Grisafi Andrea, Wilkins David M, Csányi Gábor, Ceriotti Michele
Laboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
Engineering Laboratory, University of Cambridge, Trumpington Street, Cambridge CB21PZ, United Kingdom.
Phys Rev Lett. 2018 Jan 19;120(3):036002. doi: 10.1103/PhysRevLett.120.036002.
Statistical learning methods show great promise in providing an accurate prediction of materials and molecular properties, while minimizing the need for computationally demanding electronic structure calculations. The accuracy and transferability of these models are increased significantly by encoding into the learning procedure the fundamental symmetries of rotational and permutational invariance of scalar properties. However, the prediction of tensorial properties requires that the model respects the appropriate geometric transformations, rather than invariance, when the reference frame is rotated. We introduce a formalism that extends existing schemes and makes it possible to perform machine learning of tensorial properties of arbitrary rank, and for general molecular geometries. To demonstrate it, we derive a tensor kernel adapted to rotational symmetry, which is the natural generalization of the smooth overlap of atomic positions kernel commonly used for the prediction of scalar properties at the atomic scale. The performance and generality of the approach is demonstrated by learning the instantaneous response to an external electric field of water oligomers of increasing complexity, from the isolated molecule to the condensed phase.
统计学习方法在准确预测材料和分子性质方面显示出巨大潜力，同时将对计算要求较高的电子结构计算的需求降至最低。通过将标量性质的旋转和置换不变性的基本对称性编码到学习过程中，这些模型的准确性和可转移性显著提高。然而，当参考系旋转时，张量性质的预测要求模型尊重适当的几何变换，而不是不变性。我们引入了一种形式主义，它扩展了现有方案，并使得对任意秩的张量性质以及一般分子几何结构进行机器学习成为可能。为了证明这一点，我们推导了一种适应旋转对称性的张量核，它是原子尺度上用于预测标量性质的常用原子位置平滑重叠核的自然推广。通过学习从孤立分子到凝聚相的复杂度不断增加的水低聚物对外部电场的瞬时响应，证明了该方法的性能和通用性。
