Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Benosman, Mouhacine; Borggaard, Jeff; San, Omer; Krämer, Boris

doi:10.1016/j.apm.2017.04.032

Cited by 43 publications

(37 citation statements)

References 29 publications

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“…(i) Closure Modeling: To model the effect of the discarded ROM modes, a Correction term is generally added to the standard ROM [3,29,32,37]. Given the drastic truncation used in ROMs for realistic flows, the Correction term is essential for accuracy.…”

Section: Introductionmentioning

confidence: 99%

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Continuous data assimilation reduced order models of fluid flow

Zerfas

Rebholz

Schneier

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement data into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs.

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Section: Introductionmentioning

confidence: 99%

“…(iii) Data-Driven Modeling: Recently, available numerical or experimental data have been used to construct ROM operators [30] or to determine the unknown coefficients in classical ROM operators [3,11,32].…”

Section: Introductionmentioning

confidence: 99%

Continuous data assimilation reduced order models of fluid flow

Zerfas

Rebholz

Schneier

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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show abstract

“…In our recent work [44], we have shown that this Correction term can be explicitly calculated and modeled with the available data by using the ROM projection as a spatial filter. We note that ROM spatial filtering has also been used to develop large eddy simulation ROMs, e.g., approximate deconvolution ROMs [45] and eddy viscosity ROMs [5,19,27,33,35,42]. In all these ROMs, it has been been assumed the differentiation and ROM spatial filtering commute:…”

Section: Introductionmentioning

confidence: 99%

Commutation error in reduced order modeling of fluid flows

et al. 2019

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For reduced order models (ROMs) of fluid flows, we investigate theoretically and computationally whether differentiation and ROM spatial filtering commute, i.e., whether the commutation error (CE) is nonzero. We study the CE for the Laplacian and two ROM filters: the ROM projection and the ROM differential filter. Furthermore, when the CE is nonzero, we investigate whether it has any significant effect on ROMs that are constructed by using spatial filtering. As numerical tests, we use the Burgers equation with viscosities ν = 10 −1 and ν = 10 −3 and a 2D flow past a circular cylinder at Reynolds numbers Re = 1 and Re = 100. Our investigation shows that: (i) the CE exists; and (ii) the CE has a significant effect on ROM development for low Reynolds numbers, but not so much for higher Reynolds numbers.

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“…Data‐driven ROMs (DD‐ROMs) (eg, sparse identification of nonlinear dynamics and operator inference method) use a fundamentally different strategy: they first postulate a ROM ansatz

\overset{a}{true} = \overset{A}{true} bold-italica + a^{⊤} \overset{B}{true} bold-italica,

and then, they choose the operators

\overset{A}{true}

and

\overset{B}{true}

to minimize the difference between the FOM and ansatz , ie,

\min_{\overset{A}{true}, \overset{B}{true}} {(‖‖ F O M - (\dot{bold-italica} - \hat{A} a - {bold-italica}^{⊤} \hat{B} a))}^{2} .

Both Proj‐ROMs and DD‐ROMs are facing grand challenges : one of the main roadblocks for Proj‐ROMs is that they are not accurate models for the dominant modes. In practice, a corrected Proj‐ROM is generally used instead, ie,

\overset{a}{true} = A bold-italica + a^{⊤} B bold-italica + Correction.

Thus, the ROM closure problem (ie, the modeling of the Correction term in ) needs to be addressed. DD‐ROMs, on the other hand, can be sensitive to noise in the data, since their operators

\overset{A}{true}

and

\overset{B}{true}

are obtained from an inverse problem …”

Section: Introductionmentioning

confidence: 99%

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Mohebujjaman

Rebholz

Iliescu

2018

Numerical Methods in Fluids

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Summary We have recently proposed a data‐driven correction reduced‐order model (DDC‐ROM) framework for the numerical simulation of fluid flows, which can be formally written as follows. The new DDC‐ROM was constructed by using reduced‐order model spatial filtering and data‐driven reduced‐order model closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. In this paper, we propose a physically constrained DDC‐ROM (CDDC‐ROM), which aims at improving the physical accuracy of the DDC‐ROM. The new physical constraints require that the CDDC‐ROM operators satisfy the same type of physical laws (ie, the Correction term's linear component should be dissipative, and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data‐driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC‐ROM and standard DDC‐ROM for a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. To this end, we consider a reproductive regime as well as a predictive (ie, cross‐validation) regime in which we use as little as 50% of the original training data. The numerical investigation clearly shows that the new CDDC‐ROM is significantly more accurate than the DDC‐ROM in both regimes.

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Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Cited by 43 publications

References 29 publications

Continuous data assimilation reduced order models of fluid flow

Continuous data assimilation reduced order models of fluid flow

Commutation error in reduced order modeling of fluid flows

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

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