hp-VPINNs: Variational physics-informed neural networks with domain decomposition

We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto the space of high-order polynomials. The trial space is the space o...

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Vydáno v:	Computer methods in applied mechanics and engineering Ročník 374; číslo C; s. 113547
Hlavní autoři:	Kharazmi, Ehsan, Zhang, Zhongqiang, Karniadakis, George E.M.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.02.2021 Elsevier BV Elsevier
Témata:	Algorithms Approximation Artificial neural networks Automatic differentiation Differential equations Domain decomposition Domain decomposition methods hp-refinement Machine learning Neural networks Optimization Partial differential equations Physics-informed learning Polynomials Variational neural network VPINNs Partial differential equations Variational neural network hp-refinement Physics-informed learning Domain decomposition VPINNs Automatic differentiation
ISSN:	0045-7825, 1879-2138
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Popis
Shrnutí:	We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto the space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the entire computational domain, while the test space contains piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with a local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in both accuracy and training cost for several numerical examples of function approximation and in solving differential equations. •Development of a general framework for hp-variational physics-informed neural networks•Nonlinear approximation of neural networks, projection onto space of high-order polynomials.•Domain decomposition•Comparison with other methods that use neural networks•Local and global approximations with locally/globally defined test functions.•Different loss functions based on the variational form and integration by parts.•Detailed derivation of the hp-VPINN formulation.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 USDOE
ISSN:	0045-7825 1879-2138
DOI:	10.1016/j.cma.2020.113547