I have developed a physics-assimilated convolutional autoencoder (CAE) neural network, namely, PhyAENet, to carry out nonlinear mode decomposition of the unsteady flow field around a National Advisory Committee for Aeronautics 4412 airfoil near stall conditions. The flow field snapshots are mapped into latent modes by the encoder part of the well-trained CAE, which are used for dynamic mode decomposition (DMD) analysis. The computed DMD modes are split into modes covering different frequency ranges. These high and low-frequency DMD modes are used to form reconstructed encoded sequences, which are then mapped back to generate the nonlinear decomposed spatiotemporal modes by the decoder of the CAE. As such, physics is assimilated into the neural network by incorporating the frequencies of the DMD modes into the latent modes in the latent space. The proposed PhyAENet is capable of extracting the dominant features of the flow fields, accounting for the nonlinearity of the underlying dynamics. Furthermore, the extracted nonlinear modes are evolving with time and physically interpretable. It is revealed that the nonlinear modes can be well represented when using more DMD modes for reconstruction of the encoded sequences. The energy spectrum of the nonlinear modes are obtained by ranking the Frobenius norm of the mode vector.

1.
J. L.
Lumley
, “
The structure of inhomogeneous turbulent flows
,” in
Atmospheric Turbulence and Radio Wave Propagation
(
Nauka
,
1967
), pp.
166
178
.
2.
G.
Berkooz
,
P.
Holmes
, and
J. L.
Lumley
, “
The proper orthogonal decomposition in the analysis of turbulent flows
,”
Annu. Rev. Fluid Mech.
25
(
1
),
539
575
(
1993
).
3.
O. T.
Schmidt
and
T.
Colonius
, “
Guide to spectral proper orthogonal decomposition
,”
AIAA J.
58
(
3
),
1023
1033
(
2020
).
4.
P. J.
Schmid
, “
Dynamic mode decomposition of numerical and experimental data
,”
J. Fluid Mech.
656
,
5
28
(
2010
).
5.
P. J.
Schmid
, “
Dynamic mode decomposition and its variants
,”
Annu. Rev. Fluid Mech.
54
,
225
254
(
2022
).
6.
J. N.
Kutz
,
S. L.
Brunton
,
B. W.
Brunton
, and
J. L.
Proctor
,
Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems
(
SIAM
,
2016
).
7.
M.
Sieber
,
C. O.
Paschereit
, and
K.
Oberleithner
, “
Spectral proper orthogonal decomposition
,”
J. Fluid Mech.
792
,
798
828
(
2016
).
8.
A.
Towne
,
O. T.
Schmidt
, and
T.
Colonius
, “
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
,”
J. Fluid Mech.
847
,
821
867
(
2018
).
9.
S.
Nidhan
,
O. T.
Schmidt
, and
S.
Sarkar
, “
Analysis of coherence in turbulent stratified wakes using spectral proper orthogonal decomposition
,”
J. Fluid Mech.
934
,
A12
(
2022
).
10.
M. O.
Williams
,
I. G.
Kevrekidis
, and
C. W.
Rowley
, “
A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition
,”
J. Nonlinear Sci.
25
,
1307
1346
(
2015
).
11.
Q.
Li
,
F.
Dietrich
,
E. M.
Bollt
, and
I. G.
Kevrekidis
, “
Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator
,”
Chaos
27
(
10
),
103111
(
2017
).
12.
K.
Duraisamy
,
G.
Iaccarino
, and
H.
Xiao
, “
Turbulence modeling in the age of data
,”
Annu. Rev. Fluid Mech.
51
,
357
377
(
2019
).
13.
M.
Raissi
,
A.
Yazdani
, and
G. E.
Karniadakis
, “
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
,”
Science
367
(
6481
),
1026
1030
(
2020
).
14.
S. L.
Brunton
,
B. R.
Noack
, and
P.
Koumoutsakos
, “
Machine learning for fluid mechanics
,”
Annu. Rev. Fluid Mech.
52
,
477
508
(
2020
).
15.
K.
Duraisamy
, “
Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence
,”
Phys. Rev. Fluids
6
(
5
),
050504
(
2021
).
16.
M. Z.
Yousif
,
L.
Yu
, and
H.
Lim
, “
Physics-guided deep learning for generating turbulent inflow conditions
,”
J. Fluid Mech.
936
,
A21
(
2022
).
17.
D.
Xu
,
J.
Wang
,
C.
Yu
, and
S.
Chen
, “
Artificial-neural-network-based nonlinear algebraic models for large-eddy simulation of compressible wall-bounded turbulence
,”
J. Fluid Mech.
960
,
A4
(
2023
).
18.
J.
Huang
,
H.
Liu
, and
W.
Cai
, “
Online in situ prediction of 3-D flame evolution from its history 2-D projections via deep learning
,”
J. Fluid Mech.
875
,
R2
(
2019
).
19.
M.
Raissi
,
P.
Perdikaris
, and
G. E.
Karniadakis
, “
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
,”
J. Comput. Phys.
378
,
686
707
(
2019
).
20.
G. E.
Karniadakis
,
I. G.
Kevrekidis
,
L.
Lu
,
P.
Perdikaris
,
S.
Wang
, and
L.
Yang
, “
Physics-informed machine learning
,”
Nat. Rev. Phys.
3
(
6
),
422
440
(
2021
).
21.
L.
Lu
,
P.
Jin
,
G.
Pang
,
Z.
Zhang
, and
G. E.
Karniadakis
, “
Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
,”
Nat. Mach. Intell.
3
(
3
),
218
229
(
2021
).
22.
T.
Murata
,
K.
Fukami
, and
K.
Fukagata
, “
Nonlinear mode decomposition with convolutional neural networks for fluid dynamics
,”
J. Fluid Mech.
882
,
A13
(
2020
).
23.
J.
Qu
,
W.
Cai
, and
Y.
Zhao
, “
Deep learning method for identifying the minimal representations and nonlinear mode decomposition of fluid flows
,”
Phys. Fluids
33
(
10
),
103607
(
2021
).
24.
H.
Eivazi
,
H.
Veisi
,
M. H.
Naderi
, and
V.
Esfahanian
, “
Deep neural networks for nonlinear model order reduction of unsteady flows
,”
Phys. Fluids
32
(
10
),
105104
(
2020
).
25.
H.
Csala
,
S. T.
Dawson
, and
A.
Arzani
, “
Comparing different nonlinear dimensionality reduction techniques for data-driven unsteady fluid flow modeling
,”
Phys. Fluids
34
(
11
),
117119
(
2022
).
26.
I. U.
Haq
,
T.
Iwata
, and
Y.
Kawahara
, “
Dynamic mode decomposition via convolutional autoencoders for dynamics modeling in videos
,”
Comput. Vision Image Understanding
216
,
103355
(
2022
).
27.
J.
Grosek
and
J. N.
Kutz
, “
Dynamic mode decomposition for real-time background/foreground separation in video
,” arXiv:1404.7592 (
2014
).
28.
B.
Zhang
, “
Airfoil-based convolutional autoencoder and long short-term memory neural network for predicting coherent structures evolution around an airfoil
,”
Comput. Fluids
258
,
105883
(
2023
).
29.
D.
Coles
and
A. J.
Wadcock
, “
Flying-hot-wire study of flow past an NACA 4412 airfoil at maximum lift
,”
AIAA J.
17
(
4
),
321
329
(
1979
).
30.
C. W.
Rowley
,
I.
Mezić
,
S.
Bagheri
,
P.
Schlatter
, and
D. S.
Henningson
, “
Spectral analysis of nonlinear flows
,”
J. Fluid Mech.
641
,
115
127
(
2009
).
31.
D. P.
Kingma
and
J.
Ba
, “
Adam: A method for stochastic optimization
,” arXiv:1412.6980 (
2014
).
32.
P. J.
Baddoo
,
B.
Herrmann
,
B. J.
McKeon
,
J.
Nathan Kutz
, and
S. L.
Brunton
, “
Physics-informed dynamic mode decomposition
,”
Proc. R. Soc. A
479
(
2271
),
20220576
(
2023
).
33.
S.
Kuchibhatla
and
D.
Ranjan
, “
Effect of initial conditions on Rayleigh–Taylor mixing: Modal interaction
,”
Phys. Scr.
T155
,
014057
(
2013
).
34.
S. C.
Kuchibhatla
, “
On the effect of initial conditions on Rayleigh-Taylor mixing
,” Ph.D. thesis (
Texas A&M University
,
2014
).
35.
I.
Scherl
,
B.
Strom
,
J. K.
Shang
,
O.
Williams
,
B. L.
Polagye
, and
S. L.
Brunton
, “
Robust principal component analysis for modal decomposition of corrupt fluid flows
,”
Phys. Rev. Fluids
5
(
5
),
054401
(
2020
).
You do not currently have access to this content.