The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems, the main reason being the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. Until now, the situation where for large-scale systems, we (i) only have access to partial observations (i.e., measurements, as is very common for experimental data) or (ii) deliberately perform coarse graining (for efficiency reasons) has not been treated to its full extent. In this paper, we address the pitfall associated with this situation, that the classical EDMD algorithm does not automatically provide a Koopman operator approximation for the underlying system if we do not carefully select the number of observables. Moreover, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to massively increase the model efficiency. We also briefly draw a connection to domain decomposition techniques for partial differential equations and present numerical evidence using the Kuramoto--Sivashinsky equation.

翻译：Koopman算子已成为复杂系统数据驱动分析、预测与控制的核心工具，其优势在于能够从测量数据中识别非线性动力学过程的线性函数空间表示。目前，针对大规模系统中以下两种情况尚未得到充分研究：（i）仅能获取部分观测（即测量数据，这在实验数据中非常普遍）或（ii）出于效率考虑而主动进行粗粒化处理。本文揭示了在此类情形下的关键陷阱：若未谨慎选择可观测函数的数量，经典EDMD算法无法自动提供底层系统的Koopman算子近似。此外，我们证明系统动力学中的对称性可传递至Koopman算子，从而显著提升模型效率。本文还简要讨论了与偏微分方程区域分解技术的关联，并基于Kuramoto-Sivashinsky方程给出了数值验证。