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

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 , ], an "implicit rank minimization" series of linear layers acting on the embeddings [K. Zeng , ], and a full <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>discrete</a:mi> <a:mo>+</a:mo> <a:mi>continuous</a:mi> </a:mrow> </a:math> symmetry reduction. These models are applied to two-dimensional turbulence in Kolmogorov flow for a range of Reynolds numbers <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mn>25</b:mn> <b:mo>≤</b:mo> <b:mtext>Re</b:mtext> <b:mo>≤</b:mo> <b:mn>400</b:mn> </b:mrow> </b:math> and used to estimate the dimension of the chaotic attractor, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:msub> <c:mi>d</c:mi> <c:mi mathvariant="script">A</c:mi> </c:msub> <c:mrow> <c:mo>(</c:mo> <c:mtext>Re</c:mtext> <c:mo>)</c:mo> </c:mrow> </c:mrow> </c:math> . We find that the dimension scales like <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mo>∼</e:mo> <e:msup> <e:mtext>Re</e:mtext> <e:mrow> <e:mn>1</e:mn> <e:mo>/</e:mo> <e:mn>3</e:mn> </e:mrow> </e:msup> </e:mrow> </e:math> —much weaker than known bounds on the attractor, which grow like <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msup> <f:mtext>Re</f:mtext> <f:mrow> <f:mn>4</f:mn> <f:mo>/</f:mo> <f:mn>3</f:mn> </f:mrow> </f:msup> </f:math> . 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.