Mathematics & Statistics Departmental Research Seminar

Finding Order in Chaos: Approximating Koopman eigenfunctions to understand nonlinear dynamics

Abstract:

Dynamical systems describe how the world around us evolves in time; however, these equations are often highly nonlinear and potentially chaotic. The Koopman operator K acts on observable functions ϕ, which represent some measurement of the finite-dimensional state space, and linearize a nonlinear system by K[ϕ](x_n )=ϕ(x_(n+1)). Koopman eigenfunctions are a special set of observables that evolve by multiplication with their associated eigenvalue and can be used to identify salient features of the system such as fixed points, periodic orbits, and invariant manifolds. Modern data-driven methods given powerful and efficient algorithms to approximate the Koopman operator and its eigenfunctions without a priori knowledge of the governing equations. In this talk, I will cover two prominent methods to approximate the Koopman operator, namely extended dynamic mode decomposition and Koopman autoencoders, and explore their theoretical and computational powers and limitations. In particular, I will discuss the finite-data convergence rates for Koopman operator approximations, using generalized Koopman eigenfunction approximations to predict periodic orbits, and performance guarantees for training autoencoders to approximate Koopman eigenfunctions.