We propose a framework to learn the time-dependent Hartree-Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Though the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange-correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to $60 \times 60 \times 60 \times 60$. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.

翻译：我们提出一个框架，用于从分子的电子密度动力学中学习其含时 Hartree-Fock（TDHF）电子间势。尽管整个 TDHF 哈密顿量（包括电子间势）可以从第一性原理计算得到，但我们以此问题作为测试平台，以开发可应用于学习量子动力学其他方法/近似中出现的先验未知项的策略，例如学习含时密度泛函理论中的交换关联势等新兴问题。我们开发、训练并测试了三种 TDHF 电子间势模型，每种模型均由一个尺寸高达 $60 \times 60 \times 60 \times 60$ 的四指标张量参数化。其中两个模型保持厄米对称性，而一个模型保持蕴含厄米对称性的八重置换对称性。在七个不同的分子系统中，我们发现考虑更深的八重对称性可在三个指标上获得性能最佳的模型：训练效率、测试集预测能力以及真实与学习电子间势的直接比较。所有三种模型在无外场轨迹的系综上训练后，即使在训练集之外的施加外场区域，也能生成准确的电子动力学预测。为了使我们的模型能够扩展到大型分子系统，我们推导了雅可比-向量乘积的表达式，从而实现迭代的、无矩阵的训练。