  We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of
high-fidelity Navier-Stokes simulations with a focus on the 2D fluidic pinball
problem. By varying the Reynolds number, different flow regimes emerge, ranging
from steady symmetric flows to quasi-periodic asymmetric and turbulence. We
show, that 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 (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 computed the reconstruction
error, by constructing a decoder using Geometric Harmonics. We show that the
proposed scheme outperforms the POD/PCA over the whole Reynolds 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,
thus relaxing the curse of dimensionality in numerical analysis tasks such as
bifurcation analysis, optimization and control.
