Recent evidence shows that neural networks can quantify properties of chaotic systems far better than the current state of the art. This suggests that neural networks can capture certain hidden structures or symmetries better than humans. In this subproject, we aim first at understanding these hidden features in the case of turbulence and complex fluid flows, and, second, at using them to construct efficient and effective flow control systems.

Turbulence is among the chaotic systems with highest impact on our daily lives. It comes with complex statistical properties and multi-scale correlations in space and time, which are far from being thoroughly understood. This comes with strong practical implications including heavy computational costs of fluid flow simulations due to the absence of general and accurate turbulence models and scarce capacity of efficient flow control. In recent work, the capability of machine learning models to perform highly accurate estimates of the Reynolds number of turbulent velocity signals was shown, by employing very short time-series (the typical time-scale of the phenomena). State-of-the-art methods based on statistical analyses would require, on the other hand, orders of magnitude more data to deliver such accuracy. This is a further example of the capability of machine learning to perform more with less data and a source for a new generation of fundamental turbulence studies. On this basis, this sub-project aims at the next fundamental steps, treating the following topics:

- Can we advance our understanding of the turbulence phenomenology combining state-of-the-art statistical tools and reverse engineering of self-discovered machine learning tools? To this aim, we will generalize the concept from a previous work, and connecting with sub-project 1, we will explore the issues of optimal representations (i.e. optimal bases) for turbulent data. Shell models of turbulence will be a central tool. Shell models of turbulence are 0-dimensional models of the turbulence energy cascade. While being computationally orders of magnitude cheaper than, e.g., a Direct Numerical Simulation of the Navier-Stokes equations, they retain scaling and statistical properties stunningly close to those of the Navier-Stokes equations. In combination with vectorized GPU computing, shell models enable to generate and finely tweak turbulence data to extreme statistical volumes. This opens the possibility of machine learning-based investigations at scale, with relatively rapid training cycles (hours, as the data is feature-complex but still in the order of megabytes), and without the need of super-computers.
- Can we improve, through supervised and unsupervised approaches, current models for turbulence increasing fidelity and reducing computational costs? The problem of the representation of the degrees of freedom of turbulent flows directly connects with the issue of modelling turbulence. Shell models have been used recently in this context too, and upper bounds of analytic approaches have been outlined. We expect, thanks to the capability of machine learning models to assimilate complex multiscale correlations, to overcome these limits. We will couple CNN and Recursive neural networks with shell models to accelerate the simulation of the small-time scale dynamics and evaluate the quality of the generated data through well understood turbulence observables as Lagrangian and Eulerian structure functions. This will have potential impact e.g. in large eddy numerical simulations (LES).
- Can we, using reinforcement learning, better understand the fundamental physics of the energy cascade in turbulent flows? It was recently shown, that deep reinforcement learning can effectively find out complex strategies to prevent the onset of convection stabilizing a Rayleigh-BĂ©nard cell into a conductive state. These preliminary results show the potential of RL in modifying turbulence phenomenology, providing an useful tool towards a fundamental understanding of turbulent flows.

For this subproject the PhD student Giulio Ortali has been appointed under the supervision of lead researcher Federico Toschi. Both are located at the Eindhoven University of Technology.