Learning the Latent dynamics of Fluid flows from High-Fidelity Numerical Simulations using Parsimonious Diffusion Maps

TL;DR

This paper introduces Parsimonious Diffusion Maps (PDMs) to identify low-dimensional latent dynamics in high-fidelity fluid flow simulations, outperforming traditional methods like POD/PCA in capturing the intrinsic behavior across different flow regimes.

Contribution

The paper presents a novel non-linear manifold learning approach, PDMs, that accurately discovers the intrinsic dimensions of fluid flow dynamics from simulations, improving over existing dimensionality reduction techniques.

Findings

PDMs identify 2D dynamics in oscillatory regimes.

PDMs identify 3D dynamics in chaotic regimes.

PDMs outperform POD/PCA in reconstruction accuracy.

Abstract

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.