DeepHartree: A Poisson-Coupled Neural Field for Scalable Density Functional Theory

Ab initio calculations are fundamentally bottlenecked for large systems by the steep computational scaling of solving self-consistent field (SCF) equations. While machine learning offers potential accelerations, existing methods often compromise physical rigor or rely on basis-dependent, non-transferable representations. Here, we introduce DeepHartree, a Poisson-coupled neural field that accelerates linear combination of atomic orbitals (LCAO) density functional theory (DFT). By coupling an E(3)-equivariant neural network with the Poisson equation through automatic differentiation and mitigating nuclear singularities via delta-learning, DeepHartree simultaneously predicts mutually consistent real-space electron densities and Hartree potentials. This resolves the Coulomb bottleneck by substituting $\mathcal{O}(N^4)$ analytical integrals with GPU-accelerated, near-linear $\mathcal{O}(N)$ numerical inference. Trained solely on small molecules, DeepHartree enables scalable density functional theory through a two-level transferability: for SCF convergence acceleration, it achieves robust zero-shot transferability across diverse basis sets, functionals, and systems up to 168 atoms; for predicting other density-related physical quantities, it retains zero-shot capability on small molecules while enabling precise predictions for larger systems via efficient few-shot fine-tuning. Our model accelerates standard SCF protocols by reducing iterations by up to 40.9% via high-fidelity initial density matrices, and its rigorous long-range asymptotics provide a zero-cost physical uncertainty metric prior to grid evaluation. By grounding deep learning in Poisson-coupled neural fields, DeepHartree accelerates demanding tasks -- such as near-coupled-cluster dynamic infrared simulations -- by orders of magnitude, establishing a scalable paradigm for density functional theory.