Physics-Informed Long-Range Coulomb Correction for Machine-learning Hamiltonians

Accurate atomistic simulations of materials under electric fields require long-range electrostatics and electrical-response modeling at length scales beyond the reach of conventional first-principles methods. Yet these capabilities remain difficult to reconcile with the scalability of machine-learning interatomic potentials (MLIPs). Strictly local models enable efficient spatial parallelization but neglect long-range interactions, whereas existing electrostatic extensions introduce global dependencies that hinder distributed computation. Here, we overcome this limitation by analytically separating the long-range electrostatic forces and Born effective charges (BECs) into a global electrostatics calculation and strictly local derivatives of learned latent charges. This formulation preserves the spatial decomposability of equivariant MLIPs while enabling distributed evaluation of energies, forces, stresses, and BECs across different boundary conditions. Implemented as DPA4C-LR on the nvalchemi platform, the framework scales to 5.2 million atoms on four GPUs at approximately twice the computational cost of its short-range backbone. Trained on energies and forces, DPA4C-LR accurately infers BECs and reproduces the field-dependent infrared spectra of water. We further demonstrate its capabilities by recovering the characteristic slim polarization hysteresis of relaxor \ce{Pb(Mg_{1/3}Nb_{2/3})O3}, which is obscured by finite-size effects in smaller simulations; predicting a room-temperature field-driven ionic conductivity of $6.8 \pm 0.7$~mS/cm in amorphous \ce{Li_{0.9}NbO_{0.9}Cl_{4.1}}; and modeling interfacial thermal transport across Si/diamond interfaces with slab-corrected electrostatics. By combining long-range physics, electrical-response inference, and distributed computation, this framework extends predictive atomistic modeling toward device-relevant length scales.

Equivariant graph neural networks have emerged as foundational architectures for machine-learned interatomic potentials, approaching quantum-chemical accuracy at a fraction of the computational cost. These models describe local atomic environments accurately, but finite spatial cutoffs truncate long-range information flow, and stacking message-passing layers can lead to over-smoothing and over-squashing. Existing long-range extensions either prescribe a fixed analytical propagation kernel, restrict long-range communication to scalars or degree-preserving channels, are only approximately equivariant, or incur super-linear computational cost. Combining learnable long-range equivariant transport with multiscale many-body expressivity and efficient scaling for larger systems remains a central challenge. We introduce Neural Equivariant Multipole Operators (NEMORA), a neural equivariant extension of the Fast Multipole Method (FMM) for learning long-range tensorial representations. NEMORA generalizes the FMM's analytical multipole expansion and translation operators to learned equivariant counterparts on an adaptive spatial hierarchy. Its operators couple angular degrees and form many-body interactions across length scales, retaining the FMM's hierarchical organization and analytical radial factors as physical inductive biases while learning data-dependent long-range couplings. NEMORA evaluates in linear time and memory complexity, allowing it to treat larger systems than other long-range methods reaching hundreds of thousands of atoms, and it augments both symmetry-constrained and unconstrained short-range backbones. On non-local benchmarks, it reduces force and energy errors relative to the short-range backbones by over an order of magnitude and up to three orders of magnitude, respectively, which is better than or competitive with existing long-range extensions in accuracy.

Machine-learned interatomic potentials (MLIPs) excel at in-distribution tasks, accelerating drug and material development, yet they struggle to generalize out-of-distribution. We propose to push the cost-accuracy Pareto frontier by designing observable-agnostic electronic ground-state descriptor models (GSMs) with computational costs situated between MLIPs and Kohn-Sham density functional theory (KS-DFT). We align the learning objectives and architectures of GSMs with the governing equations of KS-DFT by enforcing physical constraints and removing optimization pressure on unphysical or irrelevant degrees of freedom. In our size-extrapolation experiments from QM9 to QM40, our combined contributions OrthoNormal-Loss (ON-Loss) and Grassmann Restricted Occupied-Orbital Training (GROOT) reach a 79.1% energy and 83.4% force mean absolute error (MAE) reduction over previous state-of-the-art density GSMs. For Hamiltonian GSMs, ON-Loss and Residual Optimal-gauge Conditioning-aware KS-Eq. Training (ROCKET) together reduce the energy and force MAEs of the strongest baseline by 99.8% and 95.9%, respectively. Using a self-consistency rejection criterion, we filter out extrapolation errors on QMugs, rejecting fewer than 0.4% of predictions while reaching an energy MAE of 0.07 mHa. Finally, we demonstrate the efficiency of label-free self-consistency fine-tuning, and transfer GSMs to reactive chemistry in Transition1x, reaching energy errors below chemical accuracy.