Scalars are universal: Equivariant machine learning, structured like classical physics

TL;DR

This paper demonstrates that scalar functions can serve as a universal building block for designing neural networks respecting fundamental physical symmetries across various groups and dimensions, simplifying the construction of equivariant models.

Contribution

It introduces a universal parameterization of equivariant polynomial functions using only scalar contractions, applicable to multiple symmetry groups and dimensions.

Findings

Scalar-based functions are simple and efficient.

The approach is scalable to high dimensions.

Numerical examples confirm the method's effectiveness.

Abstract

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.