  The accurate simulation of complex dynamics in fluid flows demands a
substantial number of degrees of freedom, i.e. a high-dimensional state space.
Nevertheless, the swift attenuation of small-scale perturbations due to viscous
diffusion permits in principle the representation of these flows using a
significantly reduced dimensionality. Over time, the dynamics of such flows
evolve towards a finite-dimensional invariant manifold. Using only data from
direct numerical simulations, in the present work we identify the manifold and
determine evolution equations for the dynamics on it. We use an advanced
autoencoder framework to automatically estimate the intrinsic dimension of the
manifold and provide an orthogonal coordinate system. Then, we learn the
dynamics by determining an equation on the manifold by using both a function
space approach (approximating the Koopman operator) and a state space approach
(approximating the vector field on the manifold). We apply this method to exact
coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully
resolved simulations for these cases require O(103) and O(105) degrees of
freedom respectively, and we build models with two or three degrees of freedom
that faithfully capture the dynamics of these flows. For these examples, both
the state space and function space time evaluations provide highly accurate
predictions of the long-time dynamics in manifold coordinates.
