  Unlike with the energy, which is a scalar property, machine learning (ML)
predictions of vector or tensor properties poses the additional challenge of
achieving proper invariance (covariance) with respect to molecular rotation. If
the properties cannot be obtained by differentiation, other appropriate methods
should be applied to retain the covariance. There have been several approaches
suggested to properly treat this issue. For nonadiabatic couplings and
polarizabilities, for example, it was possible to construct virtual quantities
from which the above tensorial properties are obtained by differentiation and
thus guarantee the covariance. Here we propose a simpler alternative technique,
which does not require construction of auxiliary properties or application of
special equivariant ML techniques. We suggest a three-step approach, using the
molecular tensor of inertia. In the first step, the molecule is rotated using
the eigenvectors of this tensor to its principal axes. In the second step, the
ML procedure predicts the vector property relative to this orientation, based
on a training set where all vector properties were in this same coordinate
system. As third step, it remains to transform the ML estimate of the vector
property back to the original orientation. This rotate-predict-rotate (RPR)
procedure should thus guarantee proper covariance of a vector property and is
trivially extensible also to tensors such as polarizability. The PRP procedure
has an advantage that the accurate models can be trained very fast for
thousands of molecular configurations which might be beneficial where many
trainings are required (e.g., in active learning). We have implemented the RPR
technique, using the MLatom and Newton-X programs for ML and MD and performed
its assessment on the dipole moment along MD trajectories of
1,2-dichloroethane.
