The Adjoint Petrov-Galerkin Method for Non-Linear Model Reduction

Parish, Eric; Wentland, Christopher R.; Duraisamy, Karthik

doi:10.48550/arxiv.1810.03455

Cited by 3 publications

(7 citation statements)

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“…In the present study, we assume that the structure of the ROM closure model function g is similar to the structure of the Galerkin model function f and we utilize a least squares approach to determine the shape of g. We emphasize that, without loss of generality, our DD-VMS-ROM framework can be formulated by utilizing a supervised machine learning approach [57,64,65,66], a topic that we would like to explore in the future. Finally, we intend to explore the extension of the new DD-VMS-ROM to the Petrov-Galerkin framework [9,10,20,52].…”

Section: Discussionmentioning

confidence: 99%

“…Another ROM closure strategy that is related to the VMS-ROM framework is the adjoint Petrov-Galerkin method [52] (see [9,10,20] for related work), which is based on the Mori-Zwanzig (MZ) formalism [19,38]. In the MZ-ROM approach, the ROM closure model is represented by a memory term that depends on the temporal history of the resolved scales.…”

Section: Introductionmentioning

confidence: 99%

“…The mem-ory term is approximated to construct effective ROM closure models and, therefore, practical ROMs. The main difference between the adjoint Petrov-Galerkin method proposed in [52], and the new DD-VMS-ROM is the tool used to define the ROM closure term: The former uses a statistical tool (i.e., the MZ formalism), whereas the latter utilizes a spectral-like projection (i.e., the ROM projection).…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we note that the VMS-ROM framework proposed herein belongs to the wider class of hybrid physical/data-driven ROMs, in which data-driven modeling is used to model only the missing information (i.e., the ROM closure term) in ROMs constructed from first principles (i.e., from a Galerkin projection of the underlying equations); see, e.g., [3,11,13,18,25,40,48,52,75].…”

Section: Introductionmentioning

confidence: 99%

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Data-Driven Variational Multiscale Reduced Order Models

Mou,

Koc,

San

et al. 2020

Preprint

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We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of the 1D Burgers equation and the 2D flow past a circular cylinder at Reynolds numbers Re = 100, Re = 500, and Re = 1000. The numerical results show that the data-driven VMS-ROM is

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Data-Driven Variational Multiscale Reduced Order Models

Mou,

Koc,

San

et al. 2020

Preprint

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“…Linear basis ROMs have achieved considerable success in complex problems such as turbulent flows [12,13,14] and combustion instabilities [15,16]. However, despite the choice of optimal test spaces afforded by Petrov-Galerkin methods [17], and closure modeling [18,19,20,21], the linear trial space becomes ineffective in advection-dominated problems and many multiscale problems in general. While the associated challenges can be addressed to a certain degree by using adaptive basis [22,23,24], some of the fundamental challenges persist.…”

Section: Introductionmentioning

confidence: 99%

Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics

Xu,

Duraisamy

2019

Preprint

Self Cite

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A data-driven framework is proposed for the predictive modeling of complex spatio-temporal dynamics, leveraging nested non-linear manifolds. Three levels of neural networks are used, with the goal of predicting the future state of a system of interest in a parametric setting. A convolutional autoencoder is used as the top level to encode the high dimensional input data along spatial dimensions into a sequence of latent variables. A temporal convolutional autoencoder serves as the second level, which further encodes the output sequence from the first level along the temporal dimension, and outputs a set of latent variables that encapsulate the spatio-temporal evolution of the dynamics. A fully connected network is used as the third level to learn the mapping between these latent variables and the global parameters from training data, and predict them for new parameters. For future state predictions, the second level uses a temporal convolutional network to predict subsequent steps of the output sequence from the top level. Latent variables at the bottom-most level are decoded to obtain the dynamics in physical space at new global parameters and/or at a future time. The framework is evaluated on a range of problems involving discontinuities, wave propagation, strong transients, and coherent structures. The sensitivity of the results to different modeling choices is assessed. The results suggest that given adequate data and careful training, effective data-driven predictive models can be constructed. Perspectives are provided on the present approach and its place in the landscape of model reduction.

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