Neural-network density functional theory based on variational energy minimization

Conventional deep-learning DFT relies on supervised learning from labeled data, which separates neural network training from the physics of DFT. We flipped this paradigm: physical principles become the loss function. By backpropagating through a differentiable DFT code, the neural network learns to represent Hamiltonians without any training labels — achieving higher accuracy than supervised approaches. This physics-informed, unsupervised framework opens a new direction for developing deep-learning electronic structure methods.