Learning the latent dynamics of fluid flows from high-fidelity numerical simulations using parsimonious diffusion maps

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier-Stokes simulations with a focus on the two-dimensional (2D) fluidic pinball problem. By varying the Reynolds number Re, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and chaos. The proposed non-linear manifold learning scheme identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition/principal component analysis (POD/PCA), most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also compute the reconstruction error, by constructing a decoder using geometric harmonics (GHs). We show that the proposed scheme outperforms the POD/PCA over the whole Re number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization, and control.