# Neural Network Complexity of Chaos and Turbulence

arXiv: https://arxiv.org/abs/2211.15382 · HTML: https://mlsanity.com/paper/2211.15382/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 67 references that are in the map.
- First posted on arXiv 24 Nov 2022.

## Builds on (most cited first, 14 of 14)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Generative Modeling of Turbulence](https://mlsanity.com/paper/2112.02548/) (2021)
- [A Perspective on Machine Learning Methods in Turbulence Modelling](https://mlsanity.com/paper/2010.12226/) (2020)
- [Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database](https://mlsanity.com/paper/2006.09179/) (2020)
- [From Deep to Physics-Informed Learning of Turbulence: Diagnostics](https://mlsanity.com/paper/1810.07785/) (2018)
- [Semi-supervised Learning of Partial Differential Operators and Dynamical Flows](https://mlsanity.com/paper/2207.14366/) (2022)
- [Complexity for deep neural networks and other characteristics of deep feature representations](https://mlsanity.com/paper/2006.04791/) (2020)

## Built on by (most cited first, 3 of 3)

- [Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulations](https://mlsanity.com/paper/2311.00578/) (2023)
- [Turbulence Scaling from Deep Learning Diffusion Generative Models](https://mlsanity.com/paper/2311.06112/) (2023)
- [The Universal Statistical Structure and Scaling Laws of Chaos and Turbulence](https://mlsanity.com/paper/2311.01358/) (2023)

## Abstract (arXiv)

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.