There has been enormous progress in the last few years in designing neural
networks that respect the fundamental symmetries and coordinate freedoms of
physical law. Some of these frameworks make use of irreducible representations,
some make use of high-order tensor objects, and some apply symmetry-enforcing
constraints. Different physical laws obey different combinations of fundamental
symmetries, but a large fraction (possibly all) of classical physics is
equivariant to translation, rotation, reflection (parity), boost (relativity),
and permutations. Here we show that it is simple to parameterize universally
approximating polynomial functions that are equivariant under these symmetries,
or under the Euclidean, Lorentz, and Poincar\’e groups, at any dimensionality
$d$. The key observation is that nonlinear O($d$)-equivariant (and
related-group-equivariant) functions can be universally expressed in terms of a
lightweight collection of scalars — scalar products and scalar contractions of
the scalar, vector, and tensor inputs. We complement our theory with numerical
examples that show that the scalar-based method is simple, efficient, and
scalable.
