平稳性约束提升机器学习交换相关泛函的能量与密度精度
Stationarity Constraints Improve Both Energies and Densities of Machine-Learned Exchange-Correlation Functionals
训练交换相关泛函的老难题：能量准了密度就歪。这篇用 Brillouin 残差做惩罚项，只用 6 个能量加 11 个密度就把 SCAN 比下去了。
机器学习交换相关泛函通常只在固定轨道上拟合，自洽使用时能量变好但密度往往变差。论文证明训练后泛函在精确密度对应 Kohn-Sham 轨道上的 Brillouin 残差，一阶控制自洽密度偏移和拟合能量与自洽能量之间的差距，且训练时无需自洽场求解。仅用 6 个能量和 11 个 CCSD 密度训练，一个神经 meta-GGA 泛函在未见分子的原子化能量和密度上同时超过 SCAN。
Stationarity Constraints Improve Both Energies and Densities of Machine-Learned Exchange-Correlation Functionals
Empirical exchange-correlation functionals are fitted on fixed orbitals but used self-consistently, so better energies often come with worse densities. We show that the Brillouin residual of a trained functional on the Kohn-Sham orbitals of an accurate density controls, to first order, both the self-consistent density shift and the gap between fitted and self-consistent energies. Penalizing it requires no self-consistent-field solution during training. Trained on 6 energies and 11 CCSD densities, a neural meta-GGA surpasses SCAN in both atomization energies and densities of unseen molecules.
