# Operator algebras for closure of dynamical systems

** Speaker:** Dimitris Giannakis, Dartmouth College

** Date:** September 15, 2022

** Abstract:** In many applications of dynamical systems one is faced with the problem of approximating the evolution of fine-grained processes which are too complex to model directly. This problem is known as the closure problem, and it is tackled with a variety of techniques from asymptotic analysis, stochastic analysis, statistics, and machine learning. In this talk, we describe an approach to closure of dynamical systems which is based on ideas from the theory of operator algebras and ergodic theory. In more detail, given a system in which some components of the state are unknown, this method models the unresolved degrees of freedom as being in a time-dependent state of an operator algebra (a “quantum state”), which determines their influence on the resolved variables. The quantum state evolves over time under the action of the Koopman operator, which is the composition operator by the dynamical flow, and is updated at each timestep by the values of the resolved variables according to a quantum Bayes’ law. Moreover, kernel methods for machine learning are utilized to implement the framework numerically. We present applications of this methodology to the Lorenz 63 and multiscale Lorenz 96 systems, and show how this approach preserves important statistical and qualitative properties of the underlying chaotic systems. This is joint work with David Freeman and Joanna Slawinska.