We develop methods to learn the correlation potential for a time-dependent Kohn-Sham (TDKS) system in one spatial dimension. We start from a low-dimensional two-electron system for which we can numerically solve the time-dependent Schr\"odinger equation; this yields electron densities suitable for training models of the correlation potential. We frame the learning problem as one of optimizing a least-squares objective subject to the constraint that the dynamics obey the TDKS equation. Applying adjoints, we develop efficient methods to compute gradients and thereby learn models of the correlation potential. Our results show that it is possible to learn values of the correlation potential such that the resulting electron densities match ground truth densities. We also show how to learn correlation potential functionals with memory, demonstrating one such model that yields reasonable results for trajectories outside the training set.

翻译：我们开发了方法来学习一个基于时间的Kohn-Sham(TDKS)系统在一个空间层面的关联潜力。 我们从一个低维的双电子系统开始, 我们可以从数字上解析基于时间的 Schr\'dinger 等式; 产生适合相关潜力培训模型的电子密度。 我们把学习问题描述为优化最小方位目标的一个方法, 但要受动态服从TDKS等式的限制。 应用副接线, 我们开发高效的方法来计算梯度, 从而学习相关潜力的模型。 我们的结果表明, 能够了解相关潜力的价值, 从而让由此产生的电子密度与地面的真情密度相匹配。 我们还展示了如何学习与记忆的关联性功能, 展示一种能为训练场外的轨迹产生合理结果的模型。