Dimensionless Machine Learning: Enforcing Exact Units Equivariance

Soledad Villar, Weichi Yao, David W. Hogg, Ben Blum-Smith, Bianca Dumitrascu; 24(109):1−32, 2023.

Abstract

Units equivariance (or units covariance) arises as an exact symmetry when the relationships among measured quantities of physics relevance adhere to consistent dimensional scalings. In this study, we present this symmetry in terms of a (non-compact) group action and utilize dimensional analysis and concepts from equivariant machine learning to propose a methodology for achieving exact units-equivariant machine learning. For any given learning task, we initially construct a dimensionless version of the inputs using established results from dimensional analysis, and subsequently perform inference in the dimensionless space. Our approach can be applied to enforce units equivariance in a wide range of machine learning methods that are equivariant to rotations and other groups. We explore the potential gains in in-sample and out-of-sample prediction accuracy that can be achieved in scenarios such as symbolic regression and emulation, where symmetry plays a crucial role. We demonstrate our approach through simple numerical examples involving dynamical systems in physics and ecology.

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