Regarding sparse, nonlinear systems and our ability to learn them:

There is hope. Experimental observation is, that in most cases the coupled high dimensional dynamics almost collapses to low dimensional attractors.

The interesting thing about these is: If we apply a measurement function to their state and afterwards reconstruct a representation of their dynamics from the measurement by embedding, we get a faithful representation of the dynamics with respect to certain invariants.

Even better, suitable measurement functions are dense in function space so we can pick one at random and get a suitable one with probability one.

What can be glanced about the dynamics in terms of of these invariants can learned for certain, experience shows that we can usually also predict quite well.

There is a chain of embedding theorems by Takens and Sauer gradually broadening the scope of applicability from deterministic chaos towards stochasticly driven deterministic chaos.

Note embedding here is not what current computer science means by the word.

I spend most of my early adulthood doing theses things, would be cool to see them used once more.

What field of mathematics is this? Can you point me to some keywords/articles?