Symmetry-reduced model reduction of shift-equivariant systems via operator inference

We consider data-driven reduced-order models of fluid systems with shift equivariance. Shift-equivariant systems typically admit traveling solutions: examples in fluids include channel flows or circular pipe flows exhibiting traveling waves or relative periodic orbits. Our approach builds on previous work in which the solution is represented in a traveling reference frame, in which it can be described by a relatively small number of basis functions. Existing methods for operator inference allow one to approximate a reduced-order model directly from data, without knowledge of the full-order dynamics. Our method adds additional terms to ensure that the reduced-order model not only approximates the spatially frozen profile of the solution, but also estimates the traveling speed as a function of that profile. We have validated our approach using the Kuramoto–Sivashinsky equation, a one-dimensional partial differential equation that exhibits traveling solutions and spatiotemporal chaos. Our method performs nearly as well as Galerkin projection, but unlike Galerkin projection, our method is entirely data driven. As a next step, we are exploring integrating our approach with non-intrusive oblique (Petrov–Galerkin) projections to improve the reconstruction of transients.