Integrating essential physical priors into deep-learning first-principles methods is a critical fundamental problem. Here we demonstrate that the deep learning density functional theory Hamiltonian (DeepH) method can be substantially improved by changing the learning objective to a rotation-invariant and basis-free quantity—the real-space Kohn-Sham potential (named DeepH-R). Benefiting from the enhanced physical priors, DeepH-R achieves substantially improved prediction accuracy and generalization ability compared to previous DeepH approaches. Moreover, DeepH-R provides a more accurate and straightforward route to deep-learning density functional perturbation theory, and enables the training of foundation models of electronic structure with sub-meV accuracy, opening new opportunities for AI-driven materials discovery.

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• Electronic structure
• Density functional theory
• First-principles calculations