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. Proving a new result in the theory of orthogonal polynomials on the unit circle (OPUC), we show that in the infinite-data limit, the least-squares projection is exponentially efficient for 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, at an exponential rate. 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）是一种基于数据的动力学预测与降维工具，已在物理科学领域得到广泛应用。尽管该方法在概念上简单，但在确定性混沌系统中，其性质乃至收敛目标尚不明确。特别是，EDMD的最小二乘近似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——圆解析扩张映射为例，建立了EDMD的一般化严格理论。通过证明单位圆上正交多项式理论（OPUC）的新结果，我们表明在无限数据极限下，最小二乘投影对于多项式观测字典具有指数级效率。由此证明，在此设置下，基于EDMD产生的预测和Koopman谱数据以指数速率收敛至具有物理意义的极限。这表明即使采样测度非均匀，仅使用相对较小的多项式字典，EDMD也能非常有效。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非常有效的近似策略。