Quantum Mechanics for Closure of Dynamical Systems

TL;DR

This paper introduces a novel data-driven approach for modeling unresolved components of dynamical systems using quantum mechanics and Koopman operator theory, ensuring physical consistency and preserving system properties.

Contribution

It develops a quantum-inspired framework with kernel methods for parameterizing unresolved dynamics, a novel integration of quantum states and Koopman theory for this purpose.

Findings

Successfully applied to Lorenz 63 and Lorenz 96 systems

Preserves statistical and qualitative properties of chaotic systems

Ensures positivity and physical consistency in modeling

Abstract

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.