August 24, 2026
Linear-Scaling Density Functional Theory with Neural Operators
Properties of molecules and materials ultimately arise from the quantum-mechanical behavior of their electrons. A century ago, Erwin Schrödinger formulated his famous equation, providing a mathematical framework for describing matter at the quantum scale. Only a few years later, Paul Dirac remarked:
“the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.”
The difficulty is one of scale: a many-electron wavefunction must describe all electrons simultaneously, and every additional electron adds new dimensions to this collective quantum state. The complexity grows so rapidly that a brute-force calculation for just 100 electrons would take far longer than the age of the universe on today’s supercomputers.
Over decades, researchers have developed increasingly sophisticated approximations and algorithms to make these otherwise impossible calculations tractable. Their computational cost still severely limits the size and complexity of systems we can study. Even so, their ability to predict the behavior of matter has made electronic-structure calculations essential to developing medicines, catalysts, batteries, and electronic devices.
A major breakthrough came in 1964, when Pierre Hohenberg and Walter Kohn proved that all the information needed to determine a system’s lowest-energy properties (ground state) is contained in a much simpler quantity: the electron density. It describes where electronic charge is concentrated within a molecule or material, without tracking each electron individually. Density functional theory (DFT) is built on this insight, using the electron density rather than the many-electron wavefunction as its fundamental quantity.
The Hohenberg–Kohn theorem established that the density is enough, but not how to calculate it. Walter Kohn and Lu Jeu Sham supplied the missing step. They introduced a hypothetical system of electrons that do not interact with one another but yields the same density as the real, interacting system. The one-electron states of this auxiliary system are known as Kohn–Sham orbitals. Although much simpler than the many-electron wavefunction, their number grows with the number of electrons, and solving for them typically scales cubically with system size. DFT is therefore a theory of the density in principle, but an orbital calculation in practice. Finding a way to remove these orbitals without sacrificing the accuracy of Kohn–Sham DFT has remained an outstanding challenge for the past 60 years.
A century after Schrödinger, advances in artificial intelligence offer a new way to address this computational challenge. Existing machine-learning approaches typically take one of two shortcuts. Interatomic potentials bypass the electronic problem, predicting energies and forces directly from atomic structures. This makes them fast but leaves no explicit electronic state from which to derive other properties. Alternatively, one-shot models predict the converged electronic state directly. By compressing the entire Kohn–Sham calculation into a single prediction, however, they discard its self-consistent structure and provide no natural indication when the model has been pushed beyond its reliable regime.
We take a different approach: rather than predicting the converged electronic state in a single step, we preserve the self-consistent cycle of conventional Kohn–Sham DFT. At each iteration, we map the current Kohn–Sham potential, the effective energy landscape experienced by the electrons, to its corresponding electron density. The density then updates the potential, and the process repeats until self-consistency. Conceptually, this resembles inference-time reasoning in large language models: instead of producing a difficult answer in one step, the model refines it through a sequence of intermediate updates. Preserving this iterative structure provides an intrinsic convergence criterion and a natural indication when the model is pushed beyond its reliable regime.
The key is choosing the correct map to learn within this cycle. We learn the Kohn–Sham solution operator, the direct map from the Kohn–Sham potential to the density it produces. Most previous orbital-free approaches approximate the kinetic energy from the electron density and therefore effectively address the inverse Kohn–Sham problem, working backward from the density to the potential. This inverse map is highly sensitive: small changes in the density can correspond to large changes in the potential. The forward map from potential to density is smoother, more stable, and better suited to machine learning.
Capturing this forward map requires accounting for its nonlocal character, since the density at one point can depend on the potential throughout the system. Fourier neural operators can represent the fully nonlocal map from potential to density at quasi-linear computational cost. We developed a domain-invariant FNO that enables a single unified model to operate across both molecules and materials. After training, the model showed stable, self-consistent convergence across a wide range of biomolecules, semiconductors, and metals. From the converged density and potential, we recovered energies, forces, and band structures in close agreement with conventional DFT. The computational savings are particularly substantial for crystalline solids because the learned operator avoids the many separate electronic-state calculations that conventional DFT performs throughout the crystal’s momentum space.
We also demonstrated the scalability and extrapolation of our method by computing an unseen 82,500-electron magnesium dislocation on a single NVIDIA B300 GPU. For comparison, previous Kohn–Sham DFT calculations of a 6,614-atom magnesium dislocation used about 7,800 V100 GPUs. Together, these results open a new regime of scale for electronic-structure prediction, extending physically informed, self-consistent calculations to realistic material structures and complex biomolecular environments.
