Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems

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.

Figure 1

Learning curves of the IST components of dipole $\mathit{\mu }$ ($\lambda =1$), polarizability $\mathit{\alpha }$ ($\lambda =0$, 2), and hyperpolarizability $\mathit{\beta }$ ($\lambda =1$, 3) for water monomer (left), water dimer (center), and Zundel cation (right). For all cases the testing data set consists of 500 independent configurations. Arrows with symbols indicate the intrinsic standard deviation of the testing data set. The kernel has been computed with an environment cutoff of 4 Å for the monomer and ${\mathrm{H}}_5{\mathrm{O}}_2^{+}$, and 5 Å for the water dimer.

Figure 2

Learning curves of the IST components of water dielectric response tensors $\mathit{ϵ}$, through direct learning (red and green lines) and indirect learning going through the CM relation. The testing data set consists of 500 independent configurations. Arrows indicate the intrinsic standard deviation of the testing samples. Filled square points show the predictions for 5 ice Ih structures using the ML model trained on liquid water.