  Units equivariance (or units covariance) is the exact symmetry that follows
from the requirement that relationships among measured quantities of physics
relevance must obey self-consistent dimensional scalings. Here, we express this
symmetry in terms of a (non-compact) group action, and we employ dimensional
analysis and ideas from equivariant machine learning to provide a methodology
for exactly units-equivariant machine learning: For any given learning task, we
first construct a dimensionless version of its inputs using classic results
from dimensional analysis, and then perform inference in the dimensionless
space. Our approach can be used to impose units equivariance across a broad
range of machine learning methods which are equivariant to rotations and other
groups. We discuss the in-sample and out-of-sample prediction accuracy gains
one can obtain in contexts like symbolic regression and emulation, where
symmetry is important. We illustrate our approach with simple numerical
examples involving dynamical systems in physics and ecology.
