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    Randomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations

    Source: Journal of Engineering Mechanics:;2024:;Volume ( 150 ):;issue: 004::page 04024010-1
    Author:
    Yong Shang
    ,
    Fei Wang
    DOI: 10.1061/JENMDT.EMENG-7463
    Publisher: ASCE
    Abstract: We develop randomized neural networks (RNNs) with Petrov–Galerkin (RNN-PG) methods to solve linear elasticity and Navier–Stokes equations. RNN-PGs use the Petrov–Galerkin variational framework, where the solution is approximated by randomized neural networks and the test functions are piecewise polynomials. Unlike conventional neural networks, the parameters of the hidden layers of randomized neural networks are fixed randomly while the parameters of the output layer are determined by the least-squares method, which can effectively approximate the solution. We also develop mixed RNN-PG (M-RNN-PG) methods for linear elasticity problems to ensure symmetry of the stress tensor and avoid locking effects. For the Stokes problem, we present various M-RNN-PG methods that enforce the divergence-free constraint by different techniques. For the Navier–Stokes equations, we propose a space-time M-RNN-PG that uses Picard or Newton iteration to deal with the nonlinear term. Using several examples, we compare RNN-PG methods with the finite-element method, the mixed discontinuous Galerkin method, and the physics-informed neural network. The numerical results demonstrate that RNN-PG methods achieve higher accuracy and efficiency.
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      Randomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations

    URI
    http://yetl.yabesh.ir/yetl1/handle/yetl/4297539
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    • Journal of Engineering Mechanics

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    contributor authorYong Shang
    contributor authorFei Wang
    date accessioned2024-04-27T22:48:11Z
    date available2024-04-27T22:48:11Z
    date issued2024/04/01
    identifier other10.1061-JENMDT.EMENG-7463.pdf
    identifier urihttp://yetl.yabesh.ir/yetl1/handle/yetl/4297539
    description abstractWe develop randomized neural networks (RNNs) with Petrov–Galerkin (RNN-PG) methods to solve linear elasticity and Navier–Stokes equations. RNN-PGs use the Petrov–Galerkin variational framework, where the solution is approximated by randomized neural networks and the test functions are piecewise polynomials. Unlike conventional neural networks, the parameters of the hidden layers of randomized neural networks are fixed randomly while the parameters of the output layer are determined by the least-squares method, which can effectively approximate the solution. We also develop mixed RNN-PG (M-RNN-PG) methods for linear elasticity problems to ensure symmetry of the stress tensor and avoid locking effects. For the Stokes problem, we present various M-RNN-PG methods that enforce the divergence-free constraint by different techniques. For the Navier–Stokes equations, we propose a space-time M-RNN-PG that uses Picard or Newton iteration to deal with the nonlinear term. Using several examples, we compare RNN-PG methods with the finite-element method, the mixed discontinuous Galerkin method, and the physics-informed neural network. The numerical results demonstrate that RNN-PG methods achieve higher accuracy and efficiency.
    publisherASCE
    titleRandomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations
    typeJournal Article
    journal volume150
    journal issue4
    journal titleJournal of Engineering Mechanics
    identifier doi10.1061/JENMDT.EMENG-7463
    journal fristpage04024010-1
    journal lastpage04024010-14
    page14
    treeJournal of Engineering Mechanics:;2024:;Volume ( 150 ):;issue: 004
    contenttypeFulltext
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    DSpace software copyright © 2002-2015  DuraSpace
    نرم افزار کتابخانه دیجیتال "دی اسپیس" فارسی شده توسط یابش برای کتابخانه های ایرانی | تماس با یابش
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