  In Koopman operator theory, a finite-dimensional nonlinear system is
transformed into an infinite but linear system using a set of observable
functions. However, manually selecting observable functions that span the
invariant subspace of the Koopman operator based on prior knowledge is
inefficient and challenging, particularly when little or no information is
available about the underlying systems. Furthermore, current methodologies tend
to disregard the importance of the invertibility of observable functions, which
leads to inaccurate results. To address these challenges, we propose the
so-called FlowDMD, aka Flow-based Dynamic Mode Decomposition, that utilizes the
Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages
the intrinsically invertible characteristics of the CF-INN to learn the
invariant subspaces of the Koopman operator and accurately reconstruct state
variables. Numerical experiments demonstrate the superior performance of our
algorithm compared to state-of-the-art methodologies.
