Quantum mechanical closure of partial differential equations with symmetries

Abstract: We develop a framework for the dynamical closure of spatiotemporal dynamics governed by partial differential equations. We employ the mathematical framework of quantum mechanics to embed the original classical dynamics into an infinite dimensional quantum mechanical system, using the space of quantum states to model the unresolved degrees of freedom of the original dynamics and the technology of quantum measurement to predict their contributions to the resolved dynamics. We use a positivity preserving discretization to project the embedded dynamics to finite dimension. Combining methods from operator valued kernels and delay embedding, we derive a compressed representation of the dynamics that is invariant under the spatial symmetries of the original dynamics. We develop a data driven formulation of the scheme that can be realized numerically and apply it to a dynamical closure problem for the shallow water equations, demonstrating that our closure model can accurately predict the main features of the true dynamics, including for out of sample initial conditions.