The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations (i.e., measurements, as is very common for experimental data) has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.

翻译：库普曼算子已成为复杂系统数据驱动分析、预测与控制的重要工具。其主要原因在于，从测量数据中识别非线性动力学的线性函数空间表示具有巨大潜力。这同样适用于常微分方程、随机微分方程和偏微分方程。迄今为止，除少数例外，偏微分方程情形大多仅得到较为表面的处理，且底层动力学的特定结构在很大程度上被忽略。本文证明，系统动力学中的对称性可传递至库普曼算子，从而显著提升模型效能。此外，仅能获取部分观测（即测量数据，这在实验数据中极为常见）的情形亦未得到充分探讨。我们进一步处理了无法测量完整状态的高度相关情形，其中必须考虑替代方法（如延迟坐标）。基于嵌入理论，我们在此情形下推导了关于所需观测量数量的严格结论。我们通过包括波动方程和Kuramoto-Sivashinsky方程在内的多种数值算例提供了数值验证。