Perspectives on predicting and controlling turbulent flows through deep learning

The current revolution in the field of machine learning (ML) is leading to many interesting developments in a wide range of areas, including fluid mechanics. Here we review recent and emerging possibilities in the context of predictions, simulations and control of fluid flows, focusing on wall-bounded turbulence. A number of important areas are benefiting from ML, and it is important to identify the synergies with the existing pillars of scientific discovery, i.e. theory, experiments and simulations. It is essential to adopt a balanced approach as a community in order to harness all the positive potential of these novel methods.


Predictions in turbulence
Recent years have witnessed a renewed interest in machine-learning (ML) methods applied to the study of fluid mechanics [7].The availability of massive amounts of data, together with the possibility of employing more powerful graphics-processing-unit (GPU)-based machines and widely available ML libraries [1], are enabling significant progress at a fast pace.Following some applications to chaotic dynamical systems [8], recent studies start to focus on low-dimensional representations of turbulence [61] and even wall-bounded turbulent flows [6].In these studies the focus is on predicting the temporal dynamics of the flow, a task that can be achieved with quite some success with data-driven methods involving long-short-term-memory (LSTM) networks [27] (which are deep-learning architectures capable of exploiting the temporal patterns in the data to perform time-series predictions) and also Koopman-based frameworks with nonlinear forcing [15].Other promising approaches for such temporal predictions include reservoir computing [11] (which has been shown to effectively capture extreme events in the time series) and transformers [75] (which have the potential to perform accurate instantaneous predictions over longer time horizons than other data-driven methods).Here it is important to note that, although after a certain time horizon the various data-driven approaches start to deviate with respect to the original time series, the temporal dynamics of the system is well represented as illustrated via e.g.Poincaré maps and Lyapunov exponents [61].
Besides temporal predictions, spatial predictions in turbulence can significantly benefit from machine learning.The early work by Milano & Koumoutsakos [41] showed the feasibility of using deep neural networks to make meaningful predictions in the near-wall region of turbulent channels.Interestingly, these authors showed that by restricting to linear activation functions one can recover the well-known POD (proper-orthogonal decomposition) modes [38].Another relevant application is non-intrusive sensing, i.e. the predictions of the flow above the wall based on measurements at the wall.In this spirit, Guastoni et al. [25] showed that it is possible to use convolutional neural networks (CNNs) [34] to effectively exploit the spatial correlations in the turbulence data to perform predictions of the velocity fluctuations based on the two wall-shear-stress fields and the wall pressure.More advanced computer-vision architectures, namely generative adversarial networks (GANs) [23], have also been used to obtain robust off-wall predictions based on sparse measurements.This paves the way to obtain methodologies which can be deployed in the context of experiments.Other super-resolution approaches have been proposed by Fukami et al. [20] and by Yousif et al. [74].By exploiting the similarity between the wall-shear stress and the heat flux at the wall, Kim & Lee [29] could leverage the potential of CNNs to predict turbulent heat transfer.Another interesting application of CNNs is the prediction of the flow close to the wall based on information farther away from it [4], an approach that could potentially be used for wall modeling.

Simulations of fluid flows
When it comes to performing computations of fluid flows, ML can help to accelerate direct numerical simulations (DNS), it can improve turbulence modeling and it can help to develop more robust reduced-order models (ROMs) [66].In the first category, the interesting work by Kochkov et al. [31] suggests that it may be possible to accurately perform accurate simulations in coarse meshes.Some more recent work indicates that such an approach to accelerate DNS may also be applied to spectral methods [13].Other approaches to perform turbulence simulations at a reduced cost may be possible thanks to physics-informed neural networks (PINNs) [51,52], for instance by effectively solving the Reynolds-averaged Navier-Stokes (RANS) equations [17].In particular, Eivazi et al. [17] showed that, given adequate boundary conditions (including those of the Reynolds stresses), the incompressible RANS equations can be efficiently solved, with very good results for the Reynolds stresses, via PINNs.This datadriven approach has also been able to predict the flow from sparse measurements, as described by Sitte & Doan [57].An alternative way to perform fluid-flow simulations at a reduced computational cost is to reduce the size of the computational domain through data-driven inflow conditions [21,47,75] and far-field distributions, which can be obtained e.g.via Gaussian-process regression [43].
Turbulence modeling is also benefiting from the capabilities enabled by ML.Large-eddy simulations (LES) can rely on accurate subgrid-scale models (SGS) developed by convolutional neural networks (CNNs) together with the eddy-viscosity assumption [5].In this sense, a promising approach proposed by Novati et al. [45] relies on using reinforcement learning to determine the coefficient in the Smagorinski model [58].This has led to some success in simulations of turbulent channel.Similar ideas, based on reinforcement learning, have been proposed to develop wall models [2]; this is an area with interesting potential applications to high-Reynolds-number wall-bounded turbulent flows.Furthermore, RANS simulations have also been improved via ML, more concretely through the development of accurate and robust RANS models.Some of the earlier deep-learning-enabled RANS studies include those by Ling et al. [35] and Wu et al. [72], and as anticipated by several perspective papers [14,32], these data-driven techniques have expanded their application to turbulence modeling.Another data-driven approach to RANS modeling was proposed by Weatheritt & Sandberg [70,71], who relied on genetic programming to obtain expressions for the Reynolds stresses.The advantage of their work is the fact that they produce interpretable [54,67] models, i.e. it is possible to establish an equation relating the inputs and the outputs.In this spirit, Jiang et al. [28] have proposed a deep-learning-based RANS model which is also interpretable, a very important feature of turbulence simulations.Their method, which is shown schematically in Figure 1, is based on two different neural networks to model the structural and parametric aspects of the turbulent flow.
The last aspect of fluid-flow simulation is the development of ROMs, which can be very helpful in the context of flow control, estimation and optimization.Besides the classical approaches to obtain ROMs, namely properorthogonal decomposition (POD) [38,63] and dynamic-mode decomposition (DMD) [56] (which both rely on linear algebra), deep learning has been recently used for ROM development by leveraging the non-linearity introduced by the activation functions.Murata et al. [44] proposed the usage of CNNs, which are capable of exploiting the spatial features in the data, to produce non-linear modal decompositions of fluid flows.Moreover, Fukami et al. [22] developed an interesting approach for non-linear ROMs based on hierarchical autoencoders.The autoencoder (AE) is a deep-learning model which, through successive application of convolutional filters, can provide a compressed representation of the original data in the latent space.AEs can produce significantly compressed representations of the original data thanks to the employed nonlinearities, but they do not yield modes sorted by energy contribution, and these modes are not orthogonal.These are important properties to provide interpretable and parsimonious ROMs.The hierarchical autoencoder (HAE) [22] relies on the following process: 1.An autoencoder with dimension d = 1 in the latent space is trained, producing one latent vector r r r 1 .
2. Then, another autoencoder with d = 2 is trained such that r r r 1 is fixed, and a new latent vector r r r 2 is obtained.
The contribution of the new vector towards the reconstruction of the original data is smaller than that of the initial one.
3. Subsequently, a new autoencoder with d = 3 is trained fixing r r r 1 and r r r 2 , producing a new latent vector r r r 3 .
4. This process is repeated until the desired size of the latent space is completed.
This method produces a series of non-linear AE modes with progressively smaller contribution towards the reconstruction, and Fukami et al. [22] illustrated its usage in the flow around a two-dimensional cylinder at a Reynolds Figure 1: Schematic representation of the interpretable machine-learning framework proposed by Jiang et al. [28] for RANS modeling, which includes the three following phases: (i) design of the framework based on the domain knowledge, (ii) training strategy and (iii) performance assessment.Reprinted from Ref. [28], with permission of the publisher (AIP Publishing).
number Re = 100 based on cylinder diameter and incoming velocity.The HAE technique was assessed in the turbulent flow between two wall-mounded obstacles [33] by Eivazi et al. [16].While the method works well even in complex turbulent flows, and the modes progressively contribute less towards the reconstruction, their lack of orthogonality affects the interpretability of the ROM.An alternative approach was proposed by Eivazi et al. [16], and it focused on developing a non-linear orthogonal modal decomposition of the flow while also being able to rank the modes by their contribution towards the reconstruction of the turbulent data.Their method relies on β-variational autoencoders (βVAEs), i.e. a modification of the VAE [30], an architecture that is receiving attention in the fluidmechanics community [40].In this approach, stochasticity is introduced in the latent space, and a penalization is added to the loss function with the goal of promoting learning the minimum number of nonzero latent variables which are statistically independent.In this way, it is possible to learn a parsimonious and disentangled latent representation of the original data.The βVAE approach enables recovering almost 90% of the energy from the original data with only 5 modes, whereas the same number of POD modes leads to a recovery of around 30%.Furthermore, the βVAE modes exhibit 99.2% orthogonality, as measured by the cross-correlation matrix, a fact that shows the potential of this method for compact and interpretable ROM development.This point is illustrated in Figure 2, which shows the first 5 modes from the βVAE, POD, HAE and a standard AE based on CNNs.The first interesting observation comes from the HAE and standard AE approaches, which exhibit levels of orthogonality below 90%, where no distinct patterns or physical phenomena can be really noticed.The first POD modes show the presence of large-scale shedding, which is of course expected in this case, and interestingly this is also identified by the βVAE modes.In fact, the βVAE modes also exhibit higher-frequency content in the modes associated with turbulent fluctuations consistent with such shedding, a fact that shows the potential of this approach to significantly compress the data while retaining physical interpretability.Autoencoder-based methods have great potential to develop compact ROMs for turbulent flows, being able to exploit the nonlinear modal reconstruction, as well as potentially interpreting the latent space [3,12,36].Recent work has leveraged transformers [64] to predict the temporal dynamics of the latent space [55,59], effectively producing a ROM to advance the solution in time.

Control of turbulence
Data-driven methods for flow control have been extensively used in the literature, for instance exploiting the linear relations in the flow to control transition to turbulence [18] or forcing at the natural frequency of the shear layer [73] to control separation, a method that can be complented with harmonic resolvent analysis [46].There are however other approaches to flow control based on ML, which may exhibit interesting potential in a number of applications.For instance, Bayesian regression based on Gaussian processes [53] has been used in the context of turbulent-bounday-layer control by Mahfoze et al. [39].Another data-driven control approach which has been proved successful in controlling external flows is genetic programming [42,48], which in principle enables explor-Figure 2: The ranked spatial modes obtained from the various approaches for d = 5.Adapted from Ref. [16], with permission of the publisher (Elsevier).ing various terms in the control law through evolutionary algorithms, and has led to very promising results.One interesting aspect of these approaches is that they allow to assess a larger space of control laws by leveraging the access to large-scale databases, potentially leading to more sophisticated control strategies than the classical ones.
Another very promising approach to discover novel control strategies is deep reinforcement learning (DRL).In this framework, an agent (the neural network) interacts with an environment (the flow simulator) through actions (the control), thus changing its state.The goal of DRL is to decide the set of actions to take, given the state of the system, in order to maximize a certain reward [19].This goal is achieved through iterative interaction with the system by gathering experience on the effect of the actions, a fact that has the potential of producing novel and unexpected control mechanisms in a wide variety of flows.In this sense, one of the first studies to apply DRL for flow control was that of Rabault et al. [49], who obtained significant drag reduction in the flow around a twodimensional (2D) cylinder through active control via jets.In this context, Guastoni et al. [24] have shown that it is possible to use DRL to reduce the length of a 2D separation bubble.One of their results is the documentation of an improved result from DRL compared with classical periodic control, due to the richer range of frequencies discovered by DRL.Other interesting applications include DRL for control in Couette flow, where the policy learned in a ROM is extended to the full domain [37], the work on turbulent channels by Sonoda et al. [60] and the work on three-dimensional cylinders by Suárez et al. [62].Guastoni et al. [26] have recently documented a complete framework based on multi-agent reinforcement learning (MARL), which is illustrated in Figure 3, to perform active flow control in turbulent channels.They document a higher drag reduction through MARL (30%) than that obtained through the classical opposition control (20%) [9].An extensive account of the potential for flow control from DRL, including turbulent wings, can be found in Refs.[65,68].

Outlook
The current revolution in ML, particularly in deep learning, is leading to a number of interesting developments in the context of predicting and controlling fluid flows.It is important to highlight that these methods are not meant to replace any of the current pillars of scientific discovery, i.e. theory, experiments or simulations; rather, these techniques are enabling to solve concrete problems thanks to an improved predictive power from the available data, and therefore ML can be considered as a fourth complementing pillar instead of a competing approach.In Figure 3: Schematic representation of the multi-agent reinforcement learning (MARL) framework proposed by Guastoni et al. [26] to control turbulence in channels.The state is defined in terms of the streamwise and wallnormal fluctuations at the sensing plane, the reward is the reduction of wall-shear stress and the actions are blowing/suction at the wall.Reprinted from Ref. [26], with permission of the publisher (Springer).this sense, while very interesting progress is being observed in chaotic systems and simplified fluid flows, the applications to fully-turbulent flows are still scarce.It is in the application of ML methods to large-scale turbulent cases where the most relevant and impactful applications will be found.However, in order to obtain ML models that can effectively be trained on the large datasets from wall-bounded turbulence (on the order of many terabytes of data), it will be necessary to scale up the current algorithms.This will require to address important computerscience questions having to do with the scaling of the algorithms, and their possibility to run concurrently, in situ, with very large simulations.The current trend in high-performance-computing (HPC) facilities, combining both central-processing and graphics-processing units (CPUs and GPUs respectively), further increases the complexity of applying these methods in the future large-scale flow simulations.Despite the challenges, the possibilities in terms of prediction and control are numerous, including in experimental cases [69].Furthermore, already-trained deep-learning models may help to learn novel physical phenomena through interpretability methods [10,67].
The development of progressively larger databases and models, and the high sensitivity of the results to small hyper parameter choices in deep learning (and more crucially in DRL), make it essential to develop, as a community, better practices in terms of benchmark cases, open-source code development and data sharing.This will enable reproducibility of the results and a quicker (and more robust) advancement of the field of data-driven scientific discovery in fluid mechanics.An important implication of this is the possibility to develop deep-learning models that can generalize to new cases; transfer learning (for example from low to high Reynolds numbers [25]) is critical in this sense, as well as the framework of continual learning [50], which may enable training a deep-learning model which performs equally well in a wide range of tasks.Moreover, novel experimental design aided by ML is another very promising area of research, including data assimilation.
To conclude, there is great potential in ML for fluid mechanics in many areas, and this paradigm can complement the other existing pillars of scientific discovery.In order to properly develop the field and harness all its potential, it is important to maintain the scientific rigor as a community, keeping an informed optimism about the new developments, and avoiding exaggerated claims on the achievements of these techniques.

Funding
RV acknowledges the financial support from the Lundeqvist foundation and the ERC Grant No. "2021-CoG-101043998, DEEPCONTROL".Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union or the European Research Council.Neither the European Union nor the granting authority can be held responsible for them.