Data-driven Assimilation and Prediction of Complex Nonlinear Dynamics with Novel Quantum Mechanical Framework for Koopman Operators

A framework for data assimilation and prediction of nonlinear dynamics is presented, combining aspects of quantum mechanics, Koopman operator theory, and kernel methods for machine learning. This approach adapts the formalism of quantum dynamics and measurement to perform data assimilation (filtering), using the Koopman operator governing the evolution of observables as an analog of the Heisenberg operator in quantum mechanics, and a quantum mechanical density operator to represent the data assimilation state. The framework is implemented in a fully empirical, data-driven manner by representing the evolution and measurement operators via matrices in a basis learned from time-ordered observations. Applications to data assimilation of the Lorenz 96 multiscale system and Indo-Pacific climate show promising results. Furthermore, our framework provides a route for implementing data assimilation algorithms on quantum computers.