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. We develop for the first time 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, basic approximation 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 error is exponentially small 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 fast with respect to 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，但在确定性混沌中，其性质乃至收敛目标尚不明确。尤其需要指出的是，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。我们首次以混沌映射的最简范例（圆环解析扩张映射）为基础，建立了EDMD的一般化严格理论。为此，我们证明了单位圆正交多项式（OPUC）理论中一项新的基本近似结果，并应用了转移算子理论方法。研究表明，在无限数据极限下，对于三角多项式观测字典，最小二乘投影误差呈指数级减小。据此，我们论证了在此设定下使用EDMD生成的预测与Koopman谱数据会收敛至具有物理意义的极限，且收敛速度随字典规模增长呈指数级加快。这表明即使使用相对较小的多项式字典，EDMD也能非常有效——即便采样测度非均匀。此外，我们的OPUC结果暗示基于数据的最小二乘投影可能成为一种极具效力的近似策略。