Paper:
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition
Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.

Review:
This paper presents a method that takes advantage of MLPs to learning non-linear functions that support Koopman analysis of time-series data. The framework is based on Koopman operator theory, an observation that complex, nonlinear dynamics may be embedded with nonlinear functions g where a spectral properties of a linear operator K can be informative about both the dynamics of the system and predict future data. The paper proposes a neural network architecture and a set of loss functions suitable for learning this embedding in a Koopman invariant subspace, directly from data. The method is demonstrate both on numerical examples and on a few applications (Lorenz, Rossler, and unstable phenomena data).
The paper is exceptionally clearly written, and the use of a neural network for finding Koopman invariant subspaces, a challenging and widely applicable task, is well motivated. The figures nicely illustrate the results, and I find the examples convincing. There's a few questions I was curious about. Firstly, how well does LKIS tolerate larger magnitudes of noise? Does it converge on the right Koopman eigenfunctions, or is there a point past which it fails? Second, how do the g's found by LKIS compare with those found by EDMD and kernel DMD?
I read through the author’s rebuttal and my colleagues’ review, and decided to not change my assessment of the paper.