Abstract

Multi-fidelity modeling and calibration are data fusion tasks that ubiquitously arise in engineering design. However, there is currently a lack of general techniques that can jointly fuse multiple data sets with varying fidelity levels while also estimating calibration parameters. To address this gap, we introduce a novel approach that, using latent-map Gaussian processes (LMGPs), converts data fusion into a latent space learning problem where the relations among different data sources are automatically learned. This conversion endows our approach with some attractive advantages such as increased accuracy and reduced overall costs compared to existing techniques that need to take a combinatorial approach to fuse multiple datasets. Additionally, we have the flexibility to jointly fuse any number of data sources and the ability to visualize correlations between data sources. This visualization allows an analyst to detect model form errors or determine the optimum strategy for high-fidelity emulation by fitting LMGP only to the sufficiently correlated data sources. We also develop a new kernel that enables LMGPs to not only build a probabilistic multi-fidelity surrogate but also estimate calibration parameters with quite a high accuracy and consistency. The implementation and use of our approach are considerably simpler and less prone to numerical issues compared to alternate methods. Through analytical examples, we demonstrate the benefits of learning an interpretable latent space and fusing multiple (in particular more than two) sources of data.

References

1.
Chaudhuri
,
A.
,
Lam
,
R.
, and
Willcox
,
K.
,
2018
, “
Multifidelity Uncertainty Propagation via Adaptive Surrogates in Coupled Multidisciplinary Systems
,”
AIAA J.
,
56
(
1
), pp.
235
249
.
2.
Peherstorfer
,
B.
,
Willcox
,
K.
, and
Gunzburger
,
M.
,
2018
, “
Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization
,”
SIAM Rev.
,
60
(
3
), pp.
550
591
.
3.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2001
, “
Bayesian Calibration of Computer Models
,”
J. R. Stat. Soc. Series B Stat. Methodol.
,
63
(
3
), pp.
425
464
.
4.
Tao
,
S.
,
Apley
,
D. W.
,
Chen
,
W.
,
Garbo
,
A.
,
Pate
,
D. J.
, and
German
,
B. J.
,
2019
, “
Input Mapping for Model Calibration With Application to Wing Aerodynamics
,”
AIAA J.
,
57
(
7
), pp.
2734
2745
.
5.
Koziel
,
S.
,
Cheng
,
Q. S.
, and
Bandler
,
J. W.
,
2008
, “
Space Mapping
,”
IEEE Microwave Mag.
,
9
(
6
), pp.
105
122
.
6.
Bandler
,
J. W.
,
Biernacki
,
R. M.
,
Chen
,
S. H.
,
Grobelny
,
P. A.
, and
Hemmers
,
R. H.
,
1994
, “
Space Mapping Technique for Electromagnetic Optimization
,”
IEEE Trans. Microwave Theory Tech.
,
42
(
12
), pp.
2536
2544
.
7.
Amrit
,
A.
,
Leifsson
,
L.
, and
Koziel
,
S.
,
2020
, “
Fast Multi-Objective Aerodynamic Optimization Using Sequential Domain Patching and Multifidelity Models
,”
J. Aircr.
,
57
(
3
), pp.
388
398
.
8.
Leifsson
,
L.
, and
Koziel
,
S.
,
2015
, “
Aerodynamic Shape Optimization by Variable-Fidelity Computational Fluid Dynamics Models: A Review of Recent Progress
,”
J. Comput. Sci.
,
10
, pp.
45
54
.
9.
Koziel
,
S.
, and
Leifsson
,
L.
,
2013
, “
Multi-Level CFD-Based Airfoil Shape Optimization With Automated Low-Fidelity Model Selection
,”
Procedia Comput. Sci.
,
18
(
1
), pp.
889
898
.
10.
Forrester
,
A.
,
Sobester
,
A.
, and
Keane
,
A.
,
2008
,
Engineering Design via Surrogate Modelling: A Practical Guide
,
John Wiley & Sons
,
Hoboken, NJ
.
11.
Ng
,
L. W.-T.
, and
Eldred
,
M.
,
2012
, “
Multifidelity Uncertainty Quantification Using Non-Intrusive Polynomial Chaos and Stochastic Collocation
,”
Proceedings of the 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 20th AIAA/ASME/AHS Adaptive Structures Conference 14th AIAA
,
Honolulu, HI
,
Apr. 23–26
.
12.
Padron
,
A. S.
,
Alonso
,
J. J.
, and
Eldred
,
M. S.
,
2016
, “
Multi-Fidelity Methods in Aerodynamic Robust Optimization
,”
Proceedings of the 18th AIAA non-Deterministic Approaches Conference.
,
San Diego, CA
,
Jan. 4–8
.
13.
Zadeh
,
P. M.
,
Toropov
,
V. V.
, and
Wood
,
A. S.
,
2005
, “
Use of Moving Least Squares Method in Collaborative Optimization
,”
Proceedings of the 6th World Congresses of Structural and Multidisciplinary Optimization
,
Rio de Janeiro, Brazil
,
May 30–June 3
.
14.
Fernández-Godino
,
M. G.
,
Park
,
C.
,
Kim
,
N. H.
, and
Haftka
,
R. T.
,
2016
, “Review of Multi-Fidelity Models,”
arXiv preprint
. https://arxiv.org/abs/1609.07196
15.
Romanowicz
,
R.
,
Beven
,
K. J.
, and
Tawn
,
J.
,
1994
, “
Evaluation of Predictive Uncertainty in Nonlinear Hydrological Models Using a Bayesian Approach
,”
Stat. Environ.
,
2
, pp.
297
317
.
16.
Craig
,
P. S.
,
Goldstein
,
M.
,
Rougier
,
J. C
, and
Seheult
,
A. H.
,
2001
, “
Bayesian Forecasting Using Large Computer Models
,”
J. Am. Stat. Assoc.
,
96
(
454
), pp.
717
729
.
17.
Stainforth
,
D. A.
,
Aina
,
T.
,
Christensen
,
C.
,
Collins
,
M.
,
Faull
,
N.
,
Frame
,
D. J.
,
Kettleborough
,
J. A.
, et al
,
2005
, “
Uncertainty in Predictions of the Climate Response to Rising Levels of Greenhouse Gases
,”
Nature
,
433
(
7024
), pp.
403
406
.
18.
Zhang
,
W.
,
Bostanabad
,
R.
,
Liang
,
B.
,
Su
,
X.
,
Zeng
,
D.
,
Bessa
,
M. A.
,
Wang
,
Y.
,
Chen
,
W.
, and
Cao
,
J.
,
2019
, “
A Numerical Bayesian-Calibrated Characterization Method for Multiscale Prepreg Preforming Simulations With Tension-Shear Coupling
,”
Compos. Sci. Technol.
,
170
(
C
), pp.
15
24
.
19.
Gramacy
,
R. B.
,
Bingham
,
D.
,
Holloway
,
J. P.
,
Grosskopf
,
M. J.
,
Kuranz
,
C. C.
,
Rutter
,
E.
,
Trantham
,
M.
, and
Drake
,
R. P.
,
2015
, “
Calibrating a Large Computer Experiment Simulating Radiative Shock Hydrodynamics
,”
Ann. Appl. Stat.
,
9
(
3
), pp.
1141
1168
.
20.
Higdon
,
D.
,
Kennedy
,
M.
,
Cavendish
,
J. C.
,
Cafeo
,
J. A.
, and
Ryne
,
R. D.
,
2004
, “
Combining Field Data and Computer Simulations for Calibration and Prediction
,”
SIAM J. Sci. Comput.
,
26
(
2
), pp.
448
466
.
21.
Plumlee
,
M.
,
2017
, “
Bayesian Calibration of Inexact Computer Models
,”
J. Am. Stat. Assoc.
,
112
(
519
), pp.
1274
1285
.
22.
Apley
,
D. W.
,
Liu
,
J.
, and
Chen
,
W.
,
2006
, “
Understanding the Effects of Model Uncertainty in Robust Design With Computer Experiments
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
945
958
.
23.
Bayarri
,
M. J.
,
Berger
,
J. O
,
Paulo
,
R.
,
Sacks
,
J.
,
Cafeo
,
J. A
,
Cavendish
,
J.
,
Lin
,
C.-H.
, and
Tu
,
J.
,
2007
, “
A Framework for Validation of Computer Models
,”
Technometrics
,
49
(
2
), pp.
138
154
.
24.
Arendt
,
P. D.
,
Apley
,
D. W.
, and
Chen
,
W.
,
2012
, “
Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability
,”
ASME J. Mech. Des.
,
134
(
10
), p.
100908
.
25.
Arendt
,
P. D.
,
Apley
,
D. W.
,
Chen
,
W.
,
Lamb
,
D.
, and
Gorsich
,
D.
,
2012
, “
Improving Identifiability in Model Calibration Using Multiple Responses
,”
ASME J. Mech. Des.
,
134
(
10
), p.
100909
.
26.
Bostanabad
,
R.
,
Kearney
,
T.
,
Tao
,
S.
,
Apley
,
D. W.
, and
Chen
,
W.
,
2018
, “
Leveraging the Nugget Parameter for Efficient Gaussian Process Modeling
,”
Int. J. Numer. Methods Eng.
,
114
(
5
), pp.
501
516
.
27.
Rasmussen
,
C. E.
,
2006
,
Gaussian Processes for Machine Learning
,
The MIT Press
,
Cambridge, MA
.
28.
Tao
,
S.
,
Shintani
,
K.
,
Bostanabad
,
R.
,
Chan
,
Y. C.
,
Yang
,
G.
,
Meingast
,
H.
, and
Chen
,
W.
,
2017
, “
Enhanced Gaussian Process Metamodeling and Collaborative Optimization for Vehicle Suspension Design Optimization
,”
ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Cleveland, OH
,
Aug. 6–9
.
29.
Zhang
,
Y.
,
Tao
,
S.
,
Chen
,
W.
, and
Apley
,
D. W.
,
2019
, “
A Latent Variable Approach to Gaussian Process Modeling With Qualitative and Quantitative Factors
,”
Technometrics
,
62
(
3
), pp.
291
302
.
30.
Wang
,
Y.
,
Iyer
,
A.
,
Chen
,
W.
, and
Rondinelli
,
J. M.
,
2020
, “
Featureless Adaptive Optimization Accelerates Functional Electronic Materials Design
,”
Appl. Phys. Rev.
,
7
(
4
), p.
041403
.
31.
Qian
,
P. Z. G.
,
Wu
,
H.
, and
Wu
,
C. F. J.
,
2008
, “
Gaussian Process Models for Computer Experiments with Qualitative and Quantitative Factors
,”
Technometrics
,
50
(
3
), pp.
383
396
.
32.
Deng
,
X.
,
Lin
,
C. D.
,
Liu
,
K.-W.
, and
Rowe
,
R. K.
,
2017
, “
Additive Gaussian Process for Computer Models With Qualitative and Quantitative Factors
,”
Technometrics
,
59
(
3
), pp.
283
292
.
33.
Oune
,
N.
, and
Bostanabad
,
R.
,
2021
, “
Latent Map Gaussian Processes for Mixed Variable Metamodeling
,”
Comput. Methods Appl. Mech. Eng.
,
387
(
C
), p.
114128
.
34.
Gallager
,
R. G.
,
2013
,
Stochastic Processes: Theory for Applications
,
Cambridge University Press
,
Cambridge, UK
.
35.
Deb
,
K.
,
Pratap
,
A.
,
Agarwal
,
S.
, and
Meyarivan
,
T.
,
2002
, “
A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II
,”
IEEE Trans. Evol. Comput.
,
6
(
2
), pp.
182
197
.
36.
Toal
,
D. J. J.
,
Bressloff
,
N. W.
,
Keane
,
A. J.
, and
Holden
,
C. M. E.
,
2011
, “
The Development of a Hybridized Particle Swarm for Kriging Hyperparameter Tuning
,”
Eng. Optim.
,
43
(
6
), pp.
675
699
.
37.
Zhu
,
C.
,
Byrd
,
R. H.
,
Lu
,
P.
, and
Nocedal
,
J.
,
1997
, “
Algorithm 778: L-BFGS-B
,”
ACM Trans. Math. Softw.
,
23
(
4
), pp.
550
560
.
38.
Bostanabad
,
R.
,
Chan
,
Y.-C.
,
Wang
,
L.
,
Zhu
,
P.
, and
Chen
,
W.
,
2019
, “
Globally Approximate Gaussian Processes for Big Data With Application to Data-Driven Metamaterials Design
,”
ASME J. Mech. Des.
,
141
(
11
), p.
111402
. .
39.
Tripathy
,
R.
,
Bilionis
,
I.
, and
Gonzalez
,
M.
,
2016
, “
Gaussian Processes With Built-in Dimensionality Reduction: Applications to High-Dimensional Uncertainty Propagation
,”
J. Comput. Phys.
,
321
, pp.
191
223
.
40.
Gardner
,
J. R.
,
Pleiss
,
G.
,
Weinberger
,
K. Q.
,
Bindel
,
D.
, and
Wilson
,
A. G.
,
2018
, “
GPyTorch: Blackbox Matrix–Matrix Gaussian Process Inference with GPU Acceleration
,”
32nd Conference on Neural Information Processing Systems (NeurIPS 2018)
,
Montreal, Canada
,
Dec. 2–8
.
41.
Susiluoto
,
J.
,
Spantini,
A.
,
Haario
,
H.
,
Härkönen
,
T.
, and
Marzouk
,
Y.
,
2020
, “
Efficient Multi-Scale Gaussian Process Regression for Massive Remote Sensing Data With satGP v0.1.2
,”
Geosci. Model Dev.
,
13
(
7
), pp.
3439
3463
.
42.
Stanton
,
S.
,
Maddox
,
W.
,
Delbridge
,
I.
, and
Wilson
,
A. G.
,
2021
, “
Kernel Interpolation for Scalable Online Gaussian Processes
,”
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics
,
B.
Arindam
and
F.
Kenji
, eds.,
Online
,
Apr. 13–15
, pp.
3133
3141
.
43.
Planas
,
R.
,
Oune
,
N.
, and
Bostanabad
,
R.
,
2020
, “
Extrapolation With Gaussian Random Processes and Evolutionary Programming
,”
International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Online
,
Aug. 17–19
, American Society of Mechanical Engineers, p. V11AT11A004.
44.
Planas
,
R.
,
Oune
,
N.
, and
Bostanabad
,
R.
,
2021
, “
Evolutionary Gaussian Processes
,”
ASME J. Mech. Des.
,
143
(
11
), p.
111703
.
45.
Moon
,
H.
,
2010
,
Design and Analysis of Computer Experiments for Screening Input Variables
,
The Ohio State University
,
Columbus, OH
.
46.
Morris
,
M. D.
,
Mitchell
,
T. J.
, and
Ylvisaker
,
D.
,
1993
, “
Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction
,”
Technometrics
,
35
(
3
), pp.
243
255
.
47.
Tuo
,
R.
, and
Wu
,
C. F. J.
,
2015
, “
Efficient Calibration for Imperfect Computer Models
,”
Ann. Stat.
,
43
(
6
), pp.
2331
2352
.
48.
Tuo
,
R.
, and
Jeff Wu
,
C. F.
,
2016
, “
A Theoretical Framework for Calibration in Computer Models: Parametrization, Estimation and Convergence Properties
,”
SIAM/ASA J. Uncertain. Quantif.
,
4
(
1
), pp.
767
795
.
49.
Park
,
C.
, and
Apley
,
D.
,
2018
, “
Patchwork Kriging for Large-Scale Gaussian Process Regression
,”
J. Mach. Learn. Res.
,
19
(
1
), pp.
269
311
.
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