Data-driven prediction and control of extreme events in turbulent flows  

An extreme event is a sudden and violent change in the state of a nonlinear system. In fluid dynamics, extreme events can have adverse effects on the system's optimal design and operability, which calls for accurate methods for their prediction and control. In this paper, we propose a data-driven methodology for the prediction and control of extreme events in a chaotic shear flow.

The approach is based on echo state networks, which are a type of reservoir computing that learn temporal correlations within a time-dependent dataset. The objective is five-fold. First, we exploit ad-hoc metrics from binary classification to analyse

  1. how many of the extreme events predicted by the network actually occur in the test set (precision)
  2. how many extreme events are missed by the network (recall).

We apply a principled strategy for optimal hyperparameter selection, which is key to the networks' performance.

Second, we focus on the time-accurate prediction of extreme events. We show that echo state networks are able to predict extreme events well beyond the predictability time, i.e., up to more than five Lyapunov times.

Third, we focus on the long-term prediction of extreme events from a statistical point of view. By training the networks with datasets that contain non-converged statistics, we show that the networks are able to learn and extrapolate the flow's long-term statistics. In other words, the networks are able to extrapolate in time from relatively short time series.

Fourth, we design a simple and effective control strategy to prevent extreme events from occurring. The control strategy decreases the occurrence of extreme events up to one order of magnitude with respect to the uncontrolled system. Finally, we analyse the robustness of the results for a range of Reynolds numbers. We show that the networks perform well across a wide range of regimes. 

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Graph neural networks for unsteady Eulerian fluid dynamics

The simulation of fluid dynamics, typically by numerically solving partial differential equations, is an essential tool in many areas of science and engineering. However, the high computational cost can limit application in practice and may prohibit exploring large parameter spaces. Recent deep-learning approaches have demonstrated the potential to yield surrogate models for the simulation of fluid dynamics.

While such models exhibit lower accuracy in comparison, their low runtime makes them appealing for design-space exploration. We introduce two novel graph neural network (GNN) models, multi-scale (MuS)-GNN and rotation-equivariant (RE) MuS-GNN, for extrapolating the time evolution of the fluid flow.

In both models, previous states are processed through multiple coarsening of the graph, which enables faster information propagation through the network and improves the capture and forecast of the system state, particularly in problems encompassing phenomena spanning a range of length scales. Additionally, REMuS-GNN is architecturally equivariant to rotations, which allows the network to learn the underlying physics more efficiently, leading to improved accuracy and generalization.

We analyze these models using two canonical fluid models: advection and incompressible fluid dynamics. Our results show that the proposed GNN models can generalize from uniform advection fields to high-gradient fields on complex domains. The multi-scale graph architecture allows for inference of incompressible Navier–Stokes solutions, within a range of Reynolds numbers and design parameters, more effectively than a baseline single-scale GNN.

Simulations obtained with MuS-GNN and REMuS-GNN are between two and four orders of magnitude faster than the numerical solutions on which they were trained. 

 For further details

  • Lino, M., Fotiadis, S., Bharath, A. A., & Cantwell, C. D. (2022). Multi-scale rotation-equivariant graph neural networks for unsteady Eulerian fluid dynamics. Physics of Fluids, 34(8), 087110. 

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Flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings

The usage of neural networks (NNs) for flow reconstruction (FR) tasks from a limited number of sensors is attracting strong research interest owing to NNs’ ability to replicate high-dimensional relationships. Trained on a single flow case for a given Reynolds number or over a reduced range of Reynolds numbers, these models are unfortunately not able to handle flows around different objects without re-training.

We propose a new framework called Spatial Multi-Geometry FR (SMGFR) task, capable of reconstructing fluid flows around different two-dimensional objects without re-training, mapping the computational domain as an annulus. Different NNs for different sensor setups (where information about the flow is collected) are trained with high-fidelity simulation data for a Reynolds number equal to 300 for 64 objects randomly generated using Bezier curves. The performance of the models and sensor setups is then assessed for the flow around 16 unseen objects.

It is shown that our mapping approach improves percentage errors by up to 15% in SMGFR when compared to a more conventional approach where the models are trained on a Cartesian grid and achieves errors under 3%, 10%, and 30% for predictions of pressure, velocity, and vorticity fields, respectively. Finally, SMGFR is extended to predictions of snapshots in the future, introducing the Spatiotemporal MGFR (STMGFR) task. A novel approach is developed for STMGFR involving splitting deep neural networks into a spatial and a temporal component.

We demonstrate that this approach is able to reproduce, in time and in space, the main features of flows around arbitrary objects. 

For further details

  • Özbay, A. G., & Laizet, S. (2022). Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings. AIP Advances, 12(4), 045126. 

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