The temperature variation of the defect densities in a crystal depends on vibrational entropy. This contribution to the system thermodynamics remains computationally challenging as it requires a diagonalization of the system's Hessian which scales as $O\left(N^3\right)$ for a crystal made of $N$ atoms. Here, to circumvent such a heavy computational task and make it feasible even for systems containing millions of atoms, the harmonic vibrational entropy of point defects is estimated directly from the relaxed atomic positions through a linear-in-descriptor machine learning approach of order $O\left(N\right)$. With a size-independent descriptor dimension and fixed model parameters, an excellent predictive power is demonstrated on a wide range of defect configurations, supercell sizes, and external deformations well outside the training database. In particular, formation entropies in a range of $250k_B$ are predicted with less than $1.6k_B$ error from a training database whose formation entropies span only $25k_B$ (training error less than $1.0k_B$). This exceptional transferability is found to hold even when the training is limited to a low-energy superbasin in the phase space while the tests are performed for a different liquid-like superbasin at higher energies.

中文翻译：

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机器学习替代模型预测点缺陷振动熵

晶体中缺陷密度的温度变化取决于振动熵。对系统热力学的这种贡献在计算上仍然具有挑战性，因为它需要对系统的Hessian进行对角化，并按比例缩放$Ø（{ñ}^3）$ 制成的水晶 $ñ$原子。在这里，为了规避如此繁重的计算任务，甚至对于包含数百万个原子的系统而言，它也是可行的，可以通过阶数的线性描述符机器学习方法直接从松弛原子位置估算点缺陷的谐波振动熵$Ø（ñ）$。借助尺寸无关的描述符尺寸和固定的模型参数，可以在训练数据库之外的各种缺陷配置，超级单元尺寸和外部变形方面展现出色的预测能力。特别是，形成熵范围为$250{ķ}_{乙}$ 预测少于 $1.6{ķ}_{乙}$ 来自训练熵的信息，其形成熵仅跨越 $25{ķ}_{乙}$ （训练误差小于 $1.0{ķ}_{乙}$）。即使在相空间中将训练限于低能量的超级盆地时，也发现了这种出色的可转移性，而在更高的能量下对另一个类似液体的超级盆地进行了测试。