# Hamiltonian learning

Machine-learned Kohn–Sham Hamiltonians trained only on matrix-element error leave large errors in derived electronic properties, because the operator-to-state mapping is highly anisotropic. DSHam adds supervision on the density matrix, real-space electron density and occupied orbital energies obtained by solving the predicted Hamiltonian. On the ∇2DFT benchmark, electron-density and total-energy errors drop by factors of 40 and 500 relative to the strongest competing method, and the model extrapolates to molecules with twice the training heavy-atom count.

Machine-learned Kohn–Sham Hamiltonians trained only on matrix-element error leave large errors in derived electronic properties, because the operator-to-state mapping is highly anisotropic. DSHam adds supervision on the density matrix, real-space electron density and occupied orbital energies obtained by solving the predicted Hamiltonian. On the ∇2DFT benchmark, electron-density and total-energy errors drop by factors of 40 and 500 relative to the strongest competing method, and the model extrapolates to molecules with twice the training heavy-atom count.