# Echo State Network for two-dimensional turbulent moist Rayleigh-B'enard convection

arXiv: https://arxiv.org/abs/2101.11325 · HTML: https://mlsanity.com/paper/2101.11325/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 38 references that are in the map.
- First posted on arXiv 27 Jan 2021.

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

- [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)

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

- [Can neural networks predict dynamics they have never seen?](https://mlsanity.com/paper/2111.06783/) (2021)
- [Generative convective parametrization of dry atmospheric boundary layer](https://mlsanity.com/paper/2307.14857/) (2023)
- [Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning](https://mlsanity.com/paper/2505.24099/) (2025)
- [HFNO: an interpretable data-driven decomposition strategy for turbulent flows](https://mlsanity.com/paper/2511.01535/) (2025)

## Abstract (arXiv)

Recurrent neural networks are machine learning algorithms which are suited well to predict time series. Echo state networks are one specific implementation of such neural networks that can describe the evolution of dynamical systems by supervised machine learning without solving the underlying nonlinear mathematical equations. In this work, we apply an echo state network to approximate the evolution of two-dimensional moist Rayleigh-B'enard convection and the resulting low-order turbulence statistics. We conduct long-term direct numerical simulations in order to obtain training and test data for the algorithm. Both sets are pre-processed by a Proper Orthogonal Decomposition (POD) using the snapshot method to reduce the amount of data. The training data comprise long time series of the first 150 most energetic POD coefficients. The reservoir is subsequently fed by the data and results in predictions of future flow states. The predictions are thoroughly validated by the data of the original simulation. Our results show good agreement of the low-order statistics. This incorporates also derived statistical moments such as the cloud cover close to the top of the convection layer and the flux of liquid water across the domain. We conclude that our model is capable of learning complex dynamics which is introduced here by the tight interaction of turbulence with the nonlinear thermodynamics of phase changes between vapor and liquid water. Our work opens new ways for the dynamic parametrization of subgrid-scale transport in larger-scale circulation models.