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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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering Vol. 374; no. C; p. 113547
Main Authors: Kharazmi, Ehsan, Zhang, Zhongqiang, Karniadakis, George E.M.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.02.2021
Elsevier BV
Elsevier
Subjects:
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
Online Access:Get full text
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Summary: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.
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USDOE
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2020.113547