Impact of quantum-chemical metrics on the machine learning prediction of electron density

Machine learning (ML) algorithms have undergone an explosive development impacting every aspect of computational chemistry. To obtain reliable predictions, one needs to maintain the proper balance between the black-box nature of ML frameworks and the physics of the target properties. One of the most appealing quantum-chemical properties for regression models is the electron density, and some of us recently proposed a transferable and scalable model based on the decomposition of the density onto an atom-centered basis set. The decomposition, as well as the training of the model, is at its core a minimization of some loss function, which can be arbitrarily chosen and may lead to results of different quality. Well-studied in the context of density fitting (DF), the impact of the metric on the performance of ML models has not been analyzed yet. In this work, we compare predictions obtained using the overlap and the Coulomb repulsion metrics for both the decomposition and training. As expected, the Coulomb metric used as both the DF and ML loss functions leads to the best results for the electrostatic potential and dipole moments. The origin of this difference lies in the fact that the model is not constrained to predict densities that integrate to the exact number of electrons N. Since an a posteriori correction for the number of electrons decreases the errors, we proposed a modification of the model where N is included directly into the kernel function, which allowed to lower the errors on the test and out-of-sample sets.

机器学习（Machine Learning, ML）算法经历了爆发式发展，其影响遍及计算化学的各个领域。为获得可靠的预测结果，需在机器学习框架的黑箱特性与目标属性的物理本质之间维持恰当的平衡。对于回归模型而言，最具吸引力的量子化学属性之一便是电子密度，我们团队近期提出了一种基于将电子密度分解至以原子为中心基组的可迁移且可扩展模型。无论是密度分解还是模型训练，其核心均为对某一损失函数的最小化，损失函数可任意选取，且不同的选择会导致质量各异的预测结果。尽管在密度拟合（Density Fitting, DF）的研究背景中该指标的影响已得到充分探讨，但针对机器学习模型性能的相关分析仍未开展。本研究中，我们针对分解与训练两个环节，对比了采用重叠度量与库仑排斥度量所得到的预测结果。正如预期那般，同时将库仑度量用作密度拟合与机器学习损失函数的方案，在静电势与偶极矩的预测上取得了最优结果。该差异的根源在于，模型并未被约束为预测积分后等于精确电子数N的电子密度。由于对电子数进行后验校正可降低预测误差，我们提出了一种模型改进方案，将N直接嵌入核函数之中，该方案得以降低测试集与域外样本集上的预测误差。