Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures

Abstract

A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space [Hopf, Commun. Appl. Maths 1 , 303 (1948)]. The chaotic dynamics are shaped by the unstable simple invariant solutions populating the inertial manifold. The hope has been to turn this picture into a predictive framework where the statistics of the flow follow from a weighted sum of the statistics of each simple invariant solution. Two outstanding obstacles have prevented this goal from being achieved: 1) paucity of known solutions and 2) the lack of a rational theory for predicting the required weights. Here, we describe a method to substantially solve these problems, and thereby provide compelling evidence that the probability density functions (PDFs) of a fully developed turbulent flow can be reconstructed with a set of unstable periodic orbits. Our method for finding solutions uses automatic differentiation, with high-quality guesses constructed by minimizing a trajectory-dependent loss function. We use this approach to find hundreds of solutions in turbulent, two-dimensional Kolmogorov flow. Robust statistical predictions are then computed by learning weights after converting a turbulent trajectory into a Markov chain for which the states are individual solutions, and the nearest solution to a given snapshot is determined using a deep convolutional autoencoder. In this study, the PDFs of a spatiotemporally chaotic system have been successfully reproduced with a set of simple invariant states, and we provide a fascinating connection between self-sustaining dynamical processes and the more well-known statistical properties of turbulence.

Mixing artificial and natural intelligence: from statistical mechanics to AI and back to turbulence

“…Another interesting example of this type involves developing a famous hypothesis by Hopf made in Hopf (1948), suggesting that turbulence can be explained in terms of the unstable simple invariant solutions populating the inertial manifold of a high-dimensional space. This idea was further explored in Page et al (2024), where it was shown how to reconstruct PDFs of dissipation rate, production rate, and energy in developed two-dimensional turbulence from a set of unstable periodic orbits. A combination of AI techniques-AD and DNN (of the deep convolutional autoencoder type) -as well as Markov chain approaches were utilized to find hundreds of unstable periodic orbit solutions and to build the PDFs.…”