Ben Blum-Smith

Soledad Villar, Weichi Yao, David W. Hogg et al.

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.

Ben Blum-Smith, Soledad Villar

Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group $G$, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that $G$ is a compact Lie group.

Ben Blum-Smith, Ningyuan Huang, Marco Cuturi et al.

In this work, we present a mathematical formulation for machine learning of (1) functions on symmetric matrices that are invariant with respect to the action of permutations by conjugation, and (2) functions on point clouds that are invariant with respect to rotations, reflections, and permutations of the points. To achieve this, we provide a general construction of generically separating invariant features using ideas inspired by Galois theory. We construct $O(n^2)$ invariant features derived from generators for the field of rational functions on $n\times n$ symmetric matrices that are invariant under joint permutations of rows and columns. We show that these invariant features can separate all distinct orbits of symmetric matrices except for a measure zero set; such features can be used to universally approximate invariant functions on almost all weighted graphs. For point clouds in a fixed dimension, we prove that the number of invariant features can be reduced, generically without losing expressivity, to $O(n)$, where $n$ is the number of points. We combine these invariant features with DeepSets to learn functions on symmetric matrices and point clouds with varying sizes. We empirically demonstrate the feasibility of our approach on molecule property regression and point cloud distance prediction.

Wilson G. Gregory, David W. Hogg, Ben Blum-Smith et al.

Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors, and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet, and a UNet. In numerical experiments emulating 2D compressible Navier-Stokes, we see better accuracy and improved stability compared to baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any CNN-based method applied to an appropriate class of problems

Soledad Villar, David W. Hogg, Kate Storey-Fisher et al.

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 Poincaré 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.