Reduced order modelling relies on representing complex dynamical systems using simplified modes, which can be achieved through Koopman operator analysis. However, computing Koopman eigen pairs for high-dimensional observable data can be inefficient. This paper proposes using deep autoencoders, a type of deep learning technique, to perform non-linear geometric transformations on raw data before computing Koopman eigen vectors. The encoded data produced by the deep autoencoder is diffeomorphic to a manifold of the dynamical system, and has a significantly lower dimension than the raw data. To handle high-dimensional time series data, Takens's time delay embedding is presented as a pre-processing technique. The paper concludes by presenting examples of these techniques in action.

翻译：降阶建模依赖于使用简化模态表征复杂动力系统，这一目标可通过Koopman算子分析实现。然而，针对高维观测数据计算Koopman特征对往往效率低下。本文提出采用深度自编码器（一种深度学习技术）对原始数据进行非线性几何变换，再计算Koopman特征向量。深度自编码器生成的编码数据与动力系统流形微分同胚，且其维度显著低于原始数据。为处理高维时间序列数据，本文引入Takens时间延迟嵌入作为预处理技术。最后，通过实例展示了这些方法的具体应用效果。