  Any deterministic autonomous dynamical system may be globally linearized by
its' Koopman operator. This object is typically infinite-dimensional and can be
approximated by the so-called Dynamic Mode Decomposition (DMD). In DMD, the
central idea is to preserve a fundamental property of the Koopman operator:
linearity. This work augments DMD by preserving additional properties like
functional relationships between observables and consistency along geometric
invariants. The first set of constraints provides a framework for understanding
DMD variants like Higher-order DMD and Affine DMD. The latter set guarantees
the estimation of Koopman eigen-functions with eigen-value 1, whose level sets
are known to delineate invariant sets. These benefits are realized with only a
minimal increase in computational cost, primarily due to the linearity of
constraints.
