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

Andrew Cleary*,† and Jacob Page‡

  • School of Mathematics & Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom

  • *Contact author: andrew.cleary@ed.ac.uk
  • †Present address: Department of Mechanical Engineering, Johns Hopkins University, Baltimore MD 21218, USA.
  • ‡Contact author: jacob.page@ed.ac.uk
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Phys. Rev. E 112, 055105 – Published 12 November, 2025

DOI: https://doi.org/10.1103/cdz2-858n

Abstract

Deep autoencoder neural networks can generate highly accurate, low-order representations of turbulence. We design a family of autoencoders which are a combination of a “dense-block” encoder-decoder structure [Page et al., J. Fluid Mech. 991, A10 (2024)], an "implicit rank minimization" series of linear layers acting on the embeddings [K. Zeng et al., Mach. Learn.: Sci. Technol. 5, 025053 (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≤Re≤400 and used to estimate the dimension of the chaotic attractor, dA(Re). We find that the dimension scales like ∼Re1/3—much weaker than known bounds on the global attractor, which grow like Re4/3. 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.

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Phys. Rev. E 112, 055105– Published 12 November, 2025
  • Received 10 June 2025
  • Revised 18 September 2025
  • Accepted 21 October 2025
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