  The simulation of turbulent flow requires many degrees of freedom to resolve
all the relevant times and length scales. However, due to the dissipative
nature of the Navier-Stokes equations, the long-term dynamics are expected to
lie on a finite-dimensional invariant manifold with fewer degrees of freedom.
In this study, we build low-dimensional data-driven models of pressure-driven
flow through a circular pipe. We impose the `shift-and-reflect' symmetry to
study the system in a minimal computational cell (e.g., smallest domain size
that sustains turbulence) at a Reynolds number of 2500. We build these models
by using autoencoders to parametrize the manifold coordinates and neural ODEs
to describe their time evolution. Direct numerical simulations (DNS) typically
require on the order of O(105) degrees of freedom, while our data-driven
framework enables the construction of models with fewer than 20 degrees of
freedom. Remarkably, these reduced order models effectively capture crucial
features of the flow, including the streak breakdown. In short-time tracking,
these models accurately track the true trajectory for one Lyapunov time, while
at long-times, they successfully capture key aspects of the dynamics such as
Reynolds stresses and energy balance. Additionally, we report a library of
exact coherent states (ECS) found in the DNS with the aid of these
low-dimensional models. This approach leads to the discovery of seventeen
previously unknown solutions within the turbulent pipe flow system, notably
featuring relative periodic orbits characterized by the longest reported
periods for such flow conditions.
