  We propose a scheme for data-driven parameterization of unresolved dimensions
of dynamical systems based on the mathematical framework of quantum mechanics
and Koopman operator theory. Given a system in which some components of the
state are unknown, this method involves defining a surrogate system in a
time-dependent quantum state which determines the fluxes from the unresolved
degrees of freedom at each timestep. The quantum state is a density operator on
a finite-dimensional Hilbert space of classical observables and evolves over
time under an action induced by the Koopman operator. The quantum state also
updates with new values of the resolved variables according to a quantum Bayes'
law, implemented via an operator-valued feature map. Kernel methods are
utilized to learn data-driven basis functions and represent quantum states,
observables, and evolution operators as matrices. The resulting computational
schemes are automatically positivity-preserving, aiding in the physical
consistency of the parameterized system. We analyze the results of two
different modalities of this methodology applied to the Lorenz 63 and Lorenz 96
multiscale systems, and show how this approach preserves important statistical
and qualitative properties of the underlying chaotic systems.
