Leveraging Tangent Manifolds for Efficient Time-Dependent Basis Reduced Order Models

In this study, a novel technique is proposed for time-dependent basis reduced order modeling (TDB-ROM) of problems involving coherent structures that evolve over time. The method leverages the spatial and temporal locality of these coherent structures by utilizing a matrix differential equation to derive efficient TDB-ROMs. The approach begins by locally approximating the full-model solution in time using a reduced space based on tangent manifolds. This locally reduced tangent manifold is then updated at each time step with basis updates obtained from querying the full model at a small number of selected spatial coordinates. The TDB approach using tangent manifolds has demonstrated its effectiveness on various benchmark problems, achieving significant runtime speedups compared to full models and traditional ROMs. The use of tangent manifolds allows the reduced basis to accurately capture the local dynamical behavior of the coherent structures. Further work will explore the sensitivity of the approach to different manifold learning techniques for capturing the intrinsic local geometry.