Exact Coherent States in Minimal Flow Unit Pipe Flow

Turbulent pipe flow is governed by the Navier–Stokes equations, which admit a wide range of interacting spatial and temporal scales. Yet, despite this complexity, the dissipative nature of the system constrains the long-tiime dynamics to a low-dimensional invariant manifold. In this work, we construct reduced-order models of pressure-driven turbulence in a circular pipe in manifold coordinates. We focus on a minimal computational domain—the smallest axial and azimuthal extent capable of sustaining turbulence—at Reynolds number 2500, without imposing any symmetries. Our approach combines dimension reduction using autoencoders to uncover intrinsic manifold coordinates, and neural ordinary differential equations (neural ODEs) to capture the underlying dynamics. This framework allows us to represent the turbulent attractor with only a few degrees of freedom while preserving key physical features of the flow. Crucially, the reduced models provide good initial conditions that converge in the DNS, leading to the discovery of new exact coherent states, including the longest relative periodic orbit ever reported in this configuration.