Virtual atoms let electron density shape the interatomic potential encoder, not just a reconstructed field

Electron-density learning is recast as representation learning for interatomic potentials, with density query points and real atoms processed by the same parameterized encoder and density, energy, and force produced by separate task-specific readouts. Density reconstruction and interatomic-potential learning have generally been optimized as separate tasks, with density models reconstructing a spatial field and potentials learning atomic representations from energy and force labels alone. On the Materials Project benchmark the model reaches a lower normalized density mean absolute error than ChargE3Net, matches the best reported performance on nickel-manganese-cobalt battery cathode materials, and is competitive with dedicated density models on QM9.

Virtual atoms receive information only from nearby real atoms, so the real-atom interaction graph is unchanged and density supervision can update the shared encoder without introducing virtual interactions into the physical energy and force pathway. Probe nodes in grid-based density models also receive information from neighboring atoms, but here virtual and real centers share the same encoder parameters, so density supervision acts on the representation used for energy and force. In joint training on QM9x energy-force labels and QM9 density labels, linear-probe local-environment classification accuracy rises on the training split, the validation split, and the ORCA-relabeled QM40 subset, while energy and force MAE and RMSE fall.

The benefit of density supervision is more consistent for force prediction and grows as molecular size moves beyond the training range. Within the training distribution density supervision improves both energy and force, whereas in cross-domain evaluation energy errors move in both directions across datasets while force errors fall on all five external datasets. On the element-matched, ORCA-relabeled QM40 subset, the largest size group shows force MAE dropping from the EF model value to the EF+density value, with force RMSE falling in step.

A checkpoint pretrained on density labels alone can serve as an initialization for downstream energy-force learning. Interatomic-potential representations have generally been initialized from large energy-force pretraining, with density labels absent from downstream optimization. Under a fixed fine-tuning protocol, density-pretrained checkpoints generally give lower validation errors than random initialization at small labeled budgets on molecular tasks (drug, rMD17, H2O-PD) and solid-state tasks (AlMgCu, FerroEle, SSE-PBE).

The report covers density reconstruction, density-supervised multi-task training, linear probing, zero-shot transfer, and density-pretrained fine-tuning on molecular and periodic systems. Density reconstruction requires evaluating virtual atoms at spatial query points, which adds inference cost when the density head is used; that cost disappears once the head is discarded in the pretraining setting. The EF and EF+density experiments match energy-force sample presentations but not total computational cost, since the latter also processes density structures and uses more GPUs. The current framework treats density as a supervised target rather than coupling it to energy through a self-consistent or variational relation.

In the periodic MP/MPtrj setting the in-domain improvements are modest, and the report notes that repeated runs would be needed to distinguish them from seed-to-seed variation. In external evaluation, energy errors move in both directions across datasets, and energy MAE more than doubles on MatPES. Density accuracy and force accuracy are not in one-to-one correspondence, and accurate density reconstruction alone does not guarantee accurate forces. The authors state that the observations are consistent with, but do not establish, a microscopic causal mechanism.