# A Numerical Proof of Shell Model Turbulence Closure

arXiv: https://arxiv.org/abs/2202.09289 · HTML: https://mlsanity.com/paper/2202.09289/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 20 references that are in the map.
- First posted on arXiv 18 Feb 2022.

## Builds on (most cited first, 2 of 2)

- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [A Perspective on Machine Learning Methods in Turbulence Modelling](https://mlsanity.com/paper/2010.12226/) (2020)

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

- [How neural networks learn to classify chaotic time series](https://mlsanity.com/paper/2306.02300/) (2023)
- [Evaluating Dynamical Fidelity through Predictive Structure in Physical Representations](https://mlsanity.com/paper/2609.34627/) (2026)

## Abstract (arXiv)

The development of turbulence closure models, parametrizing the influence of small non-resolved scales on the dynamics of large resolved ones, is an outstanding theoretical challenge with vast applicative relevance. We present a closure, based on deep recurrent neural networks, that quantitatively reproduces, within statistical errors, Eulerian and Lagrangian structure functions and the intermittent statistics of the energy cascade, including those of subgrid fluxes. To achieve high-order statistical accuracy, and thus a stringent statistical test, we employ shell models of turbulence. Our results encourage the development of similar approaches for 3D Navier-Stokes turbulence.