Recursive Forward-Backward EDMD: Guaranteed Algebraic Search for Koopman Invariant Subspaces

The implementation of the Koopman operator on digital computers often relies on the approximation of its action on finite-dimensional function spaces. This approximation is generally done by orthogonally projecting on the subspace. Extended Dynamic Mode Decomposition (EDMD) is a popular, special case of this projection procedure in a data-driven setting. Importantly, the accuracy of the model obtained by EDMD depends on the quality of the finite-dimensional space, specifically on how close it is to being invariant under the Koopman operator. This paper presents a data-driven algebraic search algorithm, termed Recursive Forward-Backward EDMD, for subspaces close to being invariant under the Koopman operator. Relying on the concept of temporal consistency, which measures the quality of the subspace, our algorithm recursively decomposes the search space into two subspaces with different prediction accuracy levels. The subspace with lower level of accuracy is removed if it does not reach a satisfactory threshold. The algorithm allows for tuning the level of accuracy depending on the underlying application and is endowed with convergence and accuracy guarantees.