Numerical Solutions of Burgers Equation by Reduced-Order Modeling Based on Pseudo-Spectral Collocation Method

Seo, Jeong-Kweon; Shin, Bo Sung

doi:10.12941/jksiam.2015.19.123

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…Note that in our work, the benchmark algorithm, which is conceptually the same as the one with the algorithm introduced in Ref. 36 , i.e., a DDM model without pretraining process, was one of the recently proposed algorithms to apply the traditional DDM method calculated by preserving the flux continuity between subdomains to ANN as it is (refer to refs 36 , 37 .).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In 2020, devising a domain decomposition methodology (DDM), Jagtap et al 36 proposed conservative physics-informed neural networks to solve Burgers’ equations, incompressible Navier–Stokes equations, and compressible Euler equations. Among many variations of ANN models, in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Among many variations of ANN models, in Refs. 36 , 37 , the authors implemented PINN-based domain decomposition methodologies and showed their approximations’ effectiveness by hp -refinement. They divided their computational domain into several subdomains and constructed the main loss function by employing weight parameters on terms of boundary/initial conditions, the residual from governing equations and interfaces, and applied their method to forward and inverse problems of several challenging nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

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A pretraining domain decomposition method using artificial neural networks to solve elliptic PDE boundary value problems

Seo

2022

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Developing methods of domain decomposition (DDM) has been widely studied in the field of numerical computation to estimate solutions of partial differential equations (PDEs). Several case studies have also reported that it is feasible to use the domain decomposition approach for the application of artificial neural networks (ANNs) to solve PDEs. In this study, we devised a pretraining scheme called smoothing with a basis reconstruction process on the structure of ANNs and then implemented the classic concept of DDM. The pretraining process that is engaged at the beginning of the training epochs can make the approximation basis become well-posed on the domain so that the quality of the estimated solution is enhanced. We report that such a well-organized pretraining scheme may affect any NN-based PDE solvers as we can speed up the approximation, improve the solution’s smoothness, and so on. Numerical experiments were performed to verify the effectiveness of the proposed DDM method on ANN for estimating solutions of PDEs. Results revealed that this method could be used as a tool for tasks in general machine learning.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A pretraining domain decomposition method using artificial neural networks to solve elliptic PDE boundary value problems

Seo

2022

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Reduced order modeling of the elliptic systems of the frequency-domain method and its error estimation

Seo

2022

Numerical Heat Transfer, Part B: Fundamentals

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Reduced-order modeling using the frequency-domain method for parabolic partial differential equations

Seo

Shin

2023

MATH

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<abstract><p>This paper suggests reduced-order modeling using the Galerkin proper orthogonal decomposition (POD) to find approximate solutions for parabolic partial differential equations. We first transform a parabolic partial differential equation to the frequency-dependent elliptic equations using the Fourier integral transform in time. Such a frequency-domain method enables efficiently implementing a parallel computation to approximate the solutions because the frequency-variable elliptic equations have independent frequencies. Then, we introduce reduced-order modeling to determine approximate solutions of the frequency-variable elliptic equations quickly. A set of snapshots consists of the finite element solutions of the frequency-variable elliptic equations with some selected frequencies. The solutions are approximated using the general basis of the high-dimensional finite element space in a Hilbert space. reduced-order modeling employs the Galerkin POD for the snapshot subspace spanned by a set of snapshots. An orthonormal basis for the snapshot space can be easily computed using the spectral decomposition of the correlation matrix of the snapshots. Additionally, using an appropriate low-order basis of the snapshot space allows approximating the solutions of the frequency-variable elliptic equations quickly, where the approximate solutions are used for the inverse Fourier transforms to determine the approximated solutions in the time variable. Several numerical tests based on the finite element method are presented to asses the efficient performances of the suggested approaches.</p></abstract>

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Numerical Solutions of Burgers Equation by Reduced-Order Modeling Based on Pseudo-Spectral Collocation Method

Cited by 3 publications

References 13 publications

A pretraining domain decomposition method using artificial neural networks to solve elliptic PDE boundary value problems

A pretraining domain decomposition method using artificial neural networks to solve elliptic PDE boundary value problems

Reduced order modeling of the elliptic systems of the frequency-domain method and its error estimation

Reduced-order modeling using the frequency-domain method for parabolic partial differential equations

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