# Emulating Spatio-Temporal Realizations of Three-Dimensional Isotropic Turbulence via Deep Sequence Learning Models

arXiv: https://arxiv.org/abs/2112.03469 · HTML: https://mlsanity.com/paper/2112.03469/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 28 references that are in the map.
- First posted on arXiv 7 Dec 2021.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Improved Semantic Representations From Tree-Structured Long Short-Term Memory Networks](https://mlsanity.com/paper/1503.00075/) (2015)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [On the Origin of Deep Learning](https://mlsanity.com/paper/1702.07800/) (2017)
- [Restricted Recurrent Neural Networks](https://mlsanity.com/paper/1908.07724/) (2019)
- [Performance of Three Slim Variants of The Long Short-Term Memory (LSTM) Layer](https://mlsanity.com/paper/1901.00525/) (2019)

## Built on by (most cited first, 1 of 1)

- [QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compression](https://mlsanity.com/paper/2211.05151/) (2022)

## Abstract (arXiv)

We use a data-driven approach to model a three-dimensional turbulent flow using cutting-edge Deep Learning techniques. The deep learning framework incorporates physical constraints on the flow, such as preserving incompressibility and global statistical invariants of velocity gradient tensor. The accuracy of the model is assessed using statistical and physics-based metrics. The data set comes from Direct Numerical Simulation of an incompressible, statistically stationary, isotropic turbulent flow in a cubic box. Since the size of the dataset is memory intensive, we first generate a low-dimensional representation of the velocity data, and then pass it to a sequence prediction network that learns the spatial and temporal correlations of the underlying data. The dimensionality reduction is performed via extraction using Vector-Quantized Autoencoder (VQ-AE), which learns the discrete latent variables. For the sequence forecasting, the idea of Transformer architecture from natural language processing is used, and its performance compared against more standard Recurrent Networks (such as Convolutional LSTM). These architectures are designed and trained to perform a sequence to sequence multi-class classification task in which they take an input sequence with a fixed length (k) and predict a sequence with a fixed length (p), representing the future time instants of the flow. Our results for the short-term predictions show that the accuracy of results for both models deteriorates across predicted snapshots due to autoregressive nature of the predictions. Based on our diagnostics tests, the trained Conv-Transformer model outperforms the Conv-LSTM one and can accurately, both quantitatively and qualitatively, retain the large scales and capture well the inertial scales of flow but fails at recovering the small and intermittent fluid motions.