> Sometimes models that seem nonlinear turns linear if those nonlinearities are pushed into the basis functions, so one can still hope.

That idea was pushed to its limit by the Koopman operator theory. The argument sounds quite good at first, but unfortunately it can’t really work for all cases in its current formulation [1].

[1]: https://arxiv.org/abs/2407.08177

Quite so. Quite so indeed.

We know that under benign conditions and infinite dimensional basis must exist but finding it from finite samples is very non-trivial, we don't know how to do it in the general case.