Koopman operator learning using invertible neural networks
This work addresses inefficiencies in modeling nonlinear systems for researchers in dynamical systems and machine learning, though it appears incremental as it builds on existing invertible neural network frameworks.
The paper tackled the challenge of manually selecting observable functions in Koopman operator theory by proposing FlowDMD, which uses invertible neural networks to learn invariant subspaces and reconstruct state variables, achieving superior performance in numerical experiments compared to state-of-the-art methods.
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.