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 differentiable functions on which chaotic systems act. 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 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 projection may be a very effective approximation strategy more generally.

翻译：扩展动态模态分解（EDMD）是一种用于动力学预测和模型降阶的数据驱动工具，已在物理科学领域得到广泛应用。尽管该方法在概念上简洁，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的最小二乘逼近如何处理混沌系统作用下的可微函数类仍未厘清。我们首次以混沌映射的最简示例——圆上的解析扩张映射——为基础，建立了EDMD的通用严格理论。为此，我们证明了单位圆上正交多项式（OPUC）理论中一个新的基本逼近结果，并应用了传递算子理论的方法。研究表明，在无限数据极限下，对于三角多项式可观测量字典，最小二乘投影误差呈指数级减小。由此证明，在此框架下使用EDMD生成的预测与Koopman谱数据将收敛至具有物理意义的极限，且收敛速度随字典规模呈指数级增长。这表明，即使采样测度非均匀，仅使用较小的多项式字典即可使EDMD非常有效。此外，我们的OPUC结果暗示，基于数据的最小二乘投影可能是一种更具普适性的高效逼近策略。