[2603.24638] How unconstrained machine-learning models learn physical symmetries

Computer Science > Machine Learning arXiv:2603.24638 (cs) [Submitted on 25 Mar 2026] Title:How unconstrained machine-learning models learn physical symmetries Authors:Michelangelo Domina, Joseph William Abbott, Paolo Pegolo, Filippo Bigi, Michele Ceriotti View a PDF of the paper titled How unconstrained machine-learning models learn physical symmetries, by Michelangelo Domina and 4 other authors View PDF HTML (experimental) Abstract:The requirement of generating predictions that exactly fulfill the fundamental symmetry of the corresponding physical quantities has profoundly shaped the development of machine-learning models for physical simulations. In many cases, models are built using constrained mathematical forms that ensure that symmetries are enforced exactly. However, unconstrained models that do not obey rotational symmetries are often found to have competitive performance, and to be able to \emph{learn} to a high level of accuracy an approximate equivariant behavior with a simple data augmentation strategy. In this paper, we introduce rigorous metrics to measure the symmetry content of the learned representations in such models, and assess the accuracy by which the outputs fulfill the equivariant condition. We apply these metrics to two unconstrained, transformer-based models operating on decorated point clouds (a graph neural network for atomistic simulations and a PointNet-style architecture for particle physics) to investigate how symmetry information is process...