Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
Abstract
Nearly all modelreduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reducedbasis method, and (balanced) proper orthogonal decomposition (POD). Unfortunately, restricting the state to evolve in a linear subspace imposes a fundamental limitation to the accuracy of the resulting reducedorder model (ROM). In particular, linearsubspace ROMs can be expected to produce lowdimensional models with high accuracy only if the problem admits a fast decaying Kolmogorov nwidth (e.g., diffusiondominated problems). Unfortunately, many problems of interest exhibit a slowly decaying Kolmogorov nwidth (e.g., advectiondominated problems). To address this, we propose a novel framework for projecting dynamical systems onto nonlinear manifolds using minimumresidual formulations at the timecontinuous and timediscrete levels; the former leads to manifold Galerkin projection, while the latter leads to manifold leastsquares Petrov–Galerkin (LSPG) projection. We perform analyses that provide insight into the relationship between these proposed approaches and classical linearsubspace reducedorder models; we also derive a posteriori discretetime error bounds for the proposed approaches. In addition, we propose a computationally practical approach for computing the nonlinear manifold, which is based on convolutional autoencoders from deep learning. Lastly, we demonstrate themore »
 Authors:

 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Sponsoring Org.:
 Sandia's Advanced Simulation and Computing (ASC) Verification and Validation (V&V) Project; USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1574441
 Alternate Identifier(s):
 OSTI ID: 1691627
 Report Number(s):
 SAND201912375J; SAND20190003J
Journal ID: ISSN 00219991; 680333; TRN: US2001261
 Grant/Contract Number:
 AC0494AL85000; NA0003525
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 404; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Model reduction; Deep learning; Autoencoders; Machine learning; Nonlinear manifolds; Optimal projection
Citation Formats
Lee, Kookjin, and Carlberg, Kevin T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.108973.
Lee, Kookjin, & Carlberg, Kevin T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. United States. doi:10.1016/j.jcp.2019.108973.
Lee, Kookjin, and Carlberg, Kevin T. Fri .
"Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders". United States. doi:10.1016/j.jcp.2019.108973. https://www.osti.gov/servlets/purl/1574441.
@article{osti_1574441,
title = {Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders},
author = {Lee, Kookjin and Carlberg, Kevin T.},
abstractNote = {Nearly all modelreduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reducedbasis method, and (balanced) proper orthogonal decomposition (POD). Unfortunately, restricting the state to evolve in a linear subspace imposes a fundamental limitation to the accuracy of the resulting reducedorder model (ROM). In particular, linearsubspace ROMs can be expected to produce lowdimensional models with high accuracy only if the problem admits a fast decaying Kolmogorov nwidth (e.g., diffusiondominated problems). Unfortunately, many problems of interest exhibit a slowly decaying Kolmogorov nwidth (e.g., advectiondominated problems). To address this, we propose a novel framework for projecting dynamical systems onto nonlinear manifolds using minimumresidual formulations at the timecontinuous and timediscrete levels; the former leads to manifold Galerkin projection, while the latter leads to manifold leastsquares Petrov–Galerkin (LSPG) projection. We perform analyses that provide insight into the relationship between these proposed approaches and classical linearsubspace reducedorder models; we also derive a posteriori discretetime error bounds for the proposed approaches. In addition, we propose a computationally practical approach for computing the nonlinear manifold, which is based on convolutional autoencoders from deep learning. Lastly, we demonstrate the ability of the method to significantly outperform even the optimal linearsubspace ROM on benchmark advectiondominated problems, thereby demonstrating the method's ability to overcome the intrinsic nwidth limitations of linear subspaces.},
doi = {10.1016/j.jcp.2019.108973},
journal = {Journal of Computational Physics},
number = ,
volume = 404,
place = {United States},
year = {2019},
month = {11}
}
Web of Science
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