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Automating turbulence modelling by multi-agent reinforcement learning

A Publisher Correction to this article was published on 15 January 2021

A Publisher Correction to this article was published on 07 January 2021

This article has been updated


Turbulent flow models are critical for applications such as aircraft design, weather forecasting and climate prediction. Existing models are largely based on physical insight and engineering intuition. More recently, machine learning has been contributing to this endeavour with promising results. However, all efforts have focused on supervised learning, which is difficult to generalize beyond training data. Here we introduce multi-agent reinforcement learning as an automated discovery tool of turbulence models. We demonstrate the potential of this approach on large-eddy simulations of isotropic turbulence, using the recovery of statistical properties of direct numerical simulations as a reward. The closure model is a control policy enacted by cooperating agents, which detect critical spatio-temporal patterns in the flow field to estimate the unresolved subgrid-scale physics. Results obtained with multi-agent reinforcement learning algorithms based on experience replay compare favourably with established modelling approaches. Moreover, we show that the learned turbulence models generalize across grid sizes and flow conditions.

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Fig. 1: Schematic of the integration of MARL with the flow solver.
Fig. 2: Schematic description of the training procedure implemented with the smarties library.
Fig. 3: Statistical properties of DNS simulations of forced isotropic turbulence.
Fig. 4: Visualizations of simulations of isotropic turbulence.
Fig. 5: Comparison of SGS models for values of Reλ that were not included during the MARL training.
Fig. 6: Measurements of the reliability of the MARL model beyond the training objective.

Data availability

All the data analysed in this paper were produced with open-source software described in the code availability statement. Reference data and the scripts used to produce the data figures, as well as instructions to launch the reinforcement learning training and evaluate trained policies, are available on a GitHub repository (

Code availability

Both direct numerical simulations and large-eddy simulations were performed with the flow solver CubismUP 3D ( The data-driven SGS models were trained with the reinforcement learning library smarties ( The coupling between the two codes is also available through GitHub (

Change history

  • 07 January 2021

    A Correction to this paper has been published:

  • 15 January 2021

    A Correction to this paper has been published: <ExternalRef><RefSource></RefSource><RefTarget Address="10.1038/s42256-021-00295-1" TargetType="DOI"/></ExternalRef>.


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We are very grateful to H. J. Bae (Harvard University) and A. Leonard (Caltech) for insightful feedback on the manuscript, and to J. Canton and M. Boden (ETH Zürich) for valuable discussions throughout the course of this work. We acknowledge support by the European Research Council Advanced Investigator Award 341117. Computational resources were provided by the Swiss National Supercomputing Centre (CSCS) Project s929.

Author information




G.N. and P.K. designed the research. G.N. and H.L.L. wrote the simulation software. G.N., H.L.L. and P.K. carried out the research. G.N. and P.K. wrote the paper.

Corresponding author

Correspondence to Petros Koumoutsakos.

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The authors declare no competing interests.

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Peer review information Nature Machine Intelligence thanks Elie Hachem, Jonathan Freund and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Novati, G., de Laroussilhe, H.L. & Koumoutsakos, P. Automating turbulence modelling by multi-agent reinforcement learning. Nat Mach Intell 3, 87–96 (2021).

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