Kohn–Sham density functional theory (DFT) is the basis of modern computational approaches to electronic structures. Their accuracy heavily relies on the exchange-correlation energy functional, which encapsulates electron–electron interaction beyond the classical model. As its universal form remains undiscovered, approximated functionals constructed with heuristic approaches are used for practical studies. However, there are problems in their accuracy and transferability, while any systematic approach to improve them is yet obscure. In this study, we demonstrate that the functional can be systematically constructed using accurate density distributions and energies in reference molecules via machine learning. Surprisingly, a trial functional machine learned from only a few molecules is already applicable to hundreds of molecules comprising various first- and second-row elements with the same accuracy as the standard functionals. This is achieved by relating density and energy using a flexible feed-forward neural network, which allows us to take a functional derivative via the back-propagation algorithm. In addition, simply by introducing a nonlocal density descriptor, the nonlocal effect is included to improve accuracy, which has hitherto been impractical. Our approach thus will help enrich the DFT framework by utilizing the rapidly advancing machine-learning technique.

Kohn-Sham密度泛函理论（DFT）是现代电子结构计算方法的基础。它们的准确性在很大程度上取决于交换相关能量函数，该函数封装了经典模型以外的电子相互作用。由于尚未发现其通用形式，因此采用启发式方法构造的近似功能用于实际研究。但是，它们的准确性和可移植性存在问题，而任何用于改善它们的系统方法都还不清楚。在这项研究中，我们证明可以通过机器学习使用参考分子中的准确密度分布和能量来系统地构建功能。出奇，仅从少量分子中学到的试验功能机器已经适用于数百种分子，这些分子包含与标准功能相同的准确性的各种第一行和第二行元素。这是通过使用灵活的前馈神经网络将密度和能量相关联来实现的，这使我们能够通过反向传播算法获得功能导数。另外，仅通过引入非局部密度描述符，就包括非局部效应以提高精度，这迄今为止是不切实际的。因此，我们的方法将通过利用快速发展的机器学习技术来帮助丰富DFT框架。这使我们可以通过反向传播算法获得函数导数。另外，仅通过引入非局部密度描述符，就包括非局部效应以提高精度，这迄今为止是不切实际的。因此，我们的方法将通过利用快速发展的机器学习技术来帮助丰富DFT框架。这使我们能够通过反向传播算法获得函数导数。另外，仅通过引入非局部密度描述符，就包括非局部效应以提高精度，这迄今为止是不切实际的。因此，我们的方法将通过利用快速发展的机器学习技术来帮助丰富DFT框架。