First, let me attempt a summary in language that makes sense to me. You want to interpolate a function f:R^n to R^m. So you pick a "random non-linear dynamical system", and compose f with the output of this. To be concrete, let's say you pick a polynomial map F: R^n to R^n of degree at most k for some fixed k with random Gaussian coefficients. Let g(x) = h(1), where h' = F(h) and h(0)=x. Now, given training data (x_i, y_i), let z_i = g(x_i), and interpolate the function f* on the training data (z_i, y_i) using whatever method you like (least squares or a Neural net or whatever). Given test data x, guess f(x) = f*(g(x)).
Is this correct? Why is this plausibly a good idea? Why have the function g be the output of a dynamical system, and not a random matrix? Or if you want g to be nonlinear for some reason (what reason?), the output of a random polynomial or a random Fourier series? What happens if two clusters in the initial data which are well separated and map to different well-separated outputs in the codomain get mixed after composing with the dynamical system map g? Or what if g is not even one-to-one?