12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

COMPLETE FLOW CHARACTERIZATION FROM SNAPSHOT PIV, FASTPROBES AND PHYSICS-INFORMED NEURAL NETWORKS

Álvaro Moreno Soto, Alejandro Güemes and Stefano Discetti
Aerospace Engineering Research Group
Universidad Carlos III de Madrid
Avda. de la Universidad 30, Leganés 28911, Spainamsoto@ing.uc3m.es ∂u ∂v ∂w
 ∂ x + ∂ y + ∂ z = 0
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 ∂ w + u ∂ w + v ∂ w + w ∂ w = − ∂ p + 1 ∂ 2w + ∂ 2w + ∂ 2w ,ABSTRACTThe use of physics-informed neural networks (PINNs),
which incorporate governing laws to constrain the training ofmachine-learning algorithms, has opened the door to novel applications of artificial intelligence (AI) to model and regularize experimental data. PINNs have been recently shown to
improve the accuracy of time-resolved measurements, but its
capabilities are significantly reduced when that resolution is
not at hand. In this work, we exploit PINNs to enhance velocity measurements from non-time-resolved field measurements,
as those available for example in snapshot Particle Image Velocimetry (PIV). We propose using PINNs as a regularizerof time-resolved estimated fields from simultaneous measurements with fast pointwise probes and non-time-resolved PIV.
We use a multilayer perceptron (MLP) architecture to establish
a correspondence between probe data and the temporal coefficients of the Proper Orthogonal Decomposition of the velocity
profiles. The estimated fields are then fed to the PINNs to enhance the data and increase precision, additionally extracting
derived quantities not available in the raw data, such as thepressure distribution.

∂t

∂x

∂y

∂z

∂z

Re

∂ x2

∂ y2

∂ z2

(1)
where t, x, y, z refer to time and Cartesian coordinates, andRe = UL/ν is Reynolds number based on a characteristic velocity U, length L and kinematic viscosity ν. This set of equations is enforced as a constraint to reconstruct and regularize
fluid velocity fields.
PINNs have been recently used to obtain dense velocity fields from Lagrangian Particle Tracking measurements(Han et al. (2021); Wang et al. (2022)). The implementation of PINNs for data assimilation using sparse particle tracks
requires time-resolved data to accurately compute the time
derivatives in equation 1. This is often difficult to achieve due
to hardware limitations, and when feasible it requires in mostcases expensive equipment. This limits the use of PINNs for
non-time-resolved Particle Image Velocimetry (PIV), often referred as ‘snapshot PIV’.
To overcome the lack of time resolution, several studies
have aimed at predicting flow fields via multi-time delay linear estimators (and more recently non-linear mapping basedon neural networks) using pointwise probes, for which time
resolution is more accessible, in combination with snapshot
PIV. While this approach comes at the expense of increased
complexity of the setup, it has demonstrated being an interesting solution to obtain time-resolved flow field measurements.
Estimators based on Linear Stochastic Estimation (see for instance Tinney et al. (2008) for jet flows), Extended Proper Orthogonal Decomposition (Borée, 2003) and more recently recurrent neural networks (Deng et al., 2019; Jin et al., 2020)
have been proven effective to obtain good estimations of themost energetic flow features. On the downside, flow field features exhibiting low level of correlation with the probe data
were reconstructed with higher error. Furthermore, in mostcases data were mapped onto a lower rank space to improve
the identification of correlated features, thus inevitably introducing truncation error into the reconstructed fields.
In this work, we present a novel architecture including
PINNs as a regularizer for time-resolved flow fields estimatedfrom fast pointwise probes. This approach features several
benefits. First, we enable the use of PINNs for non-timeresolved data by introducing time-resolution through the combination of PIV with probes. Second, PINNs are expected to
improve the accuracy of the reconstructed fields by imposing

INTRODUCTION
Machine-learning (ML) algorithms are recently being embedded in an ever-growing amount of applications due to boththe increase in the computational capacity as well as the speed
at which results can be obtained. Among others applications,
ML has demonstrated being a powerful ally to recover, correct,
regularize or enhance (among other capabilities) experimental data that otherwise would not be accessible due to setup
or hardware limitations. Physics-informed neural networks(PINNs) (Raissi et al. (2019)) are an example of function approximator based on a neural network architecture trained bydirectly enforcing physical constraints. In their original implementation, PINNs were based on fully-connected layers which
incorporate governing laws as loss function and therefore areable to impose physics constraints while reconstructing field
variables from incomplete or limited measurements or data
(Mao et al. (2020); Cai et al. (2021); Yang et al. (2021)).
In the field of fluid mechanics, PINNs have been provento provide promising results regarding fluid flow estimation(Raissi et al. (2020)) using scalar field data. For this case inparticular, the normalized field variables, namely the components of the velocity vector in 3D (u, v, w) and the pressure p,follow the well-known Navier-Stokes (NS) equations, which
can be written in a simplified manner as follows:1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
both in space and time, but lacks precision due to accumulation of errors from several sources, including above all POD
truncation and inaccuracies in the temporal-mode estimation
with the MLP. To compensate for those errors and regularize
the estimated fields, PINNs are used to improve the accuracyof the fluid domain with the regularization being based on thephysical constraint of equation 1. Including boundary conditions, PINNs can also augment the data disclosing additional
quantities, such as the pressure field, which was not available
at first.
The training of the MLP and of the PINN is carried out
separately. For the MLP, a simple cost function based on theminimization of the error in the reconstruction of the temporalmodes is implemented. Regarding the PINN, the loss function
L contains three main contributions:• the residual of the Navier-Stokes equations, LNS , computed on a target refined grid to reduce truncation errors,• the error with respect to the reference PIV velocity dataon the PIV grid, LPIV ,• and, if available, a boundary condition on pressure atany given point or set of points, L p . This boundary condition
could arise from direct measurement of the pressure in pointsof the domain (not necessarily the boundaries) and / or fromreasonable assumptions on the boundaries of the domain.
The residual of the NS equations is computed by the meansquared error (mse) of each equation independently (continuity and two momentum equations for a 2D configuration suchas the case presented in this article), i.e. LNS = mse(0, e1 ) +mse(0, e2 ) + mse(0, e3 ), where 0 would be the reference residual value if NS were to be perfectly obeyed and e1 , e2 and e3
the residuals of NS from the predicted model, correspondingly.
Similarly, LPIV and L p are also calculated via mse between
the predicted quantities by the model and the reference PIV
and pressure values, respectively. It must be remarked that the
PIV data might be also affected by uncertainty, i.e. in principle, even in case of perfect assimilation and exact estimationof the velocity field, the loss function would not be zero.
In previous studies the loss function is computed as thesum of the three losses. In this work, we decided to opt
for an adaptative-weighted error function, i.e. the weightof each error contribution is tuned during the training process. More specifically, the error with the higher value isadapted with the higher weight following a direct proportionality rule, i.e. L = wNS LNS + wPIV LPIV + w p L p , wherew• = L• /(LNS + LPIV + L p ). Therefore, the total loss may
2 +L2 +
be written in a simplified manner as L = (LNSPIV
L p2 )/(LNS + LPIV + L p ). This approach is justified by theneed to compensate between two main counteractive effects
during training: on the first hand, enforcing the compliance of
NS tends to homogenize the system, whereas the input of reference values imposes that the solution is in accordance withexperimentally-accessible data (which, as discussed above, are
not necessarily accurate). In the authors’ experience, in thefirst part of the training process the effort is more dedicatedto match the estimation with the experimental data (i.e. minimizing LPIV ). As LPIV is reduced, LNS gains progressively
relative importance and the physical constraint regularizes thedata.

physical constraints. Third, with adequate boundary conditions PINNs can augment the data by providing access to additional quantities, such as the pressure field.
In the Methodology section, the flow chart of the algorithm is detailed. In the Validation and Results section, thetest case is described and the main results are outlined, respectively. Finally, the conclusions are drawn.

METHODOLOGY
The proposed algorithm is sketched in figure 1. We assume snapshot PIV data are available on a regular grid. In thesketch the input data are shown for simplicity in a planar domain, although a fully rigorous implementation would require
3D velocity measurements.
Firstly, a Proper Orthogonal Decomposition (POD) of thevelocity fields is carried out to identify a low-rank space toapproximate the velocity data. This operation is carried out
through the snapshot POD implementation (Sirovich (1987)),
corresponding to the economy-size Singular Value Decomposition:
U = ΨΣΦT(2)

In the above equation, Ψ and Φ are square unitary matrices containing respectively the temporal and spatial modes,and Σ is a diagonal matrix whose elements are the singularvalues.
To obtain time-resolved measurements, we need to collect data with time resolution from point probes. Practicallimitations of setup complexity limit the maximum number ofprobes (see for instance Discetti et al. (2019), using 5 probes
for time-resolved flow estimation in a high-Reynolds-number
pipe flow). In order to increase the available information toreconstruct each snapshot, in convection-dominated flow the
temporal information can be directly embedded to generate
virtual probes. This multi-time delay approach has been usedby several authors (Tinney et al., 2008; Hosseini et al., 2015;
Discetti et al., 2018) and is adopted also here to enhance theprobe information to reconstruct fields.
The probe data are then fed as input to a multilayer perceptron (MLP), with the output being the time coefficients ofthe most energetic temporal POD modes of the velocity fields.
Once the model is trained on a set of probe and PIV data captured simultaneously, it can encode the temporal modes of thevelocity field at any other time given the information provided
by the probes. Consequently, we are able to reconstruct an approximate time-resolved Ψ̃T R via the input of other subsequentvelocity data points at any given time, recovering from nontime-resolved PIV information a time-resolved approximationof the temporal modes of the target velocity field. The reason
for using POD as a field encoder resides in its simplicity of embedding high-dimensional information (3D PIV data can easily provide 100k vectors per snapshot) in a lower rank space,which simplifies the learning process and eases the convergence. It must be remarked that the naive approach used here
to embed the time information can be overcome by using, forinstance, long short-term memory neural networks (Hochreiter & Schmidhuber (1997); Kalchbrenner et al. (2015); Greffet al. (2017)). Here, the use of MLP is motivated by its simplicity of implementation and its integration with the PINNs
architecture, although in future works recurrent architectures
may be considered.
Once this approximate time-resolved matrix Ψ̃T R is obtained, the full velocity field may be recovered on the samegrid as the original PIV via U˜T R = Ψ̃T R ΣPIV ΦTPIV . The approximated time-resolved velocity field U˜T R is now resolved

VALIDATION DATASET AND SETTINGS
To validate our proposed architecture, we make use ofsynthetic data calculated via Direct Numerical Simulation(DNS) of a fluid flow surrounding a pinball configuration(Deng et al., 2020), i.e. three cylinders of diameter D locatedat the vertexes of an equilateral triangle with lateral distancel = 3/2D. The simulation consists of 20, 000 snapshots (time212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Time-resolved

Prediction (time-resolved)

UF P
Time-resolved

Non time-resolved

Fastprobe data
Multi-time delay
MLP

UP IV = ΨP IV ΣP IV ΦTP IV
POD

Reconstruction

Ψ̃T R

ŨT R = Ψ̃T R ΣP IV ΦTP IV

Regularization
PINN

ΨP IV

Experiments (PIV)

Figure 1.

Sketch of the proposed architecture applied to the wake of a cylinder (for illustration purposes).

interval of 0.1 simulation units) of a fluid flow at Re = 130 (i.e.
in the so-called chaotic regime) with x and y axis ranging from[−5D, 15D] and [−5D, 5D], respectively.
To compute time-resolved pointwise fast probe information, five virtual fast probes are located at the downstream edgeof the domain (x ≈ 15D), equally spaced along the vertical direction. This replicates a realistic configuration in which the
intrusiveness of point probes should be minimized.
Synthetic PIV data are obtained by randomly seeding thevelocity fields with particles and adding random noise up to
10%. The particle seeding density is set to 0.02 particles per
pixel. The domain is discretized in 1024 × 512 pixels, thus
corresponding to 51.2 pixels/D. A reasonable approximationof a PIV processing can be obtained by considering a moving
average of the velocity sampled by the particles. The synthetic
PIV interrogation window is set to 32 × 32 pixels, with a 50%
overlap.
The point probes are recording velocity and pressure information at full time resolution, i.e. the data corresponding
to the probe location is saved for the 20, 000 DNS snapshots.
Non-time resolved PIV snapshots are generated by considering
a time separation of 200 DNS time steps. This interval corresponds to one flow throughtime, i.e. a fluid parcel entering thedomain takes in average 200 snapshots to leave it. This originates from the fact that the convective dimensionless velocity
corresponds to 1D per simulation unit, the length of the fieldof view is 20D and the interval between snapshots is 0.1 simulation units. A total of 100 PIV snapshots is thus generated.
The synthetic PIV fields are analyzed with POD to extract thetemporal modes ΨPIV . In order to reduce the dimensionalityof the problem, a rank truncation is performed using the elbow
method (see e.g. Marutho et al. (2018)). For this test case, theidentified rank is R = 12. The main advantage of using POD as
an encoder is the dimensionality reduction of the output, being
it fixed at R coefficients corresponding to the most energetic
modes.
The temporal mode estimator, i.e. the MLP, is fed byprobe data corresponding to the 200 time instants past eachconsidered PIV snapshot. As detailed in the Methodology
section, in convection-dominated flows this correspond to a
multi-time delay approach. The selected timespan correspondto one convective flow throughtime, as suggested by Discetti
et al. (2018). The MLP architecture consists of a set offully-connected dense layers of decreasing number of neurons.
More specifically, the number of neurons decreases from an
initial amount of 3 × N probes × 200, where N probes = 5 corresponds to the number of fast probes installed in the domain and
200 matches the number of time instants between each PIV
snapshot, to end up in a final dense layer with dimensions thenumber of temporal modes to estimate, i.e. R = 12. We opted
for an Adam optimizer with an initial learning rate of 10−3
which is subsequently decreased parallel to the decrease of the
training loss to allow for the optimal reconstruction based on
empirical tests. This pyramidal structure allows for a fast andsteady convergence given the significant reduction of the order of magnitude between the dimensionality of the input ascompared to the output variables.
Once the temporal resolution of the probes has been embedded into the velocity fields with the MLP, we implement a
PINN to enhance the accuracy by adding physical constraints.
Our model for PINN consists on a fully-connected neural network with 10 hidden layers, each of them containing 100 nodes
for each output variable, i.e. for a 2D domain, we have 300
nodes corresponding to the two components of velocity andpressure. The same optimizer as for the MLP is implemented
here. We use for training of the PINN the outcome of the MLP
on the test dataset. This is representative of the case of direct
use of PINNs from probe data, i.e. once the flow estimator has
been trained. Clearly this same process could have been carried out by considering the training data, although in this casehigher accuracy is expected for in-sample snapshots than fortime instants in between two PIV snapshots. In addition, to allow for a full reconstruction and extraction of the pressure field(note again that this is an additional feature, not available in thefirst place from PIV measurements), we use as boundary condition the pressure values given at the position of the probes,
as if they were able to measure velocity and pressure simultaneously. We highlight that this boundary condition could
have been exchanged by any other pressure measurement at
any point within the domain.

RESULTS
Figure 2 shows the contour representation of the streamwise and crosswise components of the velocity field for an
instantaneous snapshot with its corresponding pressure field.
We include for the clarity of the comparison its associated PIV
fields, even though this information is not included in the training dataset. The synthetic PIV field is in this case projected on
the POD basis computed from the training dataset, and truncated to include the same number of modes R estimated by
the MLP. The flow field estimated with the MLP and the regularized field after applying the PINN are also included. The
pressure field of the MLP is obtained from integration of thepressure gradient computed with the momentum equation. A
robust iterative procedure is carried out, following the implementation by Chen et al. (2022). Even though the MLP isalready capable of achieving a good reconstruction of the velocity fields, significant discrepancies with respect to the refer-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2. Reference and reconstructed fields after each stage of the proposed architecture on the test dataset: (first row) DNS reference
velocity and pressure fields, (second row) reference PIV velocity field truncated to the best rank R (bear in mind that this row only exists
for reference purposes and actually no PIV information is available once the model is trained), (third row) velocity field estimation after
the MLP stage receiving fast probe measurements as input and reconstructing from the predicted temporal modes (note that pressure
has been computed directly by solving NS equations and therefore, carry the errors from the estimation using MLP), and (fourth row)
final regularized velocity and pressure fields after PINN.

error is included for each snapshot of the testing dataset and thespatiotemporally-averaged squared error of the same dataset ispresented in table 1. A clear reduction of the error with respect
to the estimation with the MLP is observed. Most remarkably, PINNs are able to smooth out snapshots with large errors,
which would have inevitably compromised the computation of
temporal derivatives. This is a clear advantage, consideringthat the estimation of derived quantities often relies on the estimation of time derivatives from temporally-resolved data (seefor instance pressure estimation (van Oudheusden, 2013)).

ence DNS arise when computing pressure. The main sourcesof error arise from the finite spatial resolution of the PIV fields
used for training (thus losing resolution in presence of large velocity gradients, especially in the near field), from truncation
to rank R of the POD coefficients, and from temporal inconsistencies due to the finite accuracy of the estimation process. On
the other hand, the PINN is able to achieve a more accuratepressure field estimation by imposing physical constraints in
the reconstruction process.
The training error with respect to each of the contributionsof the loss function is depicted in figure 3. The decrease of thetotal loss function indicates that the reconstructed fields from
the MLP are being continuously regularized by enforcing compliance with the Navier-Stokes equations with the PINN. However, the figure shows that a plateau is reached after less than
100 epochs and more iterations would not result in an improvement in accuracy. In our experience, this plateau results froma compromise between the tendency of the Navier-Stokes regularization to homogenize the field whereas the estimated fielddata from the MLP and the pressure boundary condition push
the network to comply with experimentally-accessible data. In
the end, the overall error after this regularization is reduced
significantly, as apparent in figure 2. A quantitative analysis isalso reported in figure 4, where the spatially-averaged squared

The PINN also provides additional information of thefluid flow, such as the pressure field. We observe in figure
2 that the pressure field is in qualitative excellent agreementwith the pressure field of the original simulation, with mostof the discrepancies occurring in the region between the cylinders and in the near field. Figure 4 shows indeed that a major
improvement is achieved as compared to a direct calculationof the pressure distribution from the fields estimated with the
MLP. It must be remarked, however, that the PINN implementation allows to impose directly compliance with measured
pressure in more than one point (in this case, in the 5 probe
locations). For the pressure derivation with the velocity fields
estimated by the MLP, instead, we have simply imposed thevalue of the pressure in one node for the necessary boundary

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Spatiotemporally-averaged squared error of thecomplete test dataset between the reference DNS and the estimation after the MLP and the final regularization via PINNs,
correspondingly.

Featuremse(DNS, MLP)

mse(DNS, PINN)

u

0.01740.0079v

0.02270.0097p0.04320.0045dominated flows, and increase the accuracy by embedding
physical constraints with a physics-informed neural network.
The proposed architecture leverages the capability of a MLP
to estimate temporal modes of POD and the output regularization enforced by the PINN. POD allows for the identificationof a compact low-dimensional embedding of the most relevant
features of the flow fields, thus permitting a fast and effective training with an MLP fed by probes. The input information is also enhanced using a multi-time delay approach,where temporal information collected by probes is enforced as
spatial information on grounds of the validity of Taylor’s hypothesis. This approach has been shown to work efficiently inconvection-dominated flows with estimation through extended
POD, and it is now enhanced with non-linear mapping througha MLP.
Additionally, the proposed concept allows to use PINNs
for snapshot PIV data, i.e. in absence of temporal resolution,embedding in the process also equations including temporal
derivatives (such as the momentum equations). This enables
extracting pressure fields by including proper boundary conditions in the training process. The obtained results show a significant accuracy enhancement of the predicted velocity fields
from MLP after the PINN regularization, and a reasonably
good accuracy also for pressure field estimations. Pressure
estimation could have also being achieved directly from estimated fields, although the accuracy of this process suffers due
to truncation errors of the POD basis and inaccuracies of the
mapping from probe to field data, among other error sources.
Our results show that PINNs with the complete set ofNavier-Stokes equations can be used in conjunction with probe
data and a MLP estimator to obtain velocity fields from probe
data with a reasonably good accuracy. It must be remarked,
however, that the validation test case is based on a simulation at relatively small Reynolds number of a wake flow. In
this configuration, POD is able to provide a compact basis,thus easing the task of the estimation of the velocity fields.
This concept is to be tested on flows with a richer spectrumof scales, in which the amount of modes to be reconstructed
might pose a significant challenge.

Figure 3. Loss function evolution during the different epochsof the PINN training process. Each contribution, namely,
residuals of the Navier-Stokes equations and errors with respect to velocity and pressure reference data, are indicated.
The continuous decrease of the loss function guarantees theconvergence to a more accurate solution than the one resulting
after the MLP.

Figure 4. Mean square error (mse) of the velocity fields
with respect to the DNS for the reconstruction after the MLP(dashed blue - u - and red - v - curves) and the regularization with PINNs (solid blue and red curves) over one throughtime of the domain (corresponding to 200 snapshots at DNS
time resolution). The green curves show the mse of the pressure with respect to DNS before (dashed, calculated directlyfrom integration of the pressure gradient computed imposing
the momentum equation with the velocity fields estimated via
MLP) and after (solid) the regularization via PINNs.

ACKNOWLEDGMENTS

condition for the integration of the pressure gradient.
The error distribution between MLP and PINN with respect to the DNS for an instantaneous snapshot is illustrated infigure 5, showing an excellent improvement of the accuracy of
all the reconstructed fields.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program (grant agreement No
949085) and by MCIN / AEI / 10.13039/501100011033 andthe European Union ‘NextGenerationEU/PRTR’ as part of the
grant FJC2020-044342-I. The authors warmly acknowledge
N. Deng, B. Noack, M. Morzynski and L. Pastur for providing the code for the fluidic pinball simulation.

CONCLUSIONS
We have proposed a solution to enable the estimation offlow fields from a small number of sensors in convection-

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 5. Error with respect to the DNS of each feature of the flow field after (first row) MLP reconstruction on the PIV grid and(second row) PINN regularization on a fine target grid (same as DNS). The improvement after the PINN is significant as compared tothe output of the MLP, more specifically regarding the pressure field calculated by direct application of NS to the estimated velocity
fields.

REFERENCES

reconstruction of flow field around a circular cylinder byrecurrent neural networks based on non-time-resolved particle image velocimetry measurements. Exp. Fluids 61 (114),
1–23.
Kalchbrenner, N., Danihelka, I. & Graves, A. 2015 Grid longshort term memory. arXiv .
Mao, Z., Jagtap, A. D. & Karniadakis, G. E. 2020 Physicsinformed neural networks for high-speed flows. Comput.
Methods Appl. Mech. Eng. 360 (112789), 1–26.
Marutho, D., Handaka, S. H., Wijaya, E. & Muljono 2018
The determination of cluster number at k-mean using elbow
method and purity evaluation on headline news. In 2018 International Seminar on Application for Technology of Information and Communication, pp. 533–538.
van Oudheusden, B. W. 2013 PIV-based pressure measurement. Meas. Sci. Technol. 24 (032001), 1–32.
Raissi, M., Perdikaris, P. & Karniadakis, G. E. 2019 Physicsinformed neural networks: a deep learning framework forsolving forward and inverse problems involving nonlinear
partial differential equations. J. Comput. Phys. 378, 686–
707.
Raissi, M., Yazdani, A. & Karniadakis, G. E. 2020 Hidden
fluid mechanics: learning velocity and pressure fields fromflow visualizations. Science 367 (6481), 1026–1030.
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. II. Symmetries and transformations. Quart.
Appl. Math. 45 (3), 573–582.
Tinney, C. E., Ukeiley, L. S. & Glauser, M. N. 2008 Lowdimensional characteristics of a transonic jet. Part 2. Estimate and far-field prediction. J. Fluid Mech. 625, 53–92.
Wang, H., Liu, Y. & Wang, S. 2022 Dense velocity reconstruction from particle image velocimetry/particle tracking velocimetry using a physics-informed neural network. Phys.
Fluids 34 (017116), 1–15.
Yang, L., Meng, X. & Karniadakis, G. E. 2021 B-PINNs:
bayesian physics-informed neural networks for forward andinverse PDE problems with noisy data. J. Comput. Phys.
425 (109913), 1–23.

Borée, J. 2003 Extended proper orthogonal decomposition: atool to analyse correlated events in turbulent flows. Exp.
Fluids 35, 188–192.
Cai, S., Wang, Z., Wang, S., Perdikaris, P. & Karniadakis,
G. E. 2021 Physics-informed neural networks for heat transfer problems. J. Heat Transfer 143(6) (060801), 1–15.
Chen, J., Raiola, M. & Discetti, S. 2022 Pressure from datadriven-estimated velocity fields using snapshot PIV and fastprobes. Exp. Therm. Fluid Sci. 136 (110647), 1–13.
Deng, N., Noack, B. R., Morzyński, M. & Pastur, L. R. 2020
Low-order model for successive bifurcations of the fluidic
pinball. J. Fluid Mech. 884 (A37), 1–41.
Deng, Z., Chen, Y., Liu, Y. & Kim, K. C. 2019 Time-resolved
turbulent velocity field reconstruction using a long shortterm memory (LSTM)-based artificial intelligence framework. Phys. Fluids 31 (075108), 1–12.
Discetti, S., Bellani, G., Örlü, R., Serpieri, J., Sanmiguel Vila,
C., Raiola, M., Zheng, X., Mascotelli, L., Talamelli, A.
& Ianiro, A. 2019 Characterization of very-large-scale motions in high-Re pipe flows. Exp. Therm. Fluid Sci. 104,
1–8.
Discetti, S., Raiola, M. & Ianiro, A. 2018 Estimation of timeresolved turbulent fields through correlation of non-timeresolved field measurements and time-resolved point measurements. Exp. Therm. Fluid Sci. 93, 119–130.
Greff, K., Srivastava, R. K., Koutnı́k, J., Steunebrink, B. R. &Schmidhuber, J. 2017 LSTM: a search space odyssey. IEEE
Trans. Neural Networks Learn. Syst. 28 (10), 2222–2232.
Han, J., Kim, D., Shin, H. & Kim, K. C. 2021 Enhanced data
assimilation of 4D LPT with physics informed neural networks. In 14th International Symposium on Particle Image
Velocimetry - ISPIV 2021.
Hochreiter, S. & Schmidhuber, J. 1997 Long short-term memory. Neural Comput. 9 (8), 1735–1780.
Hosseini, Z., Martinuzzi, R. J. & Noack, B. R. 2015 Sensorbased estimation of the velocity in the wake of a low-aspectratio pyramid. Exp. Fluids 56 (13), 1–16.
Jin, X., Laima, S., Chen, W.-L. & Li, H. 2020 Time-resolved

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