AI Insight
This study compares autoencoders with a classical numerical analysis method called Parsimonious Diffusion Maps (PDMs) for creating reduced-order models of fluid flows. Testing on two-dimensional flow past a rotating cylinder, the researchers found that PDMs achieved reconstruction and prediction accuracy comparable to or better than autoencoder-based models, while requiring orders of magnitude less computational time for training. The PDM approach also provides more interpretable latent coordinates and can estimate their dimension directly from data.
Why it matters
This work challenges the dominance of autoencoders in reduced-order modeling by demonstrating that classical manifold-learning methods can achieve similar or better accuracy with dramatically reduced computational costs. The findings could enable faster and more efficient simulations of complex fluid dynamics problems in engineering applications while providing more interpretable results.
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⚠️ Preprint – Noch nicht peer-reviewed
Dieser Artikel wurde noch nicht von unabhängigen Experten begutachtet. Die Ergebnisse sind vorläufig und sollten mit Vorsicht interpretiert werden.
Abstract: Autoencoders (AEs) have become a dominant approach to nonlinear latent-space construction in data-driven reduced-order modelling (ROM), with their decoders lifting latent representations back to the ambient state space. Their prominence, however, has overshadowed an established alternative: manifold-learning methods grounded in classical numerical analysis. We revisit this alternative using Parsimonious Diffusion Maps (PDMs), benchmarking them against Proper Orthogonal Decomposition (POD)-based ROMs and several convolutional AE architectures for the two-dimensional incompressible flow past a rotating cylinder —a bifurcating Navier-Stokes (NS) system organized by a codimension-2 Bogdanov-Takens point and its associated Hopf, saddle-node, and homoclinic bifurcations. Our approach uses PDMs to identify a parsimonious and interpretable set of intrinsic latent coordinates and to estimate their dimension directly from data. Gaussian process regression then learns the latent dynamics, while convex K-nearest-neighbor (K-NN) interpolation in PDMs space constructs the pre-image map, for which we establish pointwise consistency. The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to —and, in some bifurcating regimes, better than— that of AE-based ROMs. At the same time, latent-variable learning with PDMs requires orders of magnitude less computational time than AE training.
Source: Autoencoders vs. Numerical Analysis–Informed Manifold Learning for Navier–Stokes Flows