Quantum mechanical closure of partial differential equations: applications to fluid and plasma dynamics

We consider the dynamical closure of partial differential equations with applications in fluid and plasma dynamics. Our closure framework rests on the mathematical foundations of quantum mechanics, embedding the original classical dynamics into a quantum mechanical system and using a field of quantum density operators over the spatial domain to encode statistical information about the unresolved degrees of freedom of the original dynamics. The contributions (fluxes) from the density operators to the resolved dynamics are predicted using the framework of quantum measurement. The data driven implementation of our closure scheme is built around a compressed representation of the original dynamics that is invariant under the dynamical symmetries of the considered governing equations. Its numerical realization relies on kernel methods from machine learning and delay embedding from dynamical systems theory. We apply our scheme to closure problems for two models: (1) the shallow water equations on a periodic one dimensional domain; (2) the drift-reduced Braginskii equations for magnetized edge plasma turbulence on a two dimensional domain. Our numerical results demonstrate the effectiveness of our closure scheme in extracting the dominant spatiotemporal patterns of the dynamics and using them to predict the dynamics of the resolved variables for out of sample initial conditions.