5 research outputs found
Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos
Extended Dynamic Mode Decomposition (EDMD) is a data-driven tool for
forecasting and model reduction of dynamics, which has been extensively taken
up in the physical sciences. While the method is conceptually simple, in
deterministic chaos it is unclear what its properties are or even what it
converges to. In particular, it is not clear how EDMD's least-squares
approximation treats the classes of regular functions needed to make sense of
chaotic dynamics.
In this paper we develop a general, rigorous theory of EDMD on the simplest
examples of chaotic maps: analytic expanding maps of the circle. To do this, we
prove a new result in the theory of orthogonal polynomials on the unit circle
(OPUC) and apply methods from transfer operator theory. We show that in the
infinite-data limit, the least-squares projection is exponentially efficient
for trigonometric polynomial observable dictionaries. As a result, we show that
the forecasts and Koopman spectral data produced using EDMD in this setting
converge to the physically meaningful limits, exponentially quickly in the size
of the dictionary. This demonstrates that with only a relatively small
polynomial dictionary, EDMD can be very effective, even when the sampling
measure is not uniform. Furthermore, our OPUC result suggests that data-based
least-squares projections may be a very effective approximation strategy.Comment: fixed misprint in equation (1), et