# Modelling spatiotemporal turbulent dynamics with the convolutional autoencoder echo state network

arXiv: https://arxiv.org/abs/2211.11379 · HTML: https://mlsanity.com/paper/2211.11379/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 61 references that are in the map.
- First posted on arXiv 21 Nov 2022.

## Builds on (most cited first, 13 of 13)

- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems](https://mlsanity.com/paper/2005.06967/) (2020)
- [Automatic-differentiated Physics-Informed Echo State Network (API-ESN)](https://mlsanity.com/paper/2101.00002/) (2020)
- [Real-time thermoacoustic data assimilation](https://mlsanity.com/paper/2106.06409/) (2021)
- [Data-driven prediction and control of extreme events in a chaotic flow](https://mlsanity.com/paper/2204.11682/) (2022)
- [An adjoint-based approach for finding invariant solutions of Navier-Stokes equations](https://mlsanity.com/paper/1508.06363/) (2015)
- [Gradient-free optimization of chaotic acoustics with reservoir computing](https://mlsanity.com/paper/2106.09780/) (2021)
- [Statistical prediction of extreme events from small datasets](https://mlsanity.com/paper/2201.08294/) (2022)

## Built on by (most cited first, 4 of 4)

- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [Convolutional autoencoder for the spatiotemporal latent representation of turbulence](https://mlsanity.com/paper/2301.13728/) (2023)
- [Short and Straight: Geodesics on Differentiable Manifolds](https://mlsanity.com/paper/2305.15228/) (2023)
- [Uncovering solutions from data corrupted by systematic errors: A physics-constrained convolutional neural network approach](https://mlsanity.com/paper/2306.04600/) (2023)

## Abstract (arXiv)

The spatiotemporal dynamics of turbulent flows is chaotic and difficult to predict. This makes the design of accurate and stable reduced-order models challenging. The overarching objective of this paper is to propose a nonlinear decomposition of the turbulent state for a reduced-order representation of the dynamics. We divide the turbulent flow into a spatial problem and a temporal problem. First, we compute the latent space, which is the manifold onto which the turbulent dynamics live (i.e., it is a numerical approximation of the turbulent attractor). The latent space is found by a series of nonlinear filtering operations, which are performed by a convolutional autoencoder (CAE). The CAE provides the decomposition in space. Second, we predict the time evolution of the turbulent state in the latent space, which is performed by an echo state network (ESN). The ESN provides the decomposition in time. Third, by assembling the CAE and the ESN, we obtain an autonomous dynamical system: the convolutional autoncoder echo state network (CAE-ESN). This is the reduced-order model of the turbulent flow. We test the CAE-ESN on a two-dimensional flow. We show that, after training, the CAE-ESN (i) finds a latent-space representation of the turbulent flow that has less than 1% of the degrees of freedom than the physical space; (ii) time-accurately and statistically predicts the flow in both quasiperiodic and turbulent regimes; (iii) is robust for different flow regimes (Reynolds numbers); and (iv) takes less than 1% of computational time to predict the turbulent flow than solving the governing equations. This work opens up new possibilities for nonlinear decompositions and reduced-order modelling of turbulent flows from data.