# The Universal Statistical Structure and Scaling Laws of Chaos and Turbulence

arXiv: https://arxiv.org/abs/2311.01358 · HTML: https://mlsanity.com/paper/2311.01358/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 37 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 19 of 19)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [Scaling Laws and Interpretability of Learning from Repeated Data](https://mlsanity.com/paper/2205.10487/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Revisiting Neural Scaling Laws in Language and Vision](https://mlsanity.com/paper/2209.06640/) (2022)
- [A Solvable Model of Neural Scaling Laws](https://mlsanity.com/paper/2210.16859/) (2022)
- [Traditional and Heavy-Tailed Self Regularization in Neural Network Models](https://mlsanity.com/paper/1901.08276/) (2019)
- [A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent](https://mlsanity.com/paper/2006.05013/) (2020)
- [Hessian Eigenspectra of More Realistic Nonlinear Models](https://mlsanity.com/paper/2103.01519/) (2021)
- [A Denoising Diffusion Model for Fluid Field Prediction](https://mlsanity.com/paper/2301.11661/) (2023)
- [Synthetic Lagrangian Turbulence by Generative Diffusion Models](https://mlsanity.com/paper/2307.08529/) (2023)
- [Scaling Laws for Multilingual Neural Machine Translation](https://mlsanity.com/paper/2302.09650/) (2023)
- [Generative Modeling of Turbulence](https://mlsanity.com/paper/2112.02548/) (2021)
- [Scaling Laws Under the Microscope: Predicting Transformer Performance from Small Scale Experiments](https://mlsanity.com/paper/2202.06387/) (2022)
- [Random matrix analysis of deep neural network weight matrices](https://mlsanity.com/paper/2203.14661/) (2022)
- [Multiperiodic Processes: Ergodic Sources with a Sublinear Entropy](https://mlsanity.com/paper/2302.09049/) (2023)
- [The Underlying Scaling Laws and Universal Statistical Structure of Complex Datasets](https://mlsanity.com/paper/2306.14975/) (2023)
- [Physics-Constrained Generative Adversarial Networks for 3D Turbulence](https://mlsanity.com/paper/2212.00217/) (2022)
- [Neural Network Complexity of Chaos and Turbulence](https://mlsanity.com/paper/2211.15382/) (2022)

## Built on by (most cited first, 2 of 2)

- [Classifying Overlapping Gaussian Mixtures in High Dimensions: From Optimal Classifiers to Neural Nets](https://mlsanity.com/paper/2405.18427/) (2024)
- [Generative models on phase space](https://mlsanity.com/paper/2604.02415/) (2026)

## Abstract (arXiv)

Turbulence is a complex spatial and temporal structure created by the strong non-linear dynamics of fluid flows at high Reynolds numbers. Despite being an ubiquitous phenomenon that has been studied for centuries, a full understanding of turbulence remained a formidable challenge. Here, we introduce tools from the fields of quantum chaos and Random Matrix Theory (RMT) and present a detailed analysis of image datasets generated from turbulence simulations of incompressible and compressible fluid flows. Focusing on two observables: the data Gram matrix and the single image distribution, we study both the local and global eigenvalue statistics and compare them to classical chaos, uncorrelated noise and natural images. We show that from the RMT perspective, the turbulence Gram matrices lie in the same universality class as quantum chaotic rather than integrable systems, and the data exhibits power-law scalings in the bulk of its eigenvalues which are vastly different from uncorrelated classical chaos, random data, natural images. Interestingly, we find that the single sample distribution only appears as fully RMT chaotic, but deviates from chaos at larger correlation lengths, as well as exhibiting different scaling properties.