Determine the source of effectiveness in time-delay-based forced-DMD methods

Determine whether the effectiveness of the forced-linear methods of Brunton et al., Pan et al., and Khodkar et al. is attributable to time-delay embedding, the forced linear models that they construct, or the combination of both, and determine whether those methods would be equally effective when applied to the original coordinates rather than time-delay observables.

Background

The paper discusses forced-linear approaches such as HAVOK-inspired methods and the methods of Pan et al. and Khodkar et al., which commonly rely on time-delay embeddings. Time-delay coordinates can produce operators whose connection to the original physical coordinates is difficult to interpret. The authors therefore identify an unresolved issue concerning whether the observed effectiveness comes from the embedding, the forced representation, or their combination, and whether the methods retain their effectiveness without delay coordinates.