hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition

By E. Kharazmi, Zhongqiang Zhang, G. Em Karniadakis. 2020

816 citations recorded by the source corpus.

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 space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the whole computational domain, while the test space contains the piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in accuracy and training cost for several numerical examples of function approximation and solving differential equations.

Subfields: Computational Mechanics, Electronic, Optical and Magnetic Materials, Statistical and Nonlinear Physics

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, methodology): “To further improve the learning performance, the PINN framework has also been extended via incorporating variational/energy formulations of the residual physics loss function [37–40], distributed learning using domain decomposition [41, 42], and uncertainty quantification via variational/Bayesian in”
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2021, methodology): “For all cases, the GCN architecture remains the same, where the dimensions of node vector in hidden graph lays are fixed as [32 , 64 , 128 , 256 , 128 , 64 , 32].”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, background): “Some of the main contributions in this thread are based on: collocation [6–8], variational principle (deep Ritz) [9, 10], and Petrov-Galerkin domain-decomposition [11, 12].”
Parallel Physics-Informed Neural Networks via Domain Decomposition (2021, methodology): “Recently, another domain decomposition method applied to the variational formulation of PINN is proposed by Kharazmi et al. [18] and named it hp-VPINN method.”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “A variational formulation of PINNs based on the Galerkin method (hp-VPINN) was proposed to deal with PDEs with nonsmooth solutions [25].”
Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture (2020, background): “While past work has addressed this with domain decomposition [39] on the weak form of the PDEs, it requires separate basis functions within discretized sub-domains of the geometry leading to a multi-element set-up for learning.”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “Loss function can be constructed using collocation points, weighted residuals derived by the Galerkin-Method (Kharazmi et al, 2019), or energy based.”
Physics-informed neural networks for inverse problems in supersonic flows (2022, methodology): “In recent years physics-informed neural networks (PINNs) [1] emerged as an alternative simple method to solve many problems in computational science and engineering, see, for example [2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 11, 12, 13, 14, 15].”
Extended Physics-informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition based Deep Learning Framework for Nonlinear Partial Differential Equations (2020, background): “[48], where the authors developed the variational PINN framework.”
Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations (2021, background): “Stiller et al. [2020] proposed GatedPINNs, which use an auxiliary neural network to learn the domain decomposition itself, and Kharazmi et al. [2021] use domain decomposition to train variational PINNs, although only when deﬁning the space of test functions.”