# The transformative potential of machine learning for experiments in fluid
mechanics

Ricardo Vinuesa1,2∗, Steven L. Brunton3 and Beverley J. McKeon4
1FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm,
Sweden
2Swedish e-Science Research Centre (SeRC), Stockholm, Sweden
3Department of Mechanical Engineering, University of Washington, Seattle, WA,
USA
4Department of Mechanical Engineering, Stanford University, Stanford, CA 94305

###### Abstract

The field of machine learning has rapidly advanced the state of the art in
many fields of science and engineering, including experimental fluid dynamics,
which is one of the original big-data disciplines. This perspective will
highlight several aspects of experimental fluid mechanics that stand to
benefit from progress advances in machine learning, including: 1) augmenting
the fidelity and quality of measurement techniques, 2) improving experimental
design and surrogate digital-twin models and 3) enabling real-time estimation
and control. In each case, we discuss recent success stories and ongoing
challenges, along with caveats and limitations, and outline the potential for
new avenues of ML-augmented and ML-enabled experimental fluid mechanics.

††∗ Corresponding author<EMAIL_ADDRESS>
## Introduction

The current renaissance in machine learning has the potential to revolutionize
many aspects of the human experience, including scientific discovery. The
ability to learn from data is developing synergetically with the generation of
big data in a range of applications, as well as rapid advances in materials
science and additive manufacturing. In this article, we describe some
frontiers impacted or opened by machine learning (ML) applied to
experimentation in fluid mechanics, a discipline at the core of many
applications and recent technological developments in health, transportation,
energy, and the environment.

From da Vinci’s sketches of eddies forming in a pool to present-day
qualitative and quantitative observations of flow through or over complex
geometries, experiments are often either the first means of discovery, or the
only means to reach extreme flow conditions in advance of numerical
development. They may serve to complement, validate or extend theory and
simulation. However, the benefits of observing the “truth”, i.e. the real-
world manifestation of flow phenomena, are usually accompanied by challenges
associated with non-ideal experimental conditions (noise, vibration, thermal
drift, etc.) and/or diagnostic limitations (limited spatial and/or temporal
resolution, sensor accuracy, bias, etc.). For example, the study of turbulence
near walls, of fundamental importance to a range of engineering applications
as well as a foundational physics problem, is hampered by the physical
dimensions of even the smallest (intrusive or non-intrusive) probe as the
Reynolds number increases and the smallest turbulent scales decrease in size.
This particular problem has given rise to a range of empirical correlations,
which may then be used in efforts to differentiate nuanced scaling arguments.
In industrial settings, experiments form critical design and testing steps,
even in the age of advanced computational capabilities. ML has been
increasingly adopted as a core technology in the field [104, 18, 14], with a
focus on computational fluid dynamics (CFD) [109]. In this study we consider
the potential impact of ML on experimental capabilities with a particular
emphasis on the multiscale nature of turbulent flow, which may pose specific
challenges to classical approaches.

We divide the experimental-fluid-mechanics applications into three different
categories, ordered in terms of increasing novelty of the ML-enabled
capability to the field: 1) augmentation of the fidelity of measurement
techniques, for example by replacing empirical corrections with generalizable
schemes learned from data and increasing what can be learned from observation
through super-resolution or noise removal; 2) enhanced modeling, active
learning, and experimental design, including digital twins; and 3)
capabilities potentially enabled by AI/ML such as estimation and control. For
all three categories we identify various machine-learning (ML) methods which
can improve the experimental method.

A summary of the analyzed experimental methods for each of the categories is
presented in Figure 1, together with potential areas where ML can help. We
also include a classification of method difficulty, both in terms of
experimental complexity and the challenges associated with the coupled
implementation of the ML method and the experimental technique. Note that this
classification is illustrative, and it is associated with the examples
provided in the text; it is therefore not meant to be a categorical statement
regarding the merits of any of the experimental or data-driven methods.
Furthermore, the considered data points are taken as a combination of an ML
method applied to a particular experimental technique. For example, if
convolutional neural networks (CNNs) [50] are used for non-intrusive sensing,
the complexity of this ML method is lower than if CNNs are used for sensing
and control; even if the method is the same, its implementation in a more
challenging application requires a more cumbersome algorithmic solution. This
will be discussed in further detail throughout the manuscript.

Figure 1: Summary of ML impact on experimental fluid mechanics. Classification
of all the experimental applications in terms of their main category, namely
(green) measurement techniques, (blue) experimental design and modeling, as
well as (red) estimation and control. Inside each region we show in black
experimental techniques or tasks, whereas the orange bubbles indicate general
improvements brought to those tasks by ML. The boundaries between categories
are designed as arrows, reflecting the possible flows among them. All the
examples are ranked by their ML and experimental complexities, as discussed in
more detail throughout the manuscript. Note that this assessment is not
exhaustive and is meant to illustrate possible applications.

## Improvement of measurement techniques by means of machine learning

Machine learning offers the potential to improve existing measurement
techniques in a range of ways, from learning precise forms of corrections
required due to sensor imperfections and bias, to data imputation, i.e.
providing insight into data that was not measured, for example due to limited
sensor resolution. We begin by describing these examples and the ML approaches
that have been used to augment classical techniques.

One important problem of the experimental studies in fluid mechanics is the
error introduced by the instrumentation employed to perform the measurements.
ML has the potential to learn the underlying transfer function between the
variable under consideration and the observed value, even in measurements with
bias due to practical sensor limitations or noise.

One widely-used sensor to measure velocity profiles is the so-called Pitot
tube, in which the difference between total and static pressures is used to
obtain the flow velocity through the classical Bernoulli equation. These
sensors do not have a sufficiently fast frequency response for measuring the
turbulence fluctuations, but they can provide robust measurements of the mean
flow [8]. However, these sensors require a number of corrections [106], where
McKeon et al. [63] proposed corrective terms for viscous effects, shear, wall
interference and turbulence. While there is a physical motivation behind these
corrections, their functional forms are empirical [110], a fact that hinders
their usage for general purposes. Similar problems are identified when
correcting the position of the probe with respect to the wall [75, 112] and
when it comes to accounting for the filtering effect on the fluctuations
measured by a hot-wire anemometer [5], where empirical approaches are employed
[22, 67]. In the latter case, an important open question which limits the
applicability of the correction schemes to various flow cases revolves around
the lack of universality of the small scales in flows with pressure gradients.
In this sense, being able to exploit existing databases and novel machine-
learning methods to obtain more sophisticated correction strategies has the
potential to improve experimental measurements. Interpretable correction
models may be learned using symbolic-regression strategies such as genetic
programming [47] or the sparse identification of nonlinear dynamical systems
(SINDy) [17]. Alternative approaches, such as the DeepONet framework [55],
make it possible to learn nonlinear correction operators from data. In this
sense, DeepOnet could be used to generate corrected mean and fluctuating
velocity profiles as a function of a number of input parameters, based on the
boundary-layer development and integral quantities.

Non-quantitative measurement techniques provide visual information regarding
the state of the flow, but no numerical information regarding its quantities
of interest. An example of this type of measurement is the so-called smoke-
wire visualization technique [9], which enables obtaining insight into a
number of flow characteristics, such as the transition or separation
locations. One possibility to augment the measurements from smoke-wire
visualizations is to use numerical data of the same flow case to train
computer-vision tools, e.g. convolutional neural networks (CNNs) [50]. With
these databases, it would be possible to establish a mapping between the
visualization (which does not include any quantitative data) and the velocity
and pressure vectors obtained from the simulation. Due to the strong spatial
correlations present in turbulent fluid flows, CNNs can be used to effectively
predict patterns [37], being able to add the quantitative information to the
flow visualizations. In fact, other deep-learning approaches enable to first
enhance the resolution of the visualization [40, 54, 34], and subsequently add
the quantitative information.

Another technique which can benefit from data-driven methods is particle-image
velocimetry (PIV). In this technique, which enables obtaining instantaneous
flow-field measurements, the flow is seeded with tracer particles which follow
the dynamics of the stream. These particles are illuminated with a laser, and
the resulting images are used to compute the instantaneous velocity vector [2,
90, 6]. This measurement technique can be enhanced in a number of ways through
data-driven methods; for instance, Méndez et al. [64] proposed a technique
based on proper-orthogonal decomposition (POD) [56] to remove the noise and
some of the inaccuracies in PIV measurements. Robust principal component
analysis (RPCA) may also be used to remove outliers and fill in occlusions in
PIV data, as demonstrated by Scherl et al. [91]. Multi-layer perceptrons
(MLPs) and CNNs were exploited by Rabault et al. [78] to improve the post-
processing method for PIV measurements and to provide super-resolution [33].
Furthermore, autoencoders (AEs) combined with CNNs were employed by Morimoto
et al. [68] to perform robust flow estimations, which can also be used to
improve the quality of PIV measurements. A schematic representation of the
approach adopted in this work is provided in Figure 2. Other deep
convolutional-neural-network architectures have also been explored for PIV
enhancement [52].

Figure 2: Process of PIV analysis based on ML. a, Representation of the
autoencoder (AE) architecture based on convolutional neural networks (CNNs) to
predict the flow field based on the input image containing the particle
distribution. Yellow denotes convolution and max pooling, which reduce the
size of the image, while green indicates convolution and up sampling, yielding
an increase of the image size before the output. b, Example of application of
this method to the flow around a square cylinder. An artificial particle image
(API) is used as input, and the output of each of the hidden layers is shown.
The size of the image is progressively reduced until the 7th convolutional
layer; beyond this point the size is increased successively until reaching the
predicted velocity field in the output. Note that the first layers exhibit
simpler and smaller features, which are hierarchically combined into larger,
more complex elements when the output layer is approached. Figured adapted
from Ref. [68], with permission from the publisher (American Institute of
Physics).

Another possible approach to enhance PIV measurements is to exploit the
framework of physics-informed neural networks (PINNs) [80, 81], which is based
on using the machinery for training neural networks (including the important
advantages of the automatic differentiation) to solve partial differential
equations (PDEs) through minimization of the PDE residual. This framework has
exhibited great potential when solving certain PDEs, and enables embedding
certain theoretical considerations (such as perturbation theory) to enhance
its results [4]. In particular, PINNs have been shown to effectively improve
measured flow fields with noise [29], and can also increase the resolution and
quality of PIV measurements in turbulence [119]. The vortex-in-cell method
[24], which involves solving the vorticity-transport equation assuming
inviscid flow and incompressibility, has been used to improve the temporal
resolution of PIV measurements, including the resolution of the flow
structures [95]. Predictions of the coherent structures in turbulent flows
have also been conducted via deep learning, and this methodology can improve
the spatio-temporal representation of the coherent structures in PIV
measurements. Note that a complete review of recent applications of machine
learning to enhance PIV measurements can be found in the work by Discetti and
Liu [28].

## Experimental design and modeling enhanced by machine learning

The next broad area of experimental fluid mechanics that stands to benefit
from machine learning includes the design of experiments, developing reduced-
order models, and uncertainty quantification. Many of these approaches
culminate in the digital twin [44, 71, 19], which is a digital model of the
physical asset (i.e., the fluid flow experiment, or a closed-loop cyber-
physical system, etc.). These models are constantly updated with measurement
data and may incorporate a hierarchy of physics-based and machine-learned
models.

When performing wind-tunnel experiments to study canonical flow cases there is
an important aspect to take into account: how similar are the experimental
realization and the canonical definition of the flow under study. The walls in
wind-tunnel test sections introduce additional boundary layers which may
affect the flow, for instance through blockage effects [42]. Furthermore, the
corners in the test section introduce secondary flows of Prandtl’s second
kind, which may also impact the experimental results [77]. Numerical
simulations and empirical models can be used to design inserts in order to
establish the desired flow evolution and pressure-gradient conditions [103];
however, in many cases it is challenging to achieve those conditions, a fact
that can significantly impact the quality of the measurements. In these
situations it is possible to use data-driven methods to optimize the shape of
the inserts so as to achieve a particular pressure-gradient distribution. In
particular, Morita et al. [69] proposed a framework based on Gaussian
processes which enables conducting shape optimization on the inserts of a
wind-tunnel setup. In their work, the framework not only converges within few
iterations to the optimal insert design, but also decides the next case to
explore in the parameter to space to achieve the fastest possible convergence.
Note however that the Gaussian-process approach is a valid option only when a
relatively low number of parameters are to be optimized. For higher-
dimensional cases, gradient-based optimization [72] can be a suitable choice.

It is natural to synthesize experimental measurements into reduced-order
models that may be used to predict, estimate, and control the behavior of both
the fluid flow and the experimental apparatus. There is a wide variety of
data-driven modeling algorithms that may be used to characterize a system,
ranging from the linear dynamic-mode decomposition (DMD) [88, 93, 48, 87],
higher-order DMD [49] the interpretable sparse identification of nonlinear
dynamics (SINDy) [17, 89], genetic programming and symbolic regression [12,
94, 27, 26], and various neural-network-based modeling techniques [115, 76,
25, 21]. Sparse nonlinear modeling techniques have shown particular promise in
identifying interpretable and generalizable models from experimental data [84,
20, 100] and designing closure models [118, 92, 10, 11] that may be informed
by experimental data in the future. Related parsimonious models were recently
developed for an actuated D-shaped body in experiments, capturing all of the
resonance and lock-on phenomena [43].

Another essential aspect of the experimental work in fluid mechanics is
uncertainty quantification (UQ). Being able to add error bars to any results
from an experiment is critical in order to fully interpret and understand the
reported results. In the traditional approach to UQ [8] the method of
propagation of errors is used, where the uncertainty of a particular measured
quantity is essentially the addition of the uncertainties from the individual
quantities used to calculate the first one. More recent studies have proposed
comprehensive UQ frameworks that can capture more nuanced interactions among
the measured quantities [85], providing a more robust assessment of the
uncertainty. Using the example of hot-wire anemometry, Ref. [85] employs
Monte-Carlo (MC) sampling [97], polynomial-chaos expansions (PCE) [117] and
linear-perturbation methods (all providing similar results) to take into
account the correlations among variables when assessing the uncertainty. They
found that, due to error cancellation, these methods provide lower
uncertainties than those reported using propagation of errors. Furthermore,
Sobol indices [97] enable assessing the sensitivity of a particular
measurement with respect to the various intermediate quantities measured to
compute the first one. In the case of hot wires, the wire voltage is the
quantity with highest impact on the results. When it comes to oil-film
interferometry (OFI) [105], which is a technique used for direct measurement
of the wall-shear stress $\tau_{w}$, the quantity with the largest effect on
the measurement of $\tau_{w}$ is in fact the calculation of the so-called
fringe velocity. This is a complicated process [70] requiring extensive image
analysis, thus computer-vision methods such as convolutional neural networks
(CNNs) [50] may help to improve the procedure. Note that advanced UQ methods
are becoming the standard also in the context of computational fluid dynamics
(CFD) [86].

Reduced-order models, informed and updated by data with quantifiable
uncertainty, culminate in the so-called digital twin [44, 71, 19]. Digital
twins are essentially a virtual representation of a physical asset, typically
the result of a hierarchy of models of different fidelity, and constantly
being updated by various data streams from the real world. The digital twin
comprises physical, machine learning, and hybrid models. Such a digital
representation of a complex physical system may enable major advances in
optimization, iterative design, and control. A detailed schematic of the
digital-twin methodology applied to a simple aircraft, including all the
variables and the dynamic-decision network, is provided in Figure 3. Because
digital twins balance models and data, it is important to effectively sample
the system of interest, requiring modern active learning [31] and design of
experiments [60]. Further, a hierarchy of classical physics-based models and
modern data-driven reduced-order models will likely be essential for
characterizing a digital twin.

Figure 3: Schematic representation of a digital twin. a, Digital twin
illustrated in a simple aircraft, where the Observations, the Physical State
and the Control Input are quantities belonging to the Physical System, whereas
the Digital State, the Quantities of Interest and the Reward are associated
with the Digital System. b, Example of application of this framework with a
dynamic-decision network, where the quantities above the dashed line
correspond to the Physical System and the ones below to the Digital System.
Here the complete Physical State (which may not be completely observable)
defines the Observations in the Physical System (e.g. wall-shear stress or
wall pressure). The observations enable computing the Digital State through
numerical models, which in turn can be used to calculate the quantities of
interest and define the overall reward. The framework also allows eventually
evolving the solution in time without experimental measurements. Figure
adapted from Ref. [44], with permission from the publisher (Springer Nature).

## Possibilities within data-driven flow estimation and control

Finally, there are tremendous opportunities for machine learning to advance
the challenging problems of sensing/estimation and control of fluid flows. An
improved ability to model and control fluid flows may be enabler of advanced
technology, and experimental demonstrations are necessary before wide-scale
deployment. However, the field of fluid dynamics poses several challenges for
control [16, 18]. The dynamics are typically high-dimensional, nonlinear, and
multiscale in both space and time. Further, relevant timescales are often
quite fast for modern applications, requiring fast control decisions to reduce
latency. There are also often strong instabilities and time delays between
sensing and actuation. Thus, there are opportunities for better sensor and
actuator placement, improved model-based estimation from limited and noisy
sensors and enhanced closed-loop flow control, all enabled by machine
learning.

Data-driven methods are enabling a number of possibilities within sensing in
experiments. This is an essential ingredient of reactive control, where being
able to accurately assess the state of the flow based on sparse measurements
significantly improves the possibility of performing effective flow control.
Some traditional approaches to sense the flow based on information at the wall
include linear stochastic estimation (LSE) [102, 30] and extended proper-
orthogonal decomposition (EPOD) [13] (which are formally equivalent). Note
that turbulent flows are inherently nonlinear, and although some of the
energy-transfer phenomena are linear (namely, superposition [3]), the
modulation is a nonlinear phenomenon [62] which cannot be accurately
reproduced by linear methods. More recently, non-linear transfer functions
have been used to make such predictions with improved results, and the
potential of computer-vision-based methods such as convolutional neural
networks (CNNs) has been assessed in turbulent channels [37]. CNNs are able to
exploit the spatial information in the input for obtaining improved
predictions, and given the pronounced signature of the turbulence structures
at the wall, these neural networks can successfully predict the flow up over a
wide range of wall-normal locations. Using the wall-shear-stress components
and the wall pressure as inputs, Guastoni et al. [37] predicted the turbulence
fluctuations at $y^{+}=15$ (where $y^{+}$ is the wall-normal coordinate in
inner scaling) with less than $1\%$ error using CNNs, and with less than
$30\%$ at $y^{+}=100$. Some variants of this approach have been proposed in
the literature, including combining CNNs with POD [39], or using deep CNNs for
predictions based on the heat flux [45]. The limited spatial resolution
encountered in experimental settings, which may complicate the usage of CNNs,
was circumvented by Güemes et al. [40] by using generative adversarial
networks (GANs) [36]. In their proposed architecture, super resolution is used
to produce highly-resolved representations of the data at the wall, which are
in turn used to produce accurate predictions of the turbulent fluctuations
above the wall. These sensing approaches, when conducted in real time, can
guide the wall actuation to target certain scales in the flow, with the aim of
achieving drag reduction [1]. Note that, although very promising results have
been obtained from numerical data, sensing in actual experimental applications
is associated with additional challenges, e.g. due to the filtering effect of
the measurement techniques (beyond the reduced resolution), which may lead to
a nontrivial implementation of the methods developed from numerical data.

Another relevant area within sensing in experiments is the enhancement of
measured flow quantities. For instance, numerical data can be used to train
deep neural networks to correlate the flow with the polymeric stresses in non-
Newtonian flows [35]; then, these deep-learning models can be used to obtain,
in an experimental setup, those polymeric stresses based on flow quantities
(which are relatively straightforward to measure). In connection with this,
recent work based on graph neural networks has enabled learning the rheologic
parameters of complex flows through a reduced number of experiments [59].
These approaches may enable much deeper physical insight into the dynamics of
non-Newtonian flows, with very important implications in a wide range of
industrial applications. On the other hand, it is essential to establish
methodologies for optimal sensor placement, particularly in complex
geometries. Given the limited resolution and amount of sensors available in
experimental setups, one can devise strategies for optimal sensing using data-
driven methods. A very effective way to optimize sensor locations in high-
dimensional systems is the so-called QR pivoting [60], which involves reduced-
order-model (ROM) methods such as singular-value decomposition (SVD) [15].
These techniques rely on classical tools from linear algebra, and provide
excellent results in a wide range of applications. However, turbulent flows
are inherently nonlinear, which means that it is convenient to employ methods
involving nonlinearity for sensor placement. In this direction, the tools for
explainability of neural networks can provide valuable insight into optimal
sensor placement. Explainability implies being able to evaluate the parts of
the input data that contribute the most to the prediction of the output [111],
and one popular framework to evaluate this is based on the so-called SHapley
Additive exPlanations (SHAP) values. These values, which rank the features of
the input by their prediction importance, can be calculated by means of the
kernel SHAP method [57]. If a deep neural network is trained to predict flow
quantities based on features measured at the wall, the SHAP framework can
provide a map of importance of all the wall points, effectively leading to a
framework to decide the optimal location of point sensors while leveraging the
nonlinearities of deep neural networks.

Flow control, both in its active and passive forms, can also benefit from the
high-dimensional optimization capabilities of data-driven methods. On the one
hand, it is possible to use machine learning (and exploit the capabilities of
transfer learning [37]) to fine tune the correlations used to define the
geometry of the wall roughness, in order to establish effective passive-
control methods [51], with application to experimental setups. On the other
hand, several approaches have been used to optimize active flow control. For
instance, Mahfoze et al. [58] employed Bayesian optimization to identify the
best combination of control-region length and blowing amplitude to maximize
the energy savings, in an approach which also included intermittent control
regions. Furthermore, these authors also took into account the data by
Kornilov and Boiko [46] to formulate a more realistic estimate of the power
consumption by blowing. Another data-driven control approach which has been
proved successful in controlling external flows in experimental settings is
genetic programming [53, 65]. This technique enables developing novel control
strategies with a certain degree of interpretability in the resulting control
law. When it comes to turbulence control, there is indeed an interesting
question regarding which scales need to be attenuated to achieve drag
reduction. While the traditional opposition control [23] focuses on the near-
wall fluctuations, recent work by Marusic et al. [61] indicates that at higher
Reynolds numbers larger scales need to be controlled.

Reinforcement learning (RL) is a promising approach at the intersection of
machine learning and control [101, 82, 66], where it has been widely applied
to solve several challenges in gaming and general artificial intelligence [96,
83, 114]. Several RL approaches have been explored in the fluid-mechanics
literature [107, 73, 79, 116, 32, 74, 41, 113, 7, 38, 98, 108], to varying
degrees of success. Deep reinforcement learning (DRL), where deep neural
networks are used for some aspect of the RL process, has been particularly
powerful, given sufficient training resources. In this sense, DRL is an
approach that can help to establish novel and effective control strategies
[108], discerning the ranges of scales to be targeted. More recently, DRL has
been shown to provide very promising drag-reduction rates in turbulent channel
flow [38, 98]. In particular, Guastoni et al. [38] reported a larger drag
reduction ($30\%$) using DRL than that obtained via opposition control
($20\%$). A very promising (and challenging) experimental application can be
obtained by combining non-intrusive sensing [37, 40] and DRL-based flow
control [38, 98], as illustrated in Figure 4.

Figure 4: Schematic representation of the process for sensing and control. In
this figure we show a turbulent open channel, where the blue surface
represents the wall and at the upper dashed plane a symmetry condition is
imposed. Based on measurements at the wall (orange panel), which are non
intrusive, it is possible to use computer-vision methods (such as
convolutional neural networks, CNNs, or generative adversarial networks, GANs)
to predict the instantaneous velocity fluctuations above the wall (red panel).
Having this information, a deep-reinforcement-learning (DRL)-based algorithm
determines the optimal control policy to minimize the wall-shear stress. The
arrows represent active flow control by means of blowing and suction. Figure
adapted from Ref. [38].

## Conclusions and outlook

In this perspective review, we have summarized a number of promising avenues
of experimental fluid mechanics that are being transformed by techniques in
machine learning. In particular, we have explored efforts to 1) augment the
fidelity of existing measurement techniques, 2) improve experimental design
and the surrogate modeling of experiments and 3) enable real-time estimation
and control.

Augmenting measurement fidelity is a natural application of machine learning,
where similar advances have been made in image processing. For example, super-
resolution, outlier detection, and data imputation are all existing ML
techniques that have been shown to have immediate benefit in improving fluid
velocity fields from PIV. The addition of known physics, through regularizing
terms and constraints has further improved the ability to infer physically-
consistent flow fields from limited measurements. This area of research is
likely to continue rapid development, as it improves the measurement
capabilities without more costly experimental equipment, by simply leveraging
the wealth of data available from other high-fidelity measurements and
computations. The synthesis of fluid databases into larger _foundation models_
for fluid mechanics will be a crucible for these efforts.

The second area of high-priority research is in improving experimental design
and the construction of digital-twin surrogate models for experiments.
Characterizing a fluid-flow experiment across a range of relevant parameters
(e.g., Reynolds number, wing geometry, etc.), is often important for inverse
design, although it may be prohibitively expensive to sweep through all
parameters. Instead, emerging techniques in active learning are providing a
principled approach to sample these parametric spaces efficiently while
providing error bounds and uncertainty estimates on the resulting models. The
resulting surrogate models may then be used flexibly in iterative design
optimization and downstream estimation and control tasks. The development of
surrogate models is another area that is being dramatically improved with
machine learning techniques. All of these approaches culminate in the _digital
twin_ , which is a hierarchy of models (based on physics, machine learning, or
hybrid) that are pinned to measurement data and come with uncertainty
quantification. Digital-twin pipelines are still being developed and there are
many open questions. However, the tremendous industrial benefit of model-based
engineering is likely to continue driving these efforts for decades.

Finally, we have explored the potential of machine learning to improve real-
time estimation and control in fluid experiments. Estimation and control are
essential tasks, both to characterize and regulate an experiment as well as an
objective of the flow experiment itself (e.g., to minimize drag or maximize
lift). Flow control is often limited by computational latency, and machine-
learning solutions are notoriously fast, balancing accuracy and efficiency.
Although major advances have been made in the field of flow control, several
challenges remain, including future control plants with many sensors and
actuators, resulting in a high-dimensional search space. Further, strong
nonlinearities and time delays make control especially challenging, although
advances in reinforcement learning and ML-enabled model predictive control
(MPC) are promising. It is important to note that when possible, simpler
optimization and control strategies may also be advantageous, for example in
the use of simplex optimization to improve the power efficiency of a lab-scale
cross-flow turbine by over 50% [99].

To summarize the various possible applications of ML to complement and enhance
certain aspects of experimental fluid mechanics, in Table 1 we present a
selection of potentially relevant ML methods. This table is meant to inform an
experimentalist on the type of approach that might be more convenient
depending on their intended problem, so that it becomes easier to select the
most suitable ML algorithm from the vast catalog of possibilities.

Table 1: Summary of sample ML methods to use for various applications within
experimental fluid mechanics. We show in italics experimental techniques or
tasks, and in bold face relevant ML methods.

Beyond these high-profile uses of machine learning in experimental fluid
mechanics, there are several other emerging opportunities and applications.
Cloud connectivity is making it possible to have remotely-controlled,
collaborative fluid-flow facilities, perhaps providing a new generation of
shared experimental resources.

For the foreseeable future, experiments will remain the gold-standard in fluid
mechanics for capturing true multiscale, multiphysics effects for flows with
complex geometries. Increasing measurement fidelity and machine-learning
algorithms will likely make these experiments even more relevant, with the
possibility to more flexibly transfer results from one configuration to
another, which is a standing open problem. With fluids at the center of
several trillion-dollar industries, from health and defense to energy and
transportation, it is expected that these technologies will continue to mature
rapidly, driven by considerable industry investment. The aerospace industry
and biomedical applications have been particularly strong early adaptors, and
it is believed that digital twins with fluid flows will continue to develop in
these fields. Furthermore, the great relevance of fluid mechanics for the
current climate emergency further justifies the adoption of these novel
approaches to accelerate related scientific discovery.

## Acknowledgements

The authors would like to gratefully acknowledge valuable discussions with
Bernd Noack early in the development of this perspective. RV acknowledges
financial support from ERC grant no. ‘2021-CoG-101043998, DEEPCONTROL’. Views
and opinions expressed are however those of the author(s) 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. SLB acknowledges support from the National Science
Foundation AI Institute in Dynamic Systems (grant number 2112085). BJM is
grateful for the support of the U.S. ONR through a Vannevar Bush Faculty
Fellowship, N00014-17-1-3022.

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