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.

翻译：混沌与湍流是复杂的物理现象，但目前仍缺乏精确的复杂度度量来对其进行量化。本研究从深度神经网络的角度探讨混沌与湍流的相对复杂度。我们分析了一组分类问题，其中网络需要将湍流状态下的流体剖面图像与其他类别图像（如混沌状态下的流体剖面、各类噪声构造及真实世界图像）进行区分。我们研究了不可压缩及弱可压缩流体流动。通过内部特征表征的内在维度量化网络执行计算的复杂度，并计算网络用于区分类别的有效独立特征数量。该度量在提供计算复杂度数值估计的同时，还刻画了神经网络在中间及最终阶段的处理过程。我们构建对抗性样本，并利用其识别混沌与湍流涡量的两点相关谱，作为网络分类所使用的特征。