On the role of the projection fiber for modeling transient nonlinear dynamics

Recent advances enable accurate forecasting of nonlinear dynamical systems on low-dimensional curved manifolds learned from data. The use of curved manifolds has proven critical in applications to high Reynolds number fluid flows where advecting flow structures are poorly approximated in low-dimensional (flat) subspaces. We show that projecting states onto such a manifold yields accurate reconstructions, but can result in poor forecasting performance when the projection fiber fails to properly account for fast dynamics and transient amplification mechanisms associated with non-normality. To illustrate, we consider a non-parallel complex Ginzburg-Landau (CGL) equation in a regime following a supercritical Hopf bifurcation. We project onto the unstable manifold and compare various choices for the projection fiber. We show that accurate models can be obtained by removing the most quickly decaying modes, which are orthogonal to the most slowly decaying left eigenvectors. It is well-known in numerical linear algebra that computing these for high-dimensional systems requires the adjoint, making it challenging to learn appropriate projections directly from data.