Learn the Field, Construct the Matrix: Rethinking Targets for Kohn–Sham DFT

Machine learning is increasingly coupled with Kohn–Sham density functional theory (DFT), yet it remains unclear which electronic representation provides the most effective interface to the solver. We compare four candidates—charge density $\rho$, effective potential $V_{\mathrm{eff}}$, Hamiltonian $H$, and density matrix $D$—on a unified 43k-crystal benchmark, evaluating native prediction accuracy, closed-loop self-consistent-field (SCF) acceleration, and downstream property recovery. Although $H$ and $D$ lie closest to the eigensolution, their learned counterparts realize little of the large acceleration available to exact matrix states. In contrast, learned $\rho$ and $V_{\mathrm{eff}}$ accelerate SCF on 90.3% and 84.3% of test crystals, with median loop speedups of $1.18\times$ and $1.17\times$. More strikingly, matrices reconstructed from learned real-space fields outperform dedicated matrix-input routes on 97.6–99.9% of test crystals after one solver step, while residual $V_{\mathrm{eff}}$ further improves downstream recovery under matched electronic work. These results show that proximity to the eigensolution is a poor criterion for choosing a learning target. Instead, effective ML–DFT interfaces should expose quantities from which the solver can analytically reconstruct constrained electronic states, rather than requiring neural networks to learn those constraints directly.