Neural Network Complexity of Chaos and Turbulence

TL;DR

This paper investigates the complexity of chaos and turbulence by analyzing neural network internal representations and their ability to classify fluid flow images, providing a quantitative measure of their computational complexity.

Contribution

It introduces a novel complexity measure based on neural network features and applies it to distinguish chaotic and turbulent fluid flows.

Findings

Neural networks can differentiate chaos from turbulence using internal feature representations.

The intrinsic dimensionality of features correlates with the complexity of fluid phenomena.

Adversarial examples reveal distinct correlation spectra for chaotic and turbulent flows.

Abstract

Chaos and turbulence are complex physical phenomena, yet a precise definition of the complexity measure that quantifies them is still lacking. In this work we consider the relative complexity of chaos and turbulence from the perspective of deep neural networks. We analyze a set of classification problems, where the network has to distinguish images of fluid profiles in the turbulent regime from other classes of images such as fluid profiles in the chaotic regime, various constructions of noise and real world images. We analyze incompressible as well as weakly compressible fluid flows. We quantify the complexity of the computation performed by the network via the intrinsic dimensionality of the internal feature representations, and calculate the effective number of independent features which the network uses in order to distinguish between classes. In addition to providing a numerical estimate of the complexity of the computation, the measure also characterizes the neural network processing at intermediate and final stages. We construct adversarial examples and use them to identify the two point correlation spectra for the chaotic and turbulent vorticity as the feature used by the network for classification.