Data assimilation in operator algebras

Abstract

Significance Data assimilation is an essential component of numerical models for forecasting and uncertainty quantification of dynamical systems given incomplete knowledge of the state and governing equations. Here, we develop theory and computational methods for data assimilation through a combination of ideas from operator algebras, quantum information, and ergodic theory. Our approach leverages properties of noncommutative operator spaces to design computational schemes that i) preserve the sign of positive quantities such as mass in ways that are not possible with classical commutative methods and ii) are amenable to consistent data-driven approximation using machine learning. Furthermore, our framework provides a route for implementing data assimilation algorithms on quantum computers. We present applications to multiscale chaotic systems and the El Niño Southern Oscillation.