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.

通过对大规模系统的部分观测或粗粒化进行分析和数值实验，本文探讨了 Koopman 算子在非线性动力学中的线性函数空间表示，同时提出了 EDMD 算法在选取观测的数量时可能存在的问题，并揭示了系统动力学对 Koopman 算子的对称性，可大幅提高模型的效率，并提供了 Kuramoto-Sivashinsky 方程数值实例支持。