Density functional theory underlies the most successful and widely used numerical methods for electronic structure prediction of solids. However, it has the fundamental shortcoming that the universal density functional is unknown. In addition, the computational result—energy and charge density distribution of the ground state—is useful for electronic properties of solids mostly when reduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can help to free density functional theory from these limitations. We study a theory of spinless fermions on a one-dimensional lattice. The density functional is implicitly represented by a neural network, which predicts, besides the ground-state energy and density distribution, density-density correlation functions. At no point do we require a band structure interpretation. The training data, obtained via exact diagonalization, feeds into a learning scheme inspired by active learning, which minimizes the computational costs for data generation. We show that the network results are of high quantitative accuracy and, despite learning on random potentials, capture both symmetry-breaking and topological phase transitions correctly.

中文翻译：

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一维密度泛函理论的有效学习

密度泛函理论是固体电子结构预测最成功且使用最广泛的数值方法的基础。然而，它的根本缺点是通用密度泛函是未知的。此外，计算结果（基态的能量和电荷密度分布）对于固体的电子性质很有用，主要是当简化为基于Kohn-Sham方法的能带结构解释时。在这里，我们演示了机器学习算法如何帮助使密度泛函理论摆脱这些限制。我们研究一维晶格上无自旋费米子的理论。密度泛函由神经网络隐式表示，该神经网络除了预测基态能量和密度分布外，还预测密度-密度相关函数。我们绝不需要带结构的解释。通过精确的对角线化获得的训练数据将输入到受主动学习启发的学习方案中，从而最大程度地减少了数据生成的计算成本。我们表明网络结果具有很高的定量准确性，尽管学习了随机电位，但可以正确捕获对称性破坏和拓扑相变。