AEM Mechanics Seminar

Symmetry in Mechanics and AI: From Elasticity Classes to Equivariant Networks

Prof. Ryan Elliott
Department of Aerospace Engineering and Mechanics, University of Minnesota

A central challenge in modern AI for physical applications is embedding geometric structure directly into network representations. As frameworks like equivariant neural networks become standard, there is a growing need for general, algorithmic tools to characterize and manipulate invariant tensor spaces. Continuum mechanics offers a rich history and a mature mathematical toolkit for solving precisely this class of problems.

In this chalkboard lecture, we begin with the historical journey of material symmetry classification, detailing how a deeper group-theoretic perspective corrected decades of convention by reducing the number of elasticity classes from 10 to 8. We then present the general algebraic solution recently established by Olive and Auffray, demonstrating how harmonic decomposition paired with the "clips" operator (⊙) Provides a systematic procedure to algorithmically classify the symmetry of arbitrary tensor-based physical relationships. These geometric perspectives and formal classification tools offer a critical framework for the ongoing advancement of physical AI.