Practical density functional theory (DFT) owes its success to the groundbreaking work of Kohn and Sham that introduced the exact calculation of the non-interacting kinetic energy of the electrons using an auxiliary mean-field system. However, the full power of DFT will not be unleashed until the exact relationship between the electron density and the non-interacting kinetic energy is found. Various attempts have been made to approximate this functional, similar to the exchange--correlation functional, with much less success due to the larger contribution of kinetic energy and its more non-local nature. In this work we propose a new and efficient regularization method to train density functionals based on deep neural networks, with particular interest in the kinetic-energy functional. The method is tested on (effectively) one-dimensional systems, including the hydrogen chain, non-interacting electrons, and atoms of the first two periods, with excellent results. For the atomic systems, the generalizability of the regularization method is demonstrated by training also an exchange--correlation functional, and the contrasting nature of the two functionals is discussed from a machine-learning perspective.

翻译：密度泛函理论（DFT）的实际成功归功于Kohn和Sham的开创性工作，他们通过辅助平均场系统引入了电子无相互作用动能的精确计算。然而，只有在找到电子密度与无相互作用动能之间的精确关系时，DFT的全部潜力才能被释放。人们曾尝试近似该泛函（类似于交换关联泛函），但由于动能贡献更大且更具非局域性，这些尝试的成功率远低于前者。本文提出一种基于深度神经网络训练密度泛函的新型高效正则化方法，重点关注动能泛函。该方法在（有效）一维系统（包括氢链、无相互作用电子及前两个周期的原子）中进行了测试，取得了优异结果。对于原子系统，通过训练交换关联泛函进一步验证了正则化方法的泛化能力，并从机器学习视角探讨了两种泛函的对比特性。