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.

翻译：扩展动态模态分解（EDMD）是一种用于动力学系统预测与降维的数据驱动工具，已在物理科学领域得到广泛应用。尽管该方法概念简洁，但在确定性混沌系统中，其特性甚至收敛目标尚不明确。特别地，EDMD的最小二乘逼近如何处理理解混沌动力学所需的规则函数类仍不清楚。本文以最简单的混沌映射——圆周解析扩张映射为例，建立了EDMD的普适性严格理论。为此，我们证明了单位圆上正交多项式（OPUC）理论的一个新结果，并应用传递算子理论中的方法。研究表明，在无限数据极限下，三角多项式观测字典的最小二乘投影具有指数级效率。基于此，我们证明在该设定下，采用EDMD获得的预测结果与Koopman谱数据能以字典规模的指数级速度收敛至具有物理意义的极限。这表明，即使采样测度不均匀，仅需相对较小的多项式字典即可使EDMD具备高效性。此外，我们的OPUC结果提示，基于数据的最小二乘投影可能是一种极为有效的逼近策略。