the two programs here, learns the map that the expensive step computes and keeps the physics loop around it.
- The density rung. KS-FNO (Khan et al., 2026) learns the Kohn–Sham map that takes a potential in and returns a density out, precisely what an orbital diagonalization returns. A single SE(3)-equivariant Fourier neural operator on a real-space grid stands in for the
diagonalization, turning the self-consistent solve quasi-linear with no explicit orbitals. One model, trained across thousands of molecules and solids, converges systems from organic molecules to metals, up to tens of thousands of electrons on a single GPU. - The correlation rung. The fragment 2-RDM sheaf learns the two-electron reduced density matrix, the minimally sufficient object, which fixes the energy and every one- and two-body observable at any correlation strength. It learns that object fragment-local, one shared network read off overlapping neighbourhoods.
One stops at the mean-field density, whereas the other carries genuine two-body correlation. The gesture is nonetheless identical, since both fix the solution operator and evaluate it.
Why both scale by nearsightedness
The two programs are fast for the same physical reason, namely Kohn nearsightedness. What happens at a point depends only on the potential nearby, and this locality is what makes a global quantum solve learnable and near-linear.
The two halves are complementary. KS-FNO relies on nearsightedness, which is the assumption that buys the scaling. The 2-RDM sheaf instead measures it. Its consistency residual is a per-region estimator of exactly that locality, computed from the model’s own outputs, that says where locality holds and where it breaks. One leans on the principle, whereas the other supplies a certificate for it. The residual thus acts as a trust score for when a fast, local operator can be believed.
Two ways to be local
The constructions encode locality from opposite sides. KS-FNO uses a global Fourier kernel that is local only in effect, made physical by SE(3)-equivariance. The sheaf is explicitly local, carrying operator-valued data on fragments glued by restriction maps, and its Laplacian’s low modes are the discrete echo of the Fourier band the operator keeps. The two are arguably discretizations of one nearsighted operator.
The operator view, and what it opens
Both can be read as learning a solution operator for a fixed point, since the self-consistent loop is just iterating the Kohn–Sham map. This is the operator-and-observable stance of the koopman-operator. Seen that way the open questions write themselves. Can the correlated 2-RDM be given a neural-operator form and inherit the density rung’s linear scaling? Can the consistency residual flag, region by region, where a learned Kohn–Sham map has left its nearsighted regime? Are the equivariant Fourier operator and the cellular sheaf the same local operator in disguise?
A quantum corner also opens. The correlated 2-RDM is not only learnable but measurable, because the Hamiltonian is two-body. A quantum computer gets the energy by reading off the two-particle marginal (VQE) or by solving for it directly, and it pays the N-representability cost the classical model can only approximate. The learned 2-RDM and the measured 2-RDM are the same object from two sides, both bounded by the same nearsightedness.
The technical synthesis, covering the ladder, the shared engine, the spectral bridge, and the quantum corner, is in the-kohn-sham-map-and-the-2-rdm-sheaf-one-operator.