Quantum Mechanics for Parametrization of Dynamical Systems

We present a data-driven scheme for learning closures of dynamical systems (i.e. parametrizations) based on the mathematical framework of quantum mechanics and Koopman operator theory. In addition to being an approach to the problem of parametrization in general, the method has properties that may be of particular use to problems in climate modelling, including the preservation of underlying chaotic dynamics of the system being modelled and the preservation of positivity in the parameterization of positive unresolved variables. Given a system in which some components of the state are unknown, this method models the unresolved degrees of freedom as being in a time-dependent “quantum-state”, which determines their influence on the resolved variables. The quantum state is an operator on a space of observables and evolves over time under the action of the Koopman operator. The quantum state representing the unresolved degrees of freedom is updated at each timestep by the values of the resolved variables according to a quantum Bayes’ law. Moreover, kernel functions are utilized to allow the quantum Bayes’ law to be implemented numerically. We present applications of this methodology to the Lorenz 63 and multiscale Lorenz 96 systems, and show how this approach preserves important statistical and qualitative properties of the underlying chaotic systems.