Quantum mechanical closure of partial differential equations with symmetries

TL;DR

This paper introduces a novel quantum mechanics-inspired statistical framework for closing partial differential equations, effectively capturing unresolved dynamics and respecting symmetries, demonstrated on shallow water equations.

Contribution

It develops a quantum mechanical embedding approach for PDE closure, incorporating symmetry invariance and a data-driven discretization method.

Findings

Accurately predicts main features of true dynamics

Handles out-of-sample initial conditions effectively

Maintains invariance under dynamical symmetries

Abstract

We develop a statistical framework for the dynamical closure of spatiotemporal dynamics governed by partial differential equations. Employing the mathematical framework of quantum mechanics to embed the original classical dynamics into a quantum mechanical representation, we use the space of quantum density operators to model the unresolved degrees of freedom of the original dynamics in a statistical sense, and the framework of quantum measurement to predict their contributions to the resolved dynamics. The embedded dynamics is discretized by a positivity preserving process, leading to a compressed representation that is invariant under the dynamical symmetries of the resolved dynamics. We present a data based formulation of the closure scheme and apply it to a closure problem for the shallow water equations. The numerical results demonstrate that our closure model can accurately predict the main features of the true dynamics, including for out of sample initial conditions.