Nonlinear model reduction for fluid flows

Model Reduction is an essential tool that has been applied in many control applications such as control of fluid flows. Most model reduction algorithms assume linear models and fail when applied to nonlinear high dimensional systems, in particular, fluid flow problems with high Reynolds numbers. For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. However, snapshots generated by nonlinear partial differential equations (PDEs) belong to manifolds for which the geodesies do not correspond in general to the Euclidean distance. A geodesic is a curve that is locally the shortest path between points. In this paper, we propose a model reduction method which generalizes POD to nonlinear manifolds which have a differentiable structure at each of their points. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers´ equation subject to boundary control is presented and compared to the POD reduced models.