Real-Time Reduced Order Modeling of Nonlinear Sensitivities in Evolutionary Systems

We present a framework for computing nonlinear sensitivities using a model-driven low-rank approximation. To this end, we use the model's nonlinear perturbation equation (NLPE) to derive evolution equations for a low-rank orthonormal spatial basis, low-rank correlation matrix, and low-rank orthonormal parametric basis. While solving the full-rank system scales linearly with the number of perturbations, this reduced framework directly solves for the intrinsic low-dimensional manifold by leveraging correlations between perturbations on the fly. Therefore, the cost of solving the low-rank approximation scales linearly with the intrinsic rank, which allows for efficient computation of the NLPE for finite perturbations in a high-dimensional parametric space. Furthermore, in contrast to linear sensitivity equations, the NLPE remains stable for chaotic systems. We will present a case study for chaotic and non-chaotic flow with finite perturbations in the 2D compressible Navier-Stokes equations.