Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition
This addresses a bottleneck in dynamic mode decomposition for researchers analyzing nonlinear systems, offering an incremental improvement by automating observable selection.
The paper tackles the challenge of manually selecting nonlinear observables for Koopman spectral analysis in nonlinear dynamical systems by proposing a fully data-driven method that learns Koopman invariant subspaces from observed data, achieving improved performance in empirical evaluations.
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.