Bayesian operator inference for data-driven reduced-order modeling

Guo, Mengwu; McQuarrie, Shane A.; Willcox, Karen E.

doi:10.1016/j.cma.2022.115336

Title: Bayesian operator inference for data-driven reduced-order modeling

Journal Article · 14 July 2022 · Computer Methods in Applied Mechanics and Engineering

DOI: https://doi.org/10.1016/j.cma.2022.115336 · OSTI ID:1960781

^[1];

^[2];

^[2]

Univ. of Twente (The Netherlands); Oden Institute for Computational Engineering and Sciences, the University of Texas at Austin
Univ. of Texas, Austin, TX (United States)

This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inverse problem with Gaussian prior and likelihood. The resulting posterior distribution characterizes the operators defining the reduced-order model, hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The statistical moments of these predictions are estimated via a Monte Carlo sampling of the posterior distribution. Since the reduced models are fast to solve, this sampling is computationally efficient. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the regularization term that is present in the deterministic operator inference problem, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated on two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Univ. of Texas, Austin, TX (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: NA0003969

OSTI ID:: 1960781

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 402; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (42)

Optimal control of vortex shedding using low-order models. Part I?open-loop model development Graham, W. R.; Peraire, J.; Tang, K. Y. International Journal for Numerical Methods in Engineering, Vol. 44, Issue 7 https://doi.org/10.1002/(SICI)1097-0207(19990310)44:7<945::AID-NME537>3.0.CO;2-F	journal	March 1999
A computational framework for dynamic data-driven material damage control, based on Bayesian inference and model selection: SPECIAL ISSUE TO HONOUR TED BELYTSCHKO Prudencio, E. E.; Bauman, P. T.; Faghihi, D. International Journal for Numerical Methods in Engineering, Vol. 102, Issue 3-4 https://doi.org/10.1002/nme.4669	journal	April 2014
Estimation of Modeling Error in Computational Mechanics Oden, J. Tinsley; Prudhomme, Serge Journal of Computational Physics, Vol. 182, Issue 2 https://doi.org/10.1006/jcph.2002.7183	journal	November 2002
Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth Hawkins-Daarud, Andrea; Prudhomme, Serge; van der Zee, Kristoffer G. Journal of Mathematical Biology, Vol. 67, Issue 6-7 https://doi.org/10.1007/s00285-012-0595-9	journal	October 2012
An error threshold criterion for singular value decomposition modes extracted from PIV data Epps, Brenden P.; Techet, Alexandra H. Experiments in Fluids, Vol. 48, Issue 2 https://doi.org/10.1007/s00348-009-0740-4	journal	October 2009
Bayesian-based predictions of COVID-19 evolution in Texas using multispecies mixture-theoretic continuum models Jha, Prashant K.; Cao, Lianghao; Oden, J. Tinsley Computational Mechanics, Vol. 66, Issue 5 https://doi.org/10.1007/s00466-020-01889-z	journal	July 2020
Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty Oden, J. Tinsley; Babuška, Ivo; Nobile, Fabio Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 2-5 https://doi.org/10.1016/j.cma.2003.06.003	journal	February 2005
Verification and validation in computational engineering and science: basic concepts Babuska, Ivo; Oden, J. Tinsley Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 36-38 https://doi.org/10.1016/j.cma.2004.03.002	journal	September 2004
Data-driven operator inference for nonintrusive projection-based model reduction Peherstorfer, Benjamin; Willcox, Karen Computer Methods in Applied Mechanics and Engineering, Vol. 306 https://doi.org/10.1016/j.cma.2016.03.025	journal	July 2016
Data-driven reduced order modeling for time-dependent problems Guo, Mengwu; Hesthaven, Jan S. Computer Methods in Applied Mechanics and Engineering, Vol. 345 https://doi.org/10.1016/j.cma.2018.10.029	journal	March 2019
A Bayesian framework for adaptive selection, calibration, and validation of coarse-grained models of atomistic systems Farrell, Kathryn; Oden, J. Tinsley; Faghihi, Danial Journal of Computational Physics, Vol. 295 https://doi.org/10.1016/j.jcp.2015.03.071	journal	August 2015
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders Lee, Kookjin; Carlberg, Kevin T. Journal of Computational Physics https://doi.org/10.1016/j.jcp.2019.108973	journal	November 2019
Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems Qian, Elizabeth; Kramer, Boris; Peherstorfer, Benjamin Physica D: Nonlinear Phenomena, Vol. 406 https://doi.org/10.1016/j.physd.2020.132401	journal	May 2020
On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate Venturi, Daniele Journal of Fluid Mechanics, Vol. 559 https://doi.org/10.1017/S0022112006000346	journal	July 2006
Dynamic mode decomposition of numerical and experimental data Schmid, Peter J. Journal of Fluid Mechanics, Vol. 656 https://doi.org/10.1017/S0022112010001217	journal	July 2010
Adaptive multiscale predictive modelling Oden, J. Tinsley Acta Numerica, Vol. 27 https://doi.org/10.1017/S096249291800003X	journal	May 2018
Learning physics-based models from data: perspectives from inverse problems and model reduction Ghattas, Omar; Willcox, Karen Acta Numerica, Vol. 30 https://doi.org/10.1017/S0962492921000064	journal	May 2021
Model order reduction based on Runge–Kutta neural networks Zhuang, Qinyu; Lorenzi, Juan Manuel; Bungartz, Hans-Joachim Data-Centric Engineering, Vol. 2 https://doi.org/10.1017/dce.2021.15	journal	January 2021
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations Uy, Wayne Isaac Tan; Peherstorfer, Benjamin ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55, Issue 3 https://doi.org/10.1051/m2an/2021010	journal	May 2021
Proper orthogonal decomposition methods for noise reduction in particle-based transport calculations del-Castillo-Negrete, D.; Spong, D. A.; Hirshman, S. P. Physics of Plasmas, Vol. 15, Issue 9 https://doi.org/10.1063/1.2979680	journal	September 2008
Coupling between hydrodynamics, acoustics, and heat release in a self-excited unstable combustor Harvazinski, Matthew E.; Huang, Cheng; Sankaran, Venkateswaran Physics of Fluids, Vol. 27, Issue 4 https://doi.org/10.1063/1.4916673	journal	April 2015
Data-driven discovery of coordinates and governing equations Champion, Kathleen; Lusch, Bethany; Kutz, J. Nathan Proceedings of the National Academy of Sciences, Vol. 116, Issue 45 https://doi.org/10.1073/pnas.1906995116	journal	October 2019
Simplified Reaction Mechanisms for the Oxidation of Hydrocarbon Fuels in Flames Westbrook, Charles K.; Dryer, Frederick L. Combustion Science and Technology, Vol. 27, Issue 1-2 https://doi.org/10.1080/00102208108946970	journal	December 1981
Data-driven reduced-order models via regularised Operator Inference for a single-injector combustion process McQuarrie, Shane A.; Huang, Cheng; Willcox, Karen E. Journal of the Royal Society of New Zealand, Vol. 51, Issue 2 https://doi.org/10.1080/03036758.2020.1863237	journal	January 2021
Turbulence and the dynamics of coherent structures. I. Coherent structures Sirovich, Lawrence Quarterly of Applied Mathematics, Vol. 45, Issue 3 https://doi.org/10.1090/qam/910462	journal	January 1987
A Simplex Method for Function Minimization Nelder, J. A.; Mead, R. The Computer Journal, Vol. 7, Issue 4 https://doi.org/10.1093/comjnl/7.4.308	journal	January 1965
Learning partial differential equations via data discovery and sparse optimization Schaeffer, Hayden Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 473, Issue 2197 https://doi.org/10.1098/rspa.2016.0446	journal	January 2017
Robust data-driven discovery of governing physical laws with error bars Zhang, Sheng; Lin, Guang Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 474, Issue 2217 https://doi.org/10.1098/rspa.2018.0305	journal	September 2018
Localized non-intrusive reduced-order modelling in the operator inference framework Geelen, Rudy; Willcox, Karen Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 380, Issue 2229 https://doi.org/10.1098/rsta.2021.0206	journal	June 2022
A Sparse Bayesian Approach to the Identification of Nonlinear State-Space Systems Pan, Wei; Yuan, Ye; Goncalves, Jorge IEEE Transactions on Automatic Control, Vol. 61, Issue 1 https://doi.org/10.1109/TAC.2015.2426291	journal	January 2016
Tensor Decompositions and Applications Kolda, Tamara G.; Bader, Brett W. SIAM Review, Vol. 51, Issue 3 https://doi.org/10.1137/07070111X	journal	August 2009
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions Halko, N.; Martinsson, P. G.; Tropp, J. A. SIAM Review, Vol. 53, Issue 2 https://doi.org/10.1137/090771806	journal	January 2011
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems Benner, Peter; Gugercin, Serkan; Willcox, Karen SIAM Review, Vol. 57, Issue 4 https://doi.org/10.1137/130932715	journal	January 2015
Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference Peherstorfer, Benjamin SIAM Journal on Scientific Computing, Vol. 42, Issue 5 https://doi.org/10.1137/19M1292448	journal	January 2020
The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows Berkooz, G.; Holmes, P.; Lumley, J. L. Annual Review of Fluid Mechanics, Vol. 25, Issue 1 https://doi.org/10.1146/annurev.fl.25.010193.002543	journal	January 1993
Bayesian Solution Uncertainty Quantification for Differential Equations Chkrebtii, Oksana A.; Campbell, David A.; Calderhead, Ben Bayesian Analysis, Vol. 11, Issue 4 https://doi.org/10.1214/16-BA1017	journal	December 2016
Operator inference and physics-informed learning of low-dimensional models for incompressible flows Benner, Peter; Goyal, Pawan; Heiland, Jan ETNA - Electronic Transactions on Numerical Analysis, Vol. 56 https://doi.org/10.1553/etna_vol56s28	journal	January 2021
Investigations and Improvement of Robustness of Reduced-Order Models of Reacting Flow Huang, Cheng; Duraisamy, Karthik; Merkle, Charles L. AIAA Journal, Vol. 57, Issue 12 https://doi.org/10.2514/1.J058392	journal	December 2019
Learning Physics-Based Reduced-Order Models for a Single-Injector Combustion Process Swischuk, Renee; Kramer, Boris; Huang, Cheng AIAA Journal, Vol. 58, Issue 6 https://doi.org/10.2514/1.J058943	journal	June 2020
Exploration of Reduced-Order Models for Rocket Combustion Applications Huang, Cheng; Xu, Jiayang; Duraisamy, Karthik 2018 AIAA Aerospace Sciences Meeting, 2018 AIAA Aerospace Sciences Meeting https://doi.org/10.2514/6.2018-1183	conference	January 2018
Challenges in Reduced Order Modeling of Reacting Flows Huang, Cheng; Duraisamy, Karthik; Merkle, Charles 2018 Joint Propulsion Conference https://doi.org/10.2514/6.2018-4675	conference	July 2018
Performance comparison of data-driven reduced models for a single-injector combustion process Jain, Parikshit; McQuarrie, Shane; Kramer, Boris AIAA Propulsion and Energy 2021 Forum https://doi.org/10.2514/6.2021-3633	conference	July 2021

Cited By (1)

Deep Kernel Learning of Dynamical Models from High-Dimensional Noisy Data Botteghi, Nicolò; Guo, Mengwu; Brune, Christoph arXiv https://doi.org/10.48550/arxiv.2208.12975	text	January 2022

Similar Records

Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems

Journal Article · 2009 · Journal of Computational Physics · OSTI ID:21167764

Computationally efficient Bayesian inference for inverse problems.

Technical Report · 2007 · OSTI ID:1028962

Marzouk, Youssef M; Najm, Habib N; Rahn, Larry A

Bayesian inference of Stochastic reaction networks using Multifidelity Sequential Tempered Markov Chain Monte Carlo

Journal Article · 2020 · International Journal for Uncertainty Quantification · OSTI ID:1670752

Catanach, Thomas A.; Vo, Huy D.; Munsky, Brian

Related Subjects

97 MATHEMATICS AND COMPUTING
Bayesian inversion
Data-driven reduced-order modeling
Operator inference
Single-injector combustion
Tikhonov regularization
Uncertainty quantification

Title: Bayesian operator inference for data-driven reduced-order modeling

Citation Formats

References (42)

Cited By (1)

Similar Records

Related Subjects