Characterizing the Reynolds number dependence of the chaotic attractor in two-dimensional turbulence with dimension-minimizing autoencoders

TL;DR

This work develops symmetry-reduced, dimension-minimizing autoencoders (IRMAE-WD) with dense-block encoders/decoders to learn low-dimensional representations of the chaotic attractor in 2D Kolmogorov turbulence and to quantify how its dimension scales with Reynolds number. The authors demonstrate that the latent space yields an upper-bound estimate $d_{\u266aA} ilde{}\n hsim Re^{1/3}$ for $Re$ in the turbulent regime, significantly weaker than the theoretical global-attractor bound of order $Re^{4/3}$, and reveal interpretable modal structures tied to mean-flow and small-scale vortical features. They also show distinct latent-space clusters corresponding to bursting events, map pre-bursting routes, and identify gaps in coverage by libraries of unstable periodic orbits, especially at higher $Re$. These results provide a data-driven, physically interpretable view of turbulence dimensionality and dynamical pathways, with potential implications for reduced-order modeling and dynamical-systems analyses of 2D turbulence.

Abstract

Deep autoencoder neural networks can generate highly accurate, low-order representations of turbulence. We design a new family of autoencoders which are a combination of a 'dense-block' encoder-decoder structure (Page et al, J. Fluid Mech. 991, 2024), an 'implicit rank minimization' series of linear layers acting on the embeddings (Zeng et al, Mach. Learn. Sci. Tech. 5, 2024) and a full discrete+continuous symmetry reduction. These models are applied to two-dimensional turbulence in Kolmogorov flow for a range of Reynolds numbers $25 \leq Re \leq 400$, and used to estimate the dimension of the chaotic attractor, $d_{\mathcal A}(Re)$. We find that the dimension scales like $\sim Re^{1/3}$ -- much weaker than known bounds on the global attractor which grow like $Re^{4/3}$. In addition, two-dimensional maps of the latent space in our models reveal a rich structure not seen in previous studies, including multiple classes of high-dissipation events at lower $Re$ which guide bursting trajectories. We visualize the embeddings of large numbers of "turbulent" unstable periodic orbits, which the model indicates are distinct (in terms of features) from any flow snapshot in a large turbulent dataset, suggesting their dynamical irrelevance. This is in sharp contrast to their appearance in more traditional low-dimensional projections, in which they appear to lie within the turbulent attractor.

Figures (19)

• Figure 1: The distributions of dissipation rate $D$ at $Re = 100$, normalized by the laminar dissipation rate $D_{lam} = Re / (2n^2)$, is shown for the full training dataset (gray) and the resampled training dataset (blue).
• Figure 2: 16 symmetry charts of $n = 4$ Kolmogorov flow, for each of the 16 discrete symmetric copies of a state. The $x$- and $y$-axes denote $\hat{\omega}_R(0,1)$ and $\hat{\omega}_I(0,1)$ respectively. (Left) $\hat{\omega}_I(0,4) > 0$ and (right) $\hat{\omega}_I(0,4) < 0$.
• Figure 3: Schematic of the data/neural network pipeline used in this study. Symmetries are first reduced for each snapshot $\omega$, before being passed to the IRMAE network. The implicit rank minimizing layers (orange) are linear fully connected layers, while the encoder $\mathcal{E}$ and decoder $\mathcal{D}$ are fully described in Appendix \ref{['app:arch_details']}.
• Figure 4: The average reconstruction error $\varepsilon$ of the model against $Re$. A bottleneck dimension of $d_z = 1024$ is chosen for each $Re$. The error bars at each $Re$ are defined by one standard deviation of the reconstruction error.
• Figure 5: The distribution of the reconstruction error $\varepsilon$ at $Re = 40$ is shown for the best model trained with data augmentation and no symmetry reduction (green), the best model from the first AdamW optimizer stage on symmetry reduced data (red), the best model from the second Adagrad optimizer stage on symmetry reduced data (blue).