Learning physics-based reduced-order models from data using nonlinear manifolds

Geelen, Rudy; Balzano, Laura; Wright, Stephen; Willcox, Karen

doi:10.1063/5.0170105

Title: Learning physics-based reduced-order models from data using nonlinear manifolds

Journal Article · 12 March 2024 · Chaos: An Interdisciplinary Journal of Nonlinear Science

DOI: https://doi.org/10.1063/5.0170105 · OSTI ID:2340141

^[1];

^[2];

^[3];

^[4]

Univ. of Texas, Austin, TX (United States); The Oden Institute for Computational Engineering and Sciences
Univ. of Michigan, Ann Arbor, MI (United States)
Univ. of Wisconsin, Madison, WI (United States)
Univ. of Texas, Austin, TX (United States)

Here we present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

View Accepted Manuscript (DOE)

Cite

Export

Save

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

Sponsoring Organization:: USDOE; US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO)

Grant/Contract Number:: SC0019303

OSTI ID:: 2340141

Journal Information:: Chaos: An Interdisciplinary Journal of Nonlinear Science, Journal Name: Chaos: An Interdisciplinary Journal of Nonlinear Science Journal Issue: 3 Vol. 34; ISSN 1054-1500

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (50)

Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basisa posteriori error bounds Veroy, K.; Patera, A. T. International Journal for Numerical Methods in Fluids, Vol. 47, Issue 8-9 https://doi.org/10.1002/fld.867	journal	January 2005
A quadratic decoder approach to nonintrusive reduced‐order modeling of nonlinear dynamical systems Benner, Peter; Goyal, Pawan; Heiland, Jan PAMM, Vol. 23, Issue 1 https://doi.org/10.1002/pamm.202200049	journal	May 2023
A generalized solution of the orthogonal procrustes problem Schönemann, Peter H. Psychometrika, Vol. 31, Issue 1 https://doi.org/10.1007/BF02289451	journal	March 1966
Dimensionality reduction and reduced-order modeling for traveling wave physics Mendible, Ariana; Brunton, Steven L.; Aravkin, Aleksandr Y. Theoretical and Computational Fluid Dynamics, Vol. 34, Issue 4 https://doi.org/10.1007/s00162-020-00529-9	journal	May 2020
The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent Chen, Caihua; He, Bingsheng; Ye, Yinyu Mathematical Programming, Vol. 155, Issue 1-2 https://doi.org/10.1007/s10107-014-0826-5	journal	October 2014
Fast data-driven model reduction for nonlinear dynamical systems Axås, Joar; Cenedese, Mattia; Haller, George Nonlinear Dynamics, Vol. 111, Issue 9 https://doi.org/10.1007/s11071-022-08014-0	journal	November 2022
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening Allen, Samuel M.; Cahn, John W. Acta Metallurgica, Vol. 27, Issue 6 https://doi.org/10.1016/0001-6160(79)90196-2	journal	June 1979
Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis Graham, Michael D.; Kevrekidis, Ioannis G. Computers & Chemical Engineering, Vol. 20, Issue 5 https://doi.org/10.1016/0098-1354(95)00040-2	journal	May 1996
Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S. Physica D: Nonlinear Phenomena, Vol. 44, Issue 1-2 https://doi.org/10.1016/0167-2789(90)90046-R	journal	August 1990
On the computation of inertial manifolds Foias, C.; Jolly, M. S.; Kevrekidis, I. G. Physics Letters A, Vol. 131, Issue 7-8 https://doi.org/10.1016/0375-9601(88)90295-2	journal	September 1988
A reduced order method for Allen–Cahn equations Song, Huailing; Jiang, Lijian; Li, Qiuqi Journal of Computational and Applied Mathematics, Vol. 292 https://doi.org/10.1016/j.cam.2015.07.009	journal	January 2016
Operator inference with roll outs for learning reduced models from scarce and low-quality data Uy, Wayne Isaac Tan; Hartmann, Dirk; Peherstorfer, Benjamin Computers & Mathematics with Applications, Vol. 145 https://doi.org/10.1016/j.camwa.2023.06.012	journal	September 2023
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
POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition Fresca, Stefania; Manzoni, Andrea Computer Methods in Applied Mechanics and Engineering, Vol. 388 https://doi.org/10.1016/j.cma.2021.114181	journal	January 2022
Operator inference for non-intrusive model reduction with quadratic manifolds Geelen, Rudy; Wright, Stephen; Willcox, Karen Computer Methods in Applied Mechanics and Engineering, Vol. 403 https://doi.org/10.1016/j.cma.2022.115717	journal	January 2023
A quadratic manifold for model order reduction of nonlinear structural dynamics Jain, Shobhit; Tiso, Paolo; Rutzmoser, Johannes B. Computers & Structures, Vol. 188 https://doi.org/10.1016/j.compstruc.2017.04.005	journal	August 2017
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
A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder Kim, Youngkyu; Choi, Youngsoo; Widemann, David Journal of Computational Physics, Vol. 451 https://doi.org/10.1016/j.jcp.2021.110841	journal	February 2022
Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction Barnett, Joshua; Farhat, Charbel Journal of Computational Physics, Vol. 464 https://doi.org/10.1016/j.jcp.2022.111348	journal	September 2022
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
Hamiltonian operator inference: Physics-preserving learning of reduced-order models for canonical Hamiltonian systems Sharma, Harsh; Wang, Zhu; Kramer, Boris Physica D: Nonlinear Phenomena, Vol. 431 https://doi.org/10.1016/j.physd.2021.133122	journal	March 2022
Nonlinear normal modes, Part I: A useful framework for the structural dynamicist Kerschen, G.; Peeters, M.; Golinval, J. C. Mechanical Systems and Signal Processing, Vol. 23, Issue 1 https://doi.org/10.1016/j.ymssp.2008.04.002	journal	January 2009
A hierarchy of low-dimensional models for the transient and post-transient cylinder wake Noack, Bernd R.; Afanasiev, Konstantin; MorzyŃSki, Marek Journal of Fluid Mechanics, Vol. 497 https://doi.org/10.1017/S0022112003006694	journal	December 2003
Spectral analysis of nonlinear flows Rowley, Clarence W.; MeziĆ, Igor; Bagheri, Shervin Journal of Fluid Mechanics, Vol. 641 https://doi.org/10.1017/S0022112009992059	journal	November 2009
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
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
Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies Johnson, Mark E.; Jolly, Michael S.; Kevrekidis, Ioannis G. Numerical Algorithms, Vol. 14, Issue 1/3, p. 125-140 https://doi.org/10.1023/A:1019104828180	journal	April 1997
Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders Deane, A. E.; Kevrekidis, I. G.; Karniadakis, G. E. Physics of Fluids A: Fluid Dynamics, Vol. 3, Issue 10 https://doi.org/10.1063/1.857881	journal	October 1991
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
Some global dynamical properties of a class of pattern formation equations Nicolaenko, B.; Scheurer, B.; Temam, R. Communications in Partial Differential Equations, Vol. 14, Issue 2 https://doi.org/10.1080/03605308908820597	journal	January 1989
Inertial manifolds for reaction diffusion equations in higher space dimensions Mallet-Paret, John; Sell, George R. Journal of the American Mathematical Society, Vol. 1, Issue 4 https://doi.org/10.1090/S0894-0347-1988-0943276-7	journal	January 1988
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
Physics-informed dynamic mode decomposition Baddoo, Peter J.; Herrmann, Benjamin; McKeon, Beverley J. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 479, Issue 2271 https://doi.org/10.1098/rspa.2022.0576	journal	March 2023
QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems Gu, Chenjie IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 30, Issue 9 https://doi.org/10.1109/TCAD.2011.2142184	journal	September 2011
Representation Learning: A Review and New Perspectives Bengio, Y.; Courville, A.; Vincent, P. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 35, Issue 8 https://doi.org/10.1109/TPAMI.2013.50	journal	August 2013
Eigenmode Analysis in Unsteady Aerodynamics: Reduced Order Models Dowell, Earl H.; Hall, Kenneth C.; Romanowski, Michael C. Applied Mechanics Reviews, Vol. 50, Issue 6 https://doi.org/10.1115/1.3101718	journal	June 1997
Nonlinear Model Reduction via Discrete Empirical Interpolation Chaturantabut, Saifon; Sorensen, Danny C. SIAM Journal on Scientific Computing, Vol. 32, Issue 5 https://doi.org/10.1137/090766498	journal	January 2010
Two-Sided Projection Methods for Nonlinear Model Order Reduction Benner, Peter; Breiten, Tobias SIAM Journal on Scientific Computing, Vol. 37, Issue 2 https://doi.org/10.1137/14097255X	journal	January 2015
Data-Driven Discovery of Closure Models Pan, Shaowu; Duraisamy, Karthik SIAM Journal on Applied Dynamical Systems, Vol. 17, Issue 4 https://doi.org/10.1137/18M1177263	journal	January 2018
Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition Rowley, C. W. International Journal of Bifurcation and Chaos, Vol. 15, Issue 03 https://doi.org/10.1142/S0218127405012429	journal	March 2005
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator Mezić, Igor Annual Review of Fluid Mechanics, Vol. 45, Issue 1 https://doi.org/10.1146/annurev-fluid-011212-140652	journal	January 2013
Dynamic Mode Decomposition and Its Variants Schmid, Peter J. Annual Review of Fluid Mechanics, Vol. 54, Issue 1 https://doi.org/10.1146/annurev-fluid-030121-015835	journal	January 2022
Data-assisted reduced-order modeling of extreme events in complex dynamical systems Wan, Zhong Yi; Vlachas, Pantelis; Koumoutsakos, Petros PLOS ONE, Vol. 13, Issue 5 https://doi.org/10.1371/journal.pone.0197704	journal	May 2018
Uber Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse Kolmogoroff, A. The Annals of Mathematics, Vol. 37, Issue 1 https://doi.org/10.2307/1968691	journal	January 1936
Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition Kramer, Boris; Willcox, Karen E. AIAA Journal, Vol. 57, Issue 6 https://doi.org/10.2514/1.J057791	journal	June 2019
Balanced Model Reduction via the Proper Orthogonal Decomposition Willcox, K.; Peraire, J. AIAA Journal, Vol. 40, Issue 11 https://doi.org/10.2514/2.1570	journal	November 2002
Eigenmode analysis in unsteady aerodynamics - Reduced-order models Dowell, Earl H. AIAA Journal, Vol. 34, Issue 8 https://doi.org/10.2514/3.13274	journal	August 1996
Coupling of substructures for dynamic analyses. Craig, Roy R.; Bampton, Mervyn C. C. AIAA Journal, Vol. 6, Issue 7 https://doi.org/10.2514/3.4741	journal	July 1968
Reduced Basis Technique for Nonlinear Analysis of Structures Noor, Ahmed K.; Peters, Jeanne M. AIAA Journal, Vol. 18, Issue 4 https://doi.org/10.2514/3.50778	journal	April 1980
An Application of a Modified Gappy Proper Orthogonal Decomposition on Complexity Reduction of Allen-Cahn Equation Dechanubeksa, Chutipong; Chaturantabut, Saifon Algorithms, Vol. 13, Issue 6 https://doi.org/10.3390/a13060148	journal	June 2020

Similar Records

Operator inference for non-intrusive model reduction with quadratic manifolds

Journal Article · 2022 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1905877

Geelen, Rudy; Wright, Stephen; Willcox, Karen

Nonlinear manifold-based reduced order model

Technical Report · 2020 · OSTI ID:1669223

Choi, Youngsoo

Nonlinear manifold reduced order model

Software · 2024 · OSTI ID:code-142366

Choi, Youngsoo; Kim, Youngkyu; Widemann, David Paludetto

Related Subjects

97 MATHEMATICS AND COMPUTING
computational physics
dynamical systems
machine learning
mathematical modeling
partial differential equations

Title: Learning physics-based reduced-order models from data using nonlinear manifolds

Citation Formats

References (50)

Similar Records

Related Subjects