Abstract

We present an unsupervised deep learning method to perform flow denoising and super-resolution without high-resolution labels. We demonstrate the ability of a single model to reconstruct three-dimensional stenosis and aneurysm flows, with varying geometries, orientations, and boundary conditions. Ground truth data was generated using computational fluid dynamics, and then corrupted with multiplicative Gaussian noise. Auto-encoders were used to compress the representations of the flow domain geometry and the (possibly noisy and low-resolution) flow field. These representations were used to condition a physics-informed neural network. A physics-based loss was implemented to train the model to recover lost information from the noisy input by transforming the flow to a solution of the Navier–Stokes equations. Our experiments achieved mean squared errors in the true flow reconstruction of O(1.0 × 10−4), and root mean squared residuals of O(1.0 × 10−2) for the momentum and continuity equations. Our method yielded correlation coefficients of 0.971 for the hidden pressure field and 0.82 for the derived wall shear stress field. By performing point-wise predictions of the flow, the model was able to robustly denoise and super-resolve the field to 20× the input resolution.

References

1.
Updegrove
,
A.
,
Wilson
,
N. M.
,
Merkow
,
J.
,
Lan
,
H.
,
Marsden
,
A. L.
, and
Shadden
,
S. C.
,
2016
, “
SimVascular: An Open Source Pipeline for Cardiovascular Simulation
,”
Ann. Biomed. Eng.
,
45
(
3
), pp.
525
541
.10.1007/s10439-016-1762-8
2.
Izzo
,
R.
,
Steinman
,
D.
,
Manini
,
S.
, and
Antiga
,
L.
,
2018
, “
The Vascular Modeling Toolkit: A Python Library for the Analysis of Tubular Structures in Medical Images
,”
J. Open Source Software
,
3
(
25
), p.
745
.10.21105/joss.00745
3.
Markl
,
M.
,
Frydrychowicz
,
A.
,
Kozerke
,
S.
,
Hope
,
M.
, and
Wieben
,
O.
,
2012
, “
4D Flow MRI
,”
J. Magn. Reson. Imaging
,
36
(
5
), pp.
1015
1036
.10.1002/jmri.23632
4.
Xiao
,
H.
,
Wu
,
J.-L.
,
Wang
,
J.-X.
,
Sun
,
R.
, and
Roy
,
C.
,
2016
, “
Quantifying and Reducing Model-Form Uncertainties in Reynolds-Averaged Navier–Stokes Simulations: A Data-Driven, Physics-Informed Bayesian Approach
,”
J. Comput. Phys.
,
324
, pp.
115
136
.10.1016/j.jcp.2016.07.038
5.
Sun
,
L.
,
Gao
,
H.
,
Pan
,
S.
, and
Wang
,
J.-X.
,
2019
, “
Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data
,”
Comput. Methods Appl. Mech. Eng.
,
361
, p.
112732
.10.1016/j.cma.2019.112732
6.
Brunton
,
S. L.
,
Noack
,
B. R.
, and
Koumoutsakos
,
P.
,
2020
, “
Machine Learning for Fluid Mechanics
,”
Annu. Rev. Fluid Mech.
,
52
(
1
), pp.
477
508
.10.1146/annurev-fluid-010719-060214
7.
Bakhshinejad
,
A.
,
Baghaie
,
A.
,
Vali
,
A.
,
Saloner
,
D.
,
Rayz
,
V. L.
, and
D'Souza
,
R. M.
,
2017
, “
Merging Computational Fluid Dynamics and 4D Flow MRI Using Proper Orthogonal Decomposition and Ridge Regression
,”
J. Biomech.
,
58
, pp.
162
173
.10.1016/j.jbiomech.2017.05.004
8.
Fathi
,
M. F.
,
Bakhshinejad
,
A.
,
Baghaie
,
A.
,
Saloner
,
D.
,
Sacho
,
R. H.
,
Rayz
,
V. L.
, and
D'Souza
,
R. M.
,
2018
, “
Denoising and Spatial Resolution Enhancement of 4D Flow MRI Using Proper Orthogonal Decomposition and Lasso Regularization
,”
Comput. Med. Imaging Graph.
,
70
, pp.
165
172
.10.1016/j.compmedimag.2018.07.003
9.
Töger
,
J.
,
Zahr
,
M. J.
,
Aristokleous
,
N.
,
Bloch
,
K. M.
,
Carlsson
,
M.
, and
Persson
,
P.-O.
,
2020
, “
Blood Flow Imaging by Optimal Matching of Computational Fluid Dynamics to 4D-Flow Data
,”
Magn. Resonance Med.
,
84
(
4
), pp.
2231
2245
.10.1002/mrm.28269
10.
De Avila Belbute-Peres
,
F.
,
Economon
,
T.
, and
Kolter
,
Z.
, III
, and
2020
, “
Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction
,”
Proceedings of the 37th International Conference on Machine Learning
, July 12–18,
H. D. A.
Singh
, ed., Vol.
119
of Proceedings of Machine Learning Research, PMLR, pp.
2402
2411
.https://proceedings.mlr.press/v119/de-avila-belbute-peres20a.html
11.
Cai
,
S.
,
Mao
,
Z.
,
Wang
,
Z.
,
Yin
,
M.
, and
Karniadakis
,
G. E.
,
2021
, “
Physics-Informed Neural Networks (PINNs) for Fluid Mechanics: A Review
,”
Acta Mech. Sin.
,
37
(
12
), pp.
1727
1738
.https://link.springer.com/article/10.1007/s10409-021-01148-1
12.
Wang
,
R.
,
2021
, “
Physics-Guided Deep Learning for Dynamical Systems: A Survey
,”
arXiv:2107.01272
.10.48550/arXiv.2107.01272
13.
Arzani
,
A.
,
Wang
,
J.-X.
,
Sacks
,
M. S.
, and
Shadden
,
S. C.
,
2022
, “
Machine Learning for Cardiovascular Biomechanics Modeling: Challenges and Beyond
,”
Ann. Biomed. Eng.
,
50
(
6
), pp.
615
627
.10.1007/s10439-022-02967-4
14.
Subramaniam
,
A.
,
Wong
,
M. L.
,
Borker
,
R. D.
,
Nimmagadda
,
S.
, and
Lele
,
S. K.
,
2020
, “
Turbulence Enrichment Using Physics-Informed Generative Adversarial Networks
,”
arXiv:2003.01907
.10.48550/arXiv.2003.01907
15.
Gao
,
H.
,
Sun
,
L.
, and
Wang
,
J.-X.
,
2021
, “
Super-Resolution and Denoising of Fluid Flow Using Physics-Informed Convolutional Neural Networks Without High-Resolution Labels
,”
Phys. Fluids
,
33
(
7
), p.
073603
.10.1063/5.0054312
16.
Raissi
,
M.
,
Perdikaris
,
P.
, and
Karniadakis
,
G.
,
2019
, “
Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations
,”
J. Comput. Phys.
,
378
, pp.
686
707
.10.1016/j.jcp.2018.10.045
17.
Oldenburg
,
J.
,
Borowski
,
F.
,
Öner
,
A.
,
Schmitz
,
K.-P.
, and
Stiehm
,
M.
,
2022
, “Geometry Aware Physics Informed Neural Network Surrogate for Solving Navier-Stokes Equation (
GAPINN
),” Adv. Model. Simul. Eng. Sci., 9, Article No. 8.10.1186/s40323-022-00221-z
18.
Kingma
,
D. P.
, and
Welling
,
M.
,
2022
, “
Auto-Encoding Variational Bayes
,”
arXiv:1312.6114
.10.48550/arXiv.1312.6114
19.
Maleki
,
A.
,
Heyse
,
J.
,
Ranade
,
R.
,
He
,
H.
,
Kasimbeg
,
P.
, and
Pathak
,
J.
,
2021
, “
Geometry Encoding for Numerical Simulations
,”
arXiv:2104.07792
.10.48550/arXiv.2104.07792
20.
Ranade
,
R.
,
Hill
,
C.
,
Ghule
,
L.
, and
Pathak
,
J.
,
2022
, “
A Composable Machine-Learning Approach for Steady-State Simulations on High-Resolution Grids
,”
arXiv:2210.05837
.10.48550/arXiv.2210.05837
21.
Rutkowski
,
D.
,
Roldán-Alzate
,
A.
, and
Johnson
,
K.
,
2021
, “
Enhancement of Cerebrovascular 4D Flow Mri Velocity Fields Using Machine Learning and Computational Fluid Dynamics Simulation Data
,”
Sci. Rep.
,
11
(
1
), p.
05
.10.1038/s41598-021-89636-z
22.
Callaghan
,
F. M.
, and
Grieve
,
S. M.
,
2016
, “
Spatial Resolution and Velocity Field Improvement of 4D-Flow MRI
,”
Magn. Resonance Med.
,
78
(
5
), pp.
1959
1968
.10.1002/mrm.26557
23.
Zhuang
,
B.
,
Sirajuddin
,
A.
,
Zhao
,
S.
, and
Lu
,
M.
,
2021
, “
The Role of 4D Flow MRI for Clinical Applications in Cardiovascular Disease: Current Status and Future Perspectives
,”
Quant. Imaging Med. Surg.
,
11
(
9
), pp.
4193
4210
.10.21037/qims-20-1234
24.
Cherry
,
M.
,
Khatir
,
Z.
,
Khan
,
A.
, and
Bissell
,
M.
,
2022
, “
The Impact of 4D-Flow MRI Spatial Resolution on Patient-Specific CFD Simulations of the Thoracic Aorta
,”
Sci. Rep.
,
12
(
1
), p. 15128.10.1038/s41598-022-19347-6
25.
Ferdian
,
E.
,
Suinesiaputra
,
A.
,
Dubowitz
,
D. J.
,
Zhao
,
D.
,
Wang
,
A.
,
Cowan
,
B.
, and
Young
,
A. A.
,
2020
, “
4DFlownet: Super-Resolution 4D Flow MRI Using Deep Learning and Computational Fluid Dynamics
,”
Front. Phys.
,
8,
p. 138.10.3389/fphy.2020.00138
26.
Jiang
,
C.
,
Esmaeilzadeh
,
S.
,
Azizzadenesheli
,
K.
,
Kashinath
,
K.
,
Mustafa
,
M.
,
Tchelepi
,
H. A.
,
Marcus
,
P.
,
Prabhat
,
M.
, and
Anandkumar
,
A.
,
2020
, “
MESHFREEFLOWNET: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework
,”
SC20: International Conference for High Performance Computing, Networking, Storage and Analysis
, Virtual, Nov. 9–19, Article No. 9, pp.
1
15
.10.5555/3433701.3433712
27.
Fathi
,
M. F.
,
Perez-Raya
,
I.
,
Baghaie
,
A.
,
Berg
,
P.
,
Janiga
,
G.
,
Arzani
,
A.
, and
D'Souza
,
R. M.
,
2020
, “
Super-Resolution and Denoising of 4D-Flow MRI Using Physics-Informed Deep Neural Nets
,”
Comput. Methods Programs Biomedicine
,
197
, p.
105729
.10.1016/j.cmpb.2020.105729
28.
Kissas
,
G.
,
Yang
,
Y.
,
Hwuang
,
E.
,
Witschey
,
W. R.
,
Detre
,
J. A.
, and
Perdikaris
,
P.
,
2020
, “
Machine Learning in Cardiovascular Flows Modeling: Predicting Arterial Blood Pressure From Non-Invasive 4D Flow MRI Data Using Physics-Informed Neural Networks
,”
Comput. Methods Appl. Mech. Eng.
,
358
, p.
112623
.10.1016/j.cma.2019.112623
29.
Goodman
,
J. W.
,
1976
, “
Some Fundamental Properties of Speckle*
,”
J. Opt. Soc. Am.
,
66
(
11
), pp.
1145
1150
.10.1364/JOSA.66.001145
30.
Frank
,
S.
,
Lee
,
J.
,
Lantz
,
J.
,
Ebbers
,
T.
, and
Shadden
,
S.
,
2021
, “
Cardiac Kinetic Energy and Viscous Dissipation Rate From Radial Flow Data
,”
Front. Physiol.
,
12
, p.
09
.10.3389/fphys.2021.725104
31.
Huang
,
H.
,
Hu
,
X.
,
Zhao
,
Y.
,
Makkie
,
M.
,
Dong
,
Q.
,
Zhao
,
S.
,
Li
,
K.
, and
Liu
,
T.
,
2017
, “
Modeling Task fMRI Data Via Deep Convolutional Autoencoder
,”
IEEE Trans. Med. Imaging
, PP(
06
), pp.
1
1
.10.1007/978-3-319-59050-9_33
32.
Cheng
,
Z.
,
Sun
,
H.
,
Takeuchi
,
M.
, and
Katto
,
J.
,
2018
, “
Deep Convolutional Autoencoder-Based Lossy Image Compression
,” CoRR,
abs/1804.09535
.10.48550/arXiv.1804.09535
33.
Shit
,
S.
,
Zimmermann
,
J.
,
Ezhov
,
I.
,
Paetzold
,
J. C.
,
Sanches
,
A. F.
,
Pirkl
,
C.
, and
Menze
,
B. H.
,
2022
, “
SRflow: Deep Learning Based Super-Resolution of 4D-Flow MRI Data
,”
Front. Artif. Intell.
,
5
, p. 928181.10.3389/frai.2022.928181
34.
López
,
M. M.
,
Frederick
,
J. M.
, and
Ventura
,
J.
,
2021
, “
Evaluation of MRI Denoising Methods Using Unsupervised Learning
,”
Front. Artif. Intell.
,
4
, p. 642731.10.3389/frai.2021.642731
35.
Ferdian
,
E.
,
Dubowitz
,
D. J.
,
Mauger
,
C. A.
,
Wang
,
A.
, and
Young
,
A. A.
,
2022
, “
WSSNet: Aortic Wall Shear Stress Estimation Using Deep Learning on 4D Flow MRI
,”
Front. Cardiovasc. Med.
, 8, p. 769927.10.3389/fcvm.2021.769927
36.
Tan
,
M.
, and
Le
,
Q.
,
2019
, “
EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks
,”
Proceedings of the 36th International Conference on Machine Learning
, Long Beach, CA, June 9–15,
K.
Chaudhuri
, and
R.
Salakhutdinov
, eds., Vol.
97
of Proceedings of Machine Learning Research, pp.
6105
6114
.https://www.researchgate.net/publication/333444574_EfficientNet_Rethinking_Model_Scaling_for_Convolutional_Neural_Networks
You do not currently have access to this content.