  Plasmas are highly nonlinear and multi-scale, motivating a hierarchy of
models to understand and describe their behavior. However, there is a scarcity
of plasma models of lower fidelity than magnetohydrodynamics (MHD). Galerkin
models, obtained by projection of the MHD equations onto a truncated modal
basis, can furnish this gap in the lower levels of the model hierarchy. In the
present work, we develop low-dimensional Galerkin plasma models which preserve
global conservation laws by construction. This additional model structure
enables physics-constrained machine learning algorithms that can discover these
types of low-dimensional plasma models directly from data. This formulation
relies on an energy-based inner product which takes into account all of the
dynamic variables. The theoretical results here build a bridge to the extensive
Galerkin literature in fluid mechanics, and facilitate the development of
physics-constrained reduced-order models from plasma data.
