Principled selection of basis functions for Extended Dynamic Mode Decomposition

Determine a principled procedure for selecting the dictionary of basis functions used in Extended Dynamic Mode Decomposition that appropriately represents complex nonlinear dynamics when constructing finite-dimensional approximations of the Koopman operator from data.

Background

Extended Dynamic Mode Decomposition (EDMD) approximates the infinite-dimensional Koopman operator by operating on a finite set of user-chosen basis functions (a dictionary). While EDMD relaxes linear measurement assumptions, its performance critically depends on the manual choice of these basis functions.

The paper emphasizes that this manual selection is a bottleneck and explicitly notes that identifying functions capable of representing complex nonlinear dynamics remains an open problem. The authors instead pursue a data-driven approach that learns lifting functions end-to-end, but the underlying question of how to select such basis functions in a principled, general manner remains unresolved.

References

Their performance, however, is bottlenecked by the manual selection of a set of basis functions. Finding such functions that can appropriately represent the complex dynamics remains an open problem in itself.

— Selective State-Space Models for Koopman-based Data-driven Distribution System State Estimation  (2604.02273 - Alabdulrazzaq et al., 2 Apr 2026) in Subsection "Koopman Theory" (Preliminaries)

Whether the patterns generalise to other kernels, basis families, or larger-scale regressions remains for future work.

— On Basis Function Selection for Sparse Gaussian Process Regression  (2609.26624 - Soom et al., 22 Sep 2026) in Section 6, Discussion, paragraph "Limitations and future work"