Data-driven sensitivity analysis and model reduction of shear dominant flows using structured linear operators

The dynamics of a high-dimensional dissipative system often evolve on or near a low-dimensional invariant manifold, and a successful reduced-order model (ROM) must preserve the input-output properties of the full-order model. However, the challenge lies in computing and effective reduction onto the manifold without truncating dynamically important effects. The balanced proper orthogonal decomposition (BPOD) method and its extension to nonlinear systems (CoBRAS) achieve successful reduction of linear and nonlinear systems using adjoint-based Gramians and gradient covariances. However, the reliance on the adjoint makes these methods impractical when one only has access to a black-box simulation or experimental data. We propose recovering the required sensitivity information by identifying structured linear operator approximations of the underlying system from data. The extraction of the structured linear operator is based on methods such as Physics-informed dynamic mode decomposition (Pi-DMD) and matrix recovery. Then, we examine how well the sensitivity information is captured using forward data, leveraging physical property such as locality including the effect of noise. The proposed methodology is used to develop ROMs for shear dominant fluid flows with increasing complexity, (1) Complex Ginzburg Landau equation (discretized dimension ~10^3), (2) a two-dimensional compressible boundary layer (~10^5).