Comparing and Computing Invariant Manifolds Used in the Dimensionality Reduction of Dissipative Flows

Dissipative flows naturally give rise to attractors in phase space. Reduced-order models are often constructed by identifying low-dimensional invariant manifolds that contain these attractors. In this work, we compare four types of invariant manifolds commonly used for dimensionality reduction: slow manifolds, spectral submanifolds, inertial manifolds, and normally attracting invariant manifolds. To enable concrete comparisons, we compute these manifolds using established techniques and evaluate them based on the projection error of trajectories over time. One method used for computing slow manifolds involves minimizing higher-order time derivatives of the state. While this approach is formally justified for fast-slow systems, it can also be applied to systems without an explicit time-scale separation. Importantly, this technique does not require trajectory data, enabling fast manifold learning. We apply these methods to physically relevant systems, including the Kuramoto–Sivashinsky and complex Ginzburg–Landau equations, and compare the resulting manifolds.