Symmetry is present throughout nature and continues to play an increasingly central role in machine learning. In this paper, we provide a unifying theoretical and methodological framework for incorporating Lie group symmetry into machine learning models in three ways: 1. enforcing known symmetry when training a model; 2. discovering unknown symmetries of a given model or data set; and 3. promoting symmetry during training by learning a model that breaks symmetries within a user-specified candidate group only when the data provide sufficient evidence. We show that these tasks can be cast within a common mathematical framework whose central object is the Lie derivative. We extend and unify several existing results by showing that enforcing and discovering symmetry are linear-algebraic tasks that are dual under the bilinear pairing induced by the Lie derivative. We also propose a novel way to promote symmetry by introducing a class of convex regularizers, built from the Lie derivative with a nuclear-norm relaxation, that penalizes symmetry breaking during training. We explain how these ideas can be applied to a wide range of machine learning models including basis-function regression, dynamical-systems discovery, neural networks, and neural operators acting on fields.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint. In this paper, we partially explain this mystery by proving that without querying the adjoint, one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis. We then apply the result to recovering Green's functions of elliptic partial differential operators and derive an adjoint-free sample complexity bound. While existing theory justifies low sample complexity in operator learning, ours is the first adjoint-free analysis that attempts to close the gap between theory and practice.