In this work, we introduce a comprehensive machine-learning algorithm, namely, a multifidelity neural network (MFNN) architecture for data-driven constitutive metamodeling of complex fluids. The physics-based neural networks developed here are informed by the underlying rheological constitutive models through the synthetic generation of low-fidelity model-based data points. The performance of these rheologically informed algorithms is thoroughly investigated and compared against classical deep neural networks (DNNs). The MFNNs are found to recover the experimentally observed rheology of a multicomponent complex fluid consisting of several different colloidal particles, wormlike micelles, and other oil and aromatic particles. Moreover, the data-driven model is capable of successfully predicting the steady state shear viscosity of this fluid under a wide range of applied shear rates based on its constituting components. Building upon the demonstrated framework, we present the rheological predictions of a series of multicomponent complex fluids made by DNN and MFNN. We show that by incorporating the appropriate physical intuition into the neural network, the MFNN algorithms capture the role of experiment temperature, the salt concentration added to the mixture, as well as aging within and outside the range of training data parameters. This is made possible by leveraging an abundance of synthetic low-fidelity data that adhere to specific rheological models. In contrast, a purely data-driven DNN is consistently found to predict erroneous rheological behavior.
REFERENCES
1.
de Souza Mendes
, P. R.
, “Thixotropic elasto-viscoplastic model for structured fluids
,” Soft Matter
7
, 2471
–2483
(2011
). 2.
Colombo
, J.
, and E.
Del Gado
, “Stress localization, stiffening, and yielding in a model colloidal gel
,” J. Rheol.
58
, 1089
–1116
(2014
). 3.
Gurnon
, A. K.
, and N. J.
Wagner
, “Microstructure and rheology relationships for shear thickening colloidal dispersions
,” J. Fluid Mech.
769
, 242
–276
(2015
). 4.
Rogers
, S. A.
, D.
Vlassopoulos
, and P. T.
Callaghan
, “Aging, yielding, and shear banding in soft colloidal glasses
,” Phys. Rev. Lett.
100
, 128304
(2008
). 5.
Dimitriou
, C. J.
, and G. H.
McKinley
, “A comprehensive constitutive law for waxy crude oil: A thixotropic yield stress fluid
,” Soft Matter
10
, 6619
–6644
(2014
). 6.
Gelbart
, W. M.
, and A.
Ben-Shaul
, “The ‘new’ science of ‘complex fluids
,’” J. Phys. Chem.
100
, 13169
–13189
(1996
). 7.
Vermant
, J.
, and M. J.
Solomon
, “Flow-induced structure in colloidal suspensions
,” J. Phys. Condens. Matter
17
, R187
–R216
(2005
). 8.
Masschaele
, K.
, J.
Fransaer
, and J.
Vermant
, “Flow-induced structure in colloidal gels: Direct visualization of model 2D suspensions
,” Soft Matter
7
, 7717
–7726
(2011
). 9.
Herschel
, W. H.
, and R.
Bulkley
, “Konsistenzmessungen von Gummi-Benzollösungen
,” Kolloid Z.
39
, 291
–300
(1926
). 10.
Bingham
, E. C.
, “An investigation of the laws of plastic flow
,” Bull. Bureau Stand.
13
, 309
–353
(1916
). 11.
Gillespie
, T.
, “An extension of Goodeve’s impulse theory of viscosity to pseudoplastic systems
,” J. Colloid Sci.
15
, 219
–231
(1960
). 12.
Bird
, R. B.
, and O.
Hassager
, Dynamics of Polymeric Liquids: Fluid mechanics, Dynamics of Polymeric Liquids (Wiley, New York, 1987).13.
Morrison
, F. A.
, and A.
Morrison
, Understanding Rheology, Raymond F. Boyer Library Collection (Oxford University, New York, 2001).14.
Macosko
, C. W.
, Rheology: Principles, Measurements, and Applications, Advances in Interfacial Engineering Series (VCH, Weinheim, 1994).15.
Mezger
, T. G.
, The Rheology Handbook: For Users of Rotational and Oscillatory Rheometers
, 2 rev. ed. (Vincentz Network
, Hannover
, 2006
).16.
Heldman
, R. P.
, and D. R.
Singh
, Introduction to Food Engineering
, 5th ed. (Elsevier
, Amsterdam
, 2013
).17.
Coleman
, P. C.
, and M. M.
Painter
, Fundamentals of Polymer Science: An Introductory Text
, 2nd ed. (Technomic
, Lancaster, PA
, 1997
).18.
Bonn
, D.
, M. M.
Denn
, L.
Berthier
, T.
Divoux
, and S.
Manneville
, “Yield stress materials in soft condensed matter
,” Rev. Mod. Phys.
89
, 035005
(2017
). 19.
Coussot
, P.
, “Yield stress fluid flows: A review of experimental data
,” J. Nonnewton Fluid Mech.
211
, 31
–49
(2014
). 20.
Kim
, J.-Y.
, J.-Y.
Song
, E.-J.
Lee
, and S.-K.
Park
, “Rheological properties and microstructures of Carbopol gel network system
,” Colloid Polym. Sci.
281
, 614
–623
(2003
). 21.
Kaneda
, I.
, and A.
Sogabe
, “Rheological properties of water swellable microgel polymerized in a confined space
,” Colloids Surf. A
270
, 163
–170
(2005
). 22.
Petekidis
, G.
, D.
Vlassopoulos
, and P.
Pusey
, “Yielding and flow of sheared colloidal glasses
,” J. Phys.: Condens. Matter
16
, S3955
(2004
). 23.
Ghosh
, A.
, G.
Chaudhary
, J. G.
Kang
, P. V.
Braun
, R. H.
Ewoldt
, and K. S.
Schweizer
, “Linear and nonlinear rheology and structural relaxation in dense glassy and jammed soft repulsive pNIPAM microgel suspensions
,” Soft Matter
15
, 1038
–1052
(2019
). 24.
Pellet
, C.
, and M.
Cloitre
, “The glass and jamming transitions of soft polyelectrolyte microgel suspensions
,” Soft Matter
12
, 3710
–3720
(2016
). 25.
Koumakis
, N.
, A.
Pamvouxoglou
, A. S.
Poulos
, and G.
Petekidis
, “Direct comparison of the rheology of model hard and soft particle glasses
,” Soft Matter
8
, 4271
–4284
(2012
). 26.
Caggioni
, M.
, V.
Trappe
, and P. T.
Spicer
, “Variations of the Herschel–Bulkley exponent reflecting contributions of the viscous continuous phase to the shear rate-dependent stress of soft glassy materials
,” J. Rheol.
64
, 413
–422
(2020
). 27.
Hébraud
, P.
, and F.
Lequeux
, “Mode-coupling theory for the pasty rheology of soft glassy materials
,” Phys. Rev. Lett.
81
, 2934
–2937
(1998
). 28.
Bocquet
, L.
, A.
Colin
, and A.
Ajdari
, “Kinetic theory of plastic flow in soft glassy materials
,” Phys. Rev. Lett.
103
, 036001
(2009
). 29.
Seth
, J. R.
, L.
Mohan
, C.
Locatelli-Champagne
, M.
Cloitre
, and R. T.
Bonnecaze
, “A micromechanical model to predict the flow of soft particle glasses
,” Nat. Mater.
10
, 838
–843
(2011
). 30.
Helgeson
, M. E.
, S. E.
Moran
, H. Z.
An
, and P. S.
Doyle
, “Mesoporous organohydrogels from thermogelling photocrosslinkable nanoemulsions
,” Nat. Mater.
11
, 344
–352
(2012
). 31.
Mohraz
, A.
, and M. J.
Solomon
, “Orientation and rupture of fractal colloidal gels during start-up of steady shear flow
,” J. Rheol.
49
, 657
–681
(2005
). 32.
Hsiao
, L. C.
, R. S.
Newman
, S. C.
Glotzer
, and M. J.
Solomon
, “Role of isostaticity and load-bearing microstructure in the elasticity of yielded colloidal gels
,” Proc. Natl. Acad. Sci. U.S.A.
109
, 16029
–16034
(2012
). 33.
Maranzano
, B. J.
, and N. J.
Wagner
, “The effects of interparticle interactions and particle size on reversible shear thickening: Hard-sphere colloidal dispersions
,” J. Rheol.
45
, 1205
–1222
(2001
). 34.
Cipelletti
, L.
, S.
Manley
, R. C.
Ball
, and D. A.
Weitz
, “Universal aging features in the restructuring of fractal colloidal gels
,” Phys. Rev. Lett.
84
, 2275
–2278
(2000
). 35.
Zia
, R. N.
, B. J.
Landrum
, and W. B.
Russel
, “A micro-mechanical study of coarsening and rheology of colloidal gels: Cage building, cage hopping, and Smoluchowski’s ratchet
,” J. Rheol.
58
, 1121
–1157
(2014
). 36.
Landrum
, B. J.
, W. B.
Russel
, and R. N.
Zia
, “Delayed yield in colloidal gels: Creep, flow, and re-entrant solid regimes
,” J. Rheol.
60
, 783
–807
(2016
). 37.
Chu
, H. C. W.
, and R. N.
Zia
, “Active microrheology of hydrodynamically interacting colloids: Normal stresses and entropic energy density
,” J. Rheol.
60
, 755
–781
(2016
). 38.
Jamali
, S.
, G. H.
McKinley
, and R. C.
Armstrong
, “Microstructural rearrangements and their rheological implications in a model thixotropic elastoviscoplastic fluid
,” Phys. Rev. Lett.
118
, 048003
(2017
). 39.
Boromand
, A.
, S.
Jamali
, and J. M.
Maia
, “Structural fingerprints of yielding mechanisms in attractive colloidal gels
,” Soft Matter
13
, 458
–473
(2017
). 40.
Kim
, J.
, D.
Merger
, M.
Wilhelm
, and M. E.
Helgeson
, “Microstructure and nonlinear signatures of yielding in a heterogeneous colloidal gel under large amplitude oscillatory shear
,” J. Rheol.
58
, 1359
–1390
(2014
). 41.
Gao
, Y.
, J.
Kim
, and M. E.
Helgeson
, “Microdynamics and arrest of coarsening during spinodal decomposition in thermoreversible colloidal gels
,” Soft Matter
11
, 6360
–6370
(2015
). 42.
Min Kim
, J.
, A. P. R.
Eberle
, A.
Kate Gurnon
, L.
Porcar
, and N. J.
Wagner
, “The microstructure and rheology of a model, thixotropic nanoparticle gel under steady shear and large amplitude oscillatory shear (LAOS)
,” J. Rheol.
58
, 1301
–1328
(2014
). 43.
Solomon
, M. J.
, and P. T.
Spicer
, “Microstructural regimes of colloidal rod suspensions, gels, and glasses
,” Soft Matter
6
, 1391
–1400
(2010
). 44.
Dibble
, C. J.
, M.
Kogan
, and M. J.
Solomon
, “Structure and dynamics of colloidal depletion gels: Coincidence of transitions and heterogeneity
,” Phys. Rev. E
74
, 041403
(2006
). 45.
Furst
, E. M.
, and J. P.
Pantina
, “Yielding in colloidal gels due to nonlinear microstructure bending mechanics
,” Phys. Rev. E
75
, 050402
(2007
). 46.
Johnson
, L. C.
, B. J.
Landrum
, and R. N.
Zia
, “Yield of reversible colloidal gels during flow start-up: Release from kinetic arrest
,” Soft Matter
14
, 5048
–5068
(2018
). 47.
Johnson
, L. C.
, R. N.
Zia
, E.
Moghimi
, and G.
Petekidis
, “Influence of structure on the linear response rheology of colloidal gels
,” J. Rheol.
63
, 583
–608
(2019
). 48.
Lin
, N. Y. C.
, B. M.
Guy
, M.
Hermes
, C.
Ness
, J.
Sun
, W. C. K.
Poon
, and I.
Cohen
, “Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions
,” Phys. Rev. Lett.
115
, 228304
(2015
). 49.
Berret
, J.-F.
, “Rheology of wormlike micelles: Equilibrium properties and shear banding transition,” arXiv:0406681 [cond-mat] (2004).50.
Rothstein
, J. P.
, “Transient extensional rheology of wormlike micelle solutions
,” J. Rheol.
47
, 1227
–1247
(2003
). 51.
Padding
, J. T.
, E. S.
Boek
, and W. J.
Briels
, “Rheology of wormlike micellar fluids from Brownian and molecular dynamics simulations
,” J. Phys.: Condens. Matter
17
, S3347
–S3353
(2005
). 52.
Padding
, J. T.
, E. S.
Boek
, and W. J.
Briels
, “Dynamics and rheology of wormlike micelles emerging from particulate computer simulations
,” J. Chem. Phys.
129
, 074903
(2008
). 53.
Padding
, J. T.
, W. J.
Briels
, M. R.
Stukan
, and E. S.
Boek
, “Review of multi-scale particulate simulation of the rheology of wormlike micellar fluids
,” Soft Matter
5
, 4367
–4375
(2009
). 54.
Zhao
, Y.
, S. J.
Haward
, and A. Q.
Shen
, “Rheological characterizations of wormlike micellar solutions containing cationic surfactant and anionic hydrotropic salt
,” J. Rheol.
59
, 1229
–1259
(2015
). 55.
Larson
, R. G.
, “Constitutive equations for thixotropic fluids
,” J. Rheol.
59
, 595
–611
(2015
). 56.
Jamali
, S.
, R. C.
Armstrong
, and G. H.
McKinley
, “Multiscale nature of thixotropy and rheological hysteresis in attractive colloidal suspensions under shear
,” Phys. Rev. Lett.
123
, 248003
(2019
). 57.
Jamali
, S.
, R. C.
Armstrong
, and G. H.
McKinley
, “Time-rate-transformation framework for targeted assembly of short-range attractive colloidal suspensions
,” Mater. Today Adv.
5
, 100026
(2020
). 58.
Divoux
, T.
, V.
Grenard
, and S.
Manneville
, “Rheological hysteresis in soft glassy materials
,” Phys. Rev. Lett.
110
, 018304
(2013
). 59.
Mewis
, J.
, and N. J.
Wagner
, “Thixotropy
,” Adv. Colloid Interface Sci.
147-148
, 214
–227
(2009
). 60.
de Souza Mendes
, P. R.
, and R. L.
Thompson
, “A critical overview of elasto-viscoplastic thixotropic modeling
,” J. Nonnewton Fluid Mech.
187-188
, 8
–15
(2012
). 61.
Wei
, Y.
, M. J.
Solomon
, and R. G.
Larson
, “Quantitative nonlinear thixotropic model with stretched exponential response in transient shear flows
,” J. Rheol.
60
, 1301
–1315
(2016
). 62.
Coussot
, P.
, Q. D.
Nguyen
, H. T.
Huynh
, and D.
Bonn
, “Viscosity bifurcation in thixotropic, yielding fluids
,” J. Rheol.
46
, 573
–589
(2002
). 63.
Goodeve
, C. F.
, and G. W.
Whitfield
, “The measurement of thixotropy in absolute units
,” Trans. Faraday Soc.
34
, 511
–520
(1938
). 64.
Janes
, K. A.
, and M. B.
Yaffe
, “Data-driven modelling of signal-transduction networks
,” Nat. Rev. Mol. Cell Biol.
7
, 820
–828
(2006
). 65.
Solomatine
, D. P.
, and A.
Ostfeld
, “Data-driven modelling: Some past experiences and new approaches
,” J. Hydroinf.
10
, 3
–22
(2008
). 66.
Solomatine
, D.
, L.
See
, and R.
Abrahart
, Data-driven modelling: Concepts, approaches and experiences, in Practical Hydroinformatics (Springer, Berlin), pp. 17–30.67.
Bishop
, C.
, Pattern Recognition and Machine Learning
(Springer
, New York
, 2006
), p. 738
.68.
Brunton
, S. L.
, B. R.
Noack
, and P.
Koumoutsakos
, “Machine learning for fluid mechanics
,” Annu. Rev. Fluid Mech.
52
, 477
–508
(2020
). 69.
Chang
, W.
, X.
Chu
, A. F. B.
S. Fareed
, S.
Pandey
, J.
Luo
, B.
Weigand
, and E.
Laurien
, “Heat transfer prediction of supercritical water with artificial neural networks
,” Appl. Therm. Eng.
131
, 815
–824
(2018
). 70.
Mahmoudabadbozchelou
, M.
, A.
Eghtesad
, S.
Jamali
, and H.
Afshin
, “Entropy analysis and thermal optimization of nanofluid impinging jet using artificial neural network and genetic algorithm
,” Int. Commun. Heat Mass Transfer
119
, 104978
(2020
). 71.
Mohanraj
, M.
, S.
Jayaraj
, and C.
Muraleedharan
, “Applications of artificial neural networks for thermal analysis of heat exchangers—A review
,” Int. J. Therm. Sci.
90
, 150
–172
(2015
). 72.
Mahmoudabadbozchelou
, M.
, N.
Rabiei
, and M.
Bazargan
, “Numerical and experimental investigation of the optimization of vehicle speed and inter-vehicle distance in an automated highway car platoon to minimize fuel consumption
,” SAE Int. J. CAV
1
, 3
–12
(2018
). 73.
Rabault
, J.
, J.
Kolaas
, and A.
Jensen
, “Performing particle image velocimetry using artificial neural networks: A proof-of-concept
,” Meas. Sci. Technol.
28
, 125301
(2017
). 74.
Eghtesad
, A.
, M.
Mahmoudabadbozchelou
, and H.
Afshin
, “Heat transfer optimization of twin turbulent sweeping impinging jets
,” Int. J. Therm. Sci.
146
, 106064
(2019
). 75.
Xie
, G.
, B.
Sunden
, Q.
Wang
, and L.
Tang
, “Performance predictions of laminar and turbulent heat transfer and fluid flow of heat exchangers having large tube-diameter and large tube-row by artificial neural networks
,” Int. J. Heat Mass Transfer
52
, 2484
–2497
(2009
). 76.
Sirignano
, J.
, and K.
Spiliopoulos
, “DGM: A deep learning algorithm for solving partial differential equations
,” J. Comput. Phys.
375
, 1339
–1364
(2018
). 77.
Poplin
, R.
, A. V.
Varadarajan
, K.
Blumer
, Y.
Liu
, M. V.
McConnell
, G. S.
Corrado
, L.
Peng
, and D. R.
Webster
, “Prediction of cardiovascular risk factors from retinal fundus photographs via deep learning
,” Nat. Biomed. Eng.
2
, 158
–164
(2018
). 78.
Kim
, B.
, V. C.
Azevedo
, N.
Thuerey
, T.
Kim
, M.
Gross
, and B.
Solenthaler
, “Deep fluids: A generative network for parameterized fluid simulations
,” Comput. Graph. Forum
38
, 59
–70
(2019
). 79.
Geneva
, N.
, and N.
Zabaras
, “Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks
,” J. Comput. Phys.
383
, 125
–147
(2019
). 80.
Lu
, D.
, M.
Heisler
, S.
Lee
, G.
Ding
, M. V.
Sarunic
, and M. F.
Beg
, “Retinal fluid segmentation and detection in optical coherence tomography images using fully convolutional neural network,” arXiv:1710.04778 (2017).81.
Murata
, T.
, K.
Fukami
, and K.
Fukagata
, “Nonlinear mode decomposition with convolutional neural networks for fluid dynamics
,” J. Fluid Mech.
882
, A13
(2020
). 82.
Rashno
, A.
, D. D.
Koozekanani
, and K. K.
Parhi
, OCT fluid segmentation using graph shortest path and convolutional neural network*, in 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) (IEEE, New York, 2018), pp. 3426–3429.83.
Smith
, C.
, J.
Doherty
, and Y.
Jin
, “Multi-objective evolutionary recurrent neural network ensemble for prediction of computational fluid dynamic simulations,” in 2014 IEEE Congress on Evolutionary Computation (CEC) (IEEE, New York, 2014), pp. 2609–2616.84.
Liao
, C.
, K.
Wang
, M.
Yu
, and W.
Chen
, “Modeling of magneto-rheological fluid damper employing recurrent neural networks,” in 2005 International Conference on Neural Networks and Brain (IEEE, New York, 2005), Vol. 2, pp. 616–620.85.
Raissi
, M.
, P.
Perdikaris
, and G.
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
). 86.
Meng
, X.
, Z.
Li
, D.
Zhang
, and G. E.
Karniadakis
, “PPINN: Parareal physics-informed neural network for time-dependent PDEs,” arXiv:1909.10145 (2019), pp. 1–17.87.
Meng
, X.
, and G. E.
Karniadakis
, “A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems
,” J. Comput. Phys.
401
, 109020
(2020
). 88.
Pang
, G.
, L.
Lu
, and G. E.
Karniadakis
, “fPINNs: Fractional physics-informed neural networks
,” SIAM J. Sci. Comput.
35
, 225
–253
(2018
).89.
Pang
, G.
, M.
D’Elia
, M.
Parks
, and G. E.
Karniadakis
, “nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal laplacian operator. Algorithms and applications,” arXiv:2004.04276 (2020).90.
Lu
, L.
, P.
Jin
, and G. E.
Karniadakis
, “DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators,” arXiv:1910.03193 (2019), pp. 1–22.91.
Lu
, L.
, X.
Meng
, Z.
Mao
, and G. E.
Karniadakis
, “DeepXDE: A deep learning library for solving differential equations,” arXiv:1907.04502 (2019), pp. 1–17.92.
Wang
, J. X.
, J. L.
Wu
, and H.
Xiao
, “Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data
,” Phys. Rev. Fluids
2
, 1
–22
(2017
). 93.
Wu
, J. L.
, H.
Xiao
, and E.
Paterson
, “Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework
,” Phys. Rev. Fluids
7
, 1
–28
(2018
). 94.
Swischuk
, R.
, L.
Mainini
, B.
Peherstorfer
, and K.
Willcox
, “Projection-based model reduction: Formulations for physics-based machine learning
,” Comput. Fluids
179
, 704
–717
(2019
). 95.
Jia
, X.
, J.
Willard
, A.
Karpatne
, J. S.
Read
, J. A.
Zwart
, M.
Steinbach
, and V.
Kumar
, “Physics-guided machine learning for scientific discovery: An application in simulating lake temperature profiles,” arXiv:2001.11086 (2020), pp. 1–25.96.
Rackauckas
, C.
, Y.
Ma
, J.
Martensen
, C.
Warner
, K.
Zubov
, R.
Supekar
, D.
Skinner
, and A.
Ramadhan
, “Universal differential equations for scientific machine learning,” arXiv:2001.04385 (2020).97.
Piazza
, R.
, and G. D.
Pietro
, “Phase separation and gel-like structures in mixtures of colloids and surfactant
,” Europhys. Lett.
28
, 445
–450
(1994
). 98.
Petekidis
, G.
, L.
Galloway
, S.
Egelhaaf
, M.
Cates
, and W.
Poon
, “Mixtures of colloids and wormlike micelles: Phase behavior and kinetics
,” Langmuir
18
, 4248
–4257
(2002
). 99.
Fernández-Godino
, M. G.
, C.
Park
, N.-H.
Kim
, and R. T.
Haftka
, “Review of multi-fidelity models,” arXiv:1609.07196 (2016).100.
Zhang
, D.
, L.
Lu
, L.
Guo
, and G. E.
Karniadakis
, “Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
,” J. Comput. Phys.
397
, 108850
(2019
). 101.
Foody
, G. M.
, and M. K.
Arora
, “An evaluation of some factors affecting the accuracy of classification by an artificial neural network
,” Int. J. Remote Sens.
18
, 799
–810
(1997
). © 2021 The Society of Rheology.
2021
The Society of Rheology
You do not currently have access to this content.
