Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning

Nikolaj T. Mücke; Sander M. Bohté; Cornelis W. Oosterlee

doi:10.1016/j.jocs.2021.101408

Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning

Nikolaj T. Mücke^*
, Sander M. Bohté
, Cornelis W. Oosterlee

^*Corresponding author for this work

Centrum voor Wiskunde en Informatica
extern

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: a nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. To show the variety of problems the ROM can handle, the methodology is demonstrated on the advection equation, and the flow past a cylinder problem modeled by the incompressible Navier–Stokes equations.

Original language	English
Article number	101408
Journal	Journal of Computational Science
Volume	53
DOIs	https://doi.org/10.1016/j.jocs.2021.101408
Publication status	Published - Jul 2021

Bibliographical note

Funding Information:
This work is supported by the Dutch National Science Foundation NWO under the grant number 629.002.213 , which is a cooperative projects with IISC Bangalore and Shell Research as project partners. The authors furthermore acknowledge fruitful discussions with Dr. B. Sanderse. Lastly, the authors thank Peter Piontek for proofreading.

Publisher Copyright:
© 2021 The Author(s)

Funding

This work is supported by the Dutch National Science Foundation NWO under the grant number 629.002.213 , which is a cooperative projects with IISC Bangalore and Shell Research as project partners. The authors furthermore acknowledge fruitful discussions with Dr. B. Sanderse. Lastly, the authors thank Peter Piontek for proofreading.

Keywords

Deep learning
Parameterized PDEs
Reduced order modeling
Spatio-temporal dynamics

Access to Document

10.1016/j.jocs.2021.101408Licence: CC BY

1-s2.0-S1877750321000934-mainFinal published version, 7.51 MBLicence: CC BY

Cite this

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title = "Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning",

abstract = "We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: a nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. To show the variety of problems the ROM can handle, the methodology is demonstrated on the advection equation, and the flow past a cylinder problem modeled by the incompressible Navier–Stokes equations.",

keywords = "Deep learning, Parameterized PDEs, Reduced order modeling, Spatio-temporal dynamics",

author = "M{\"u}cke, \{Nikolaj T.\} and Boht{\'e}, \{Sander M.\} and Oosterlee, \{Cornelis W.\}",

note = "Funding Information: This work is supported by the Dutch National Science Foundation NWO under the grant number 629.002.213 , which is a cooperative projects with IISC Bangalore and Shell Research as project partners. The authors furthermore acknowledge fruitful discussions with Dr. B. Sanderse. Lastly, the authors thank Peter Piontek for proofreading. Publisher Copyright: {\textcopyright} 2021 The Author(s)",

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language = "English",

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AU - Mücke, Nikolaj T.

AU - Bohté, Sander M.

AU - Oosterlee, Cornelis W.

N1 - Funding Information: This work is supported by the Dutch National Science Foundation NWO under the grant number 629.002.213 , which is a cooperative projects with IISC Bangalore and Shell Research as project partners. The authors furthermore acknowledge fruitful discussions with Dr. B. Sanderse. Lastly, the authors thank Peter Piontek for proofreading. Publisher Copyright: © 2021 The Author(s)

PY - 2021/7

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N2 - We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: a nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. To show the variety of problems the ROM can handle, the methodology is demonstrated on the advection equation, and the flow past a cylinder problem modeled by the incompressible Navier–Stokes equations.

AB - We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: a nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. To show the variety of problems the ROM can handle, the methodology is demonstrated on the advection equation, and the flow past a cylinder problem modeled by the incompressible Navier–Stokes equations.

KW - Deep learning

KW - Parameterized PDEs

KW - Reduced order modeling

KW - Spatio-temporal dynamics

UR - https://www.scopus.com/pages/publications/85108799696

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DO - 10.1016/j.jocs.2021.101408

M3 - Article

AN - SCOPUS:85108799696

SN - 1877-7503

VL - 53

JO - Journal of Computational Science

JF - Journal of Computational Science

M1 - 101408

ER -

Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning

Abstract

Bibliographical note

Funding

Keywords

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