发表机构
School of Physical Sciences, National Institute of Science Education and Research; Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń（国立科学教育与研究所物理科学学院; 托伦哥白尼大学物理、天文与信息学院物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将半局域机器学习修正应用于四种密度泛函，发现其学习到的原子化能修正具有强尺寸依赖且依赖于父泛函的误差结构，揭示了尺寸依赖误差累积是评估ML增强泛函的关键因素。
AI 中文摘要
密度泛函近似（DFA）的机器学习（ML）修正提供了一条在保留半局域泛函效率的同时改进电子结构预测的途径。一个重要的问题是，此类修正学到的是可迁移的改进，还是仅仅补偿了父泛函特有的误差。在此，我们通过将Wang等人[J. Chem. Phys. 158, 154107 (2023)]的半局域ML修正应用于PBE、B3LYP、SCAN和r²SCAN来回答这一问题，同时保持网络架构、损失函数、训练集和优化协议不变。我们发现，对于所有四种父泛函，学习到的修正都会对原子化能产生系统性的尺寸依赖贡献，在正烷烃系列中线性拟合的Pearson系数|r|≥0.9999。其大小和符号系统地依赖于父泛函：该修正部分补偿了PBE和B3LYP的尺寸依赖误差，但对SCAN和r²SCAN引入了显著的尺寸依赖贡献，而这两种泛函的父误差几乎没有尺寸依赖性。相比之下，电离势、电子亲和能和异构化能基本不受影响。空间分解进一步揭示了这种广延贡献的不同来源，其中键合区域主导B3LYP、SCAN和r²SCAN的贡献，而核心区域主导PBE的贡献。这些结果表明，全局能量ML修正强烈依赖于其父DFA的误差结构，凸显了尺寸依赖误差累积作为评估和开发ML增强密度泛函的关键考量因素。
英文摘要
Machine-learning (ML) corrections to density functional approximations (DFAs) offer a route to improving electronic-structure predictions while retaining the efficiency of semilocal functionals. An important question is whether such corrections learn transferable improvements or instead compensate for errors specific to the parent functional. Here, we address this question by applying the semilocal ML correction of Wang \textit{et al.} [J.~Chem.~Phys.~\textbf{158}, 154107 (2023)] to PBE, B3LYP, SCAN, and r$^2$SCAN, while keeping the network architecture, loss function, training set, and optimization protocol unchanged. We find that the learned correction develops a systematic size-dependent contribution to atomization energies for all four parents, with $|r|\geq0.9999$ along the $n$-alkane series, where r is the Pearson coefficient of the linear fit. Its magnitude and sign depend systematically on the parent: the correction partially compensates the size-dependent errors of PBE and B3LYP, but introduces substantial size-dependent contributions for SCAN and r$^2$SCAN, whose parent errors show little size dependence. In contrast, ionization potentials, electron affinities, and isomerization energies are largely unaffected. Spatial decomposition further reveals distinct origins of this extensive contribution, with bonding regions dominating for B3LYP, SCAN, and r$^2$SCAN and core regions dominating for PBE. These results show that the global-energy ML correction is strongly dependent on the error structure of its parent DFA, highlighting size-dependent error accumulation as a critical consideration in the assessment and development of ML-enhanced density functionals.
