Symmetry Considerations for Machine Learning Algorithms Operating on 3D Geometry and Physical Data

Symmetry is a pervading and multi-facetted concept in physics: The symmetry of space gives rise to mathematical objects that transform predictably under a change in coordinate system (geometry and geometric tensors). The symmetry of physical systems determines which properties are allowed or require symmetry breaking mechanisms to occur (Curie's Principle). The differentiable symmetries of physics give rise to conservation laws (Noether's theorem). In this talk, I will discuss methods for merging the rigors of these many flavors of symmetry with the flexibility of machine learning methods. As a demonstrative example, I'll focus on the properties of Euclidean Neural Networks which are constructed to preserve 3D Euclidean symmetry. These principles can be extended to other spaces and their relevant symmetries. Perhaps unsurprisingly, symmetry preserving machine learning algorithms are extremely data-efficient; they are able to achieve better results with less training data. More unexpectedly, they also act as "symmetry-compilers": they can only learn tasks that are symmetrically well-posed and in the case of differentiable methods they can also help uncover when there is symmetry implied missing information. I'll give examples of these properties and how they can be used to craft useful training tasks for physical data. To conclude, I'll highlight some open questions in merging symmetry and machine learning techniques particularly relevant to representing physical systems.