This paper describes a method for learning low-dimensional approximations of
nonlinear dynamical systems, based on neural-network approximations of the
underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD)
provides a useful data-driven approximation of the Koopman operator for
analyzing dynamical systems. This paper addresses a fundamental problem
associated with EDMD: a trade-off between representational capacity of the
dictionary and over-fitting due to insufficient data. A new neural network
architecture combining an autoencoder with linear recurrent dynamics in the
encoded state is used to learn a low-dimensional and highly informative
Koopman-invariant subspace of observables. A method is also presented for
balanced model reduction of over-specified EDMD systems in feature space.
Nonlinear reconstruction using partially linear multi-kernel regression aims to
improve reconstruction accuracy from the low-dimensional state when the data
has complex but intrinsically low-dimensional structure. The techniques
demonstrate the ability to identify Koopman eigenfunctions of the unforced
Duffing equation, create accurate low-dimensional models of an unstable
cylinder wake flow, and make short-time predictions of the chaotic
Kuramoto-Sivashinsky equation.

本文提出了使用神经网络逼近 Koopman 算子的方法来学习非线性动态系统的低维逼近，并且讨论了与这种方法相关的数据表示问题和过拟合问题，提出了一种结合自编码器和线性递归动力学的新型神经网络架构来学习 Koopman 不变子空间。同时，还提出了在特征空间下过度规定的 EDMD 系统的均衡模型缩减方法和使用多核回归算法来提高低维状态下数据重构的精度，最后通过案例研究验证了这些技术的有效性。