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 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier B.V
01.02.2021
Elsevier BV Elsevier |
| Témata: | |
| ISSN: | 0045-7825, 1879-2138 |
| On-line přístup: | Získat plný text |
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| Abstract | 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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| AbstractList | 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. 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. |
| ArticleNumber | 113547 |
| Author | Karniadakis, George E.M. Kharazmi, Ehsan Zhang, Zhongqiang |
| Author_xml | – sequence: 1 givenname: Ehsan orcidid: 0000-0002-3680-5500 surname: Kharazmi fullname: Kharazmi, Ehsan email: ehsan_kharazmi@brown.edu organization: Division of Applied Mathematics, Brown University, 170 Hope St, Providence, RI 02906, USA – sequence: 2 givenname: Zhongqiang surname: Zhang fullname: Zhang, Zhongqiang organization: Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA – sequence: 3 givenname: George E.M. surname: Karniadakis fullname: Karniadakis, George E.M. organization: Division of Applied Mathematics, Brown University, 170 Hope St, Providence, RI 02906, USA |
| BackLink | https://www.osti.gov/biblio/1776284$$D View this record in Osti.gov |
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| References | Li, Tang, Wu, Liao (b8) 2019 Raissi, Perdikaris, Karniadakis (b3) 2019; 378 Raissi, Perdikaris, Karniadakis (b52) 2017; 348 Novak, Ritter (b40) 1996; 75 Jagtap, Kharazmi, Karniadakis (b37) 2020; 365 E.J. Candès, et al. Compressive sampling, in: Proceedings of the International Congress of Mathematicians, Vol. 3, Madrid, Spain, 2006, pp. 1433–1452. Ohlsson, Yang, Dong, Sastry (b14) 2013 Berg, Nyström (b1) 2018; 317 Liao, Ming (b7) 2019 Sirignano, Spiliopoulos (b23) 2018; 375 Yang, Zhang, Karniadakis (b24) 2018 Jagtap, Kawaguchi, Karniadakis (b30) 2019 Bao, Ye, Zang, Zhou (b35) 2020 Daubechies (b11) 1992 Candès, Wakin (b16) 2008; 25 Mao, Jagtap, Karniadakis (b4) 2020; 360 Tariyal, Majumdar, Singh, Vatsa (b12) 2016 Xu, Tartakovsky, Burghardt, Darve (b58) 2020 E, Yu (b2) 2018; 6 Kharazmi, Zhang, Karniadakis (b25) 2019 Davis (b13) 1994 Tromp, Tape, Liu (b47) 2005; 160 Zang, Bao, Ye, Zhou (b34) 2020 Al-Aradi, Correia, Naiff, Jardim, Saporito (b33) 2019 Khodayi-Mehr, Zavlanos (b26) 2019 van Leeuwen, Herrmann (b51) 2015; 32 Meng, Karniadakis (b55) 2020; 401 Smolyak (b39) 1963; 4 Shin, Darbon, Karniadakis (b60) 2020 Kingma, Ba (b42) 2014 Bradley (b45) 2010 Shin, Zhang, Karniadakis (b61) 2020 Finlayson, Scriven (b22) 1966; 19 Haghighat, Juanes (b32) 2020 Jagtap, Kawaguchi, Karniadakis (b31) 2019 Morokoff, Caflisch (b38) 1995; 122 Khodayi-mehr, Zavlanos (b36) 2018 Kharazmi, Zayernouri (b48) 2019; 80 Raissi, Babaee, Givi (b28) 2019; 4 Mhaskar, Poggio (b20) 2019 Cyr, Gulian, Patel, Perego, Trask (b41) 2019 Cao, Li, Petzold, Serban (b46) 2003; 24 Hangelbroek, Ron (b18) 2010; 259 DeVore (b10) 2009 Karniadakis, Sherwin (b43) 2013 Raissi, Karniadakis (b53) 2018; 357 Khoo, Lu, Ying (b5) 2019; 6 Xu, Darve (b57) 2020 Samaniego, Anitescu, Goswami, Nguyen-Thanh, Guo, Hamdia, Zhuang, Rabczuk (b6) 2020; 362 DeVore (b9) 1998; 7 Mojtabi, Deville (b59) 2015; 107 Mhaskar, Micchelli (b19) 1992; 13 Daubechies, DeVore, Foucart, Hanin, Petrova (b21) 2019 Yang, Meng, Karniadakis (b56) 2020 Allaire (b50) 2015; 836 DeVore, Ron (b17) 2010; 362 Pang, Lu, Karniadakis (b29) 2019; 41 M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, et al. Tensorflow: A system for large-scale machine learning, in: 12th {USENIX} Symposium on Operating Systems Design and Implementation, {OSDI} 16, 2016, pp. 265–283. Plessix (b49) 2006; 167 Lu, Meng, Mao, Karniadakis (b44) 2019 Wang, Teng, Perdikaris (b27) 2020 Ohlsson (10.1016/j.cma.2020.113547_b14) 2013 Mao (10.1016/j.cma.2020.113547_b4) 2020; 360 Daubechies (10.1016/j.cma.2020.113547_b21) 2019 Shin (10.1016/j.cma.2020.113547_b61) 2020 Lu (10.1016/j.cma.2020.113547_b44) 2019 Jagtap (10.1016/j.cma.2020.113547_b31) 2019 Raissi (10.1016/j.cma.2020.113547_b53) 2018; 357 Kingma (10.1016/j.cma.2020.113547_b42) 2014 Khoo (10.1016/j.cma.2020.113547_b5) 2019; 6 Haghighat (10.1016/j.cma.2020.113547_b32) 2020 Smolyak (10.1016/j.cma.2020.113547_b39) 1963; 4 Plessix (10.1016/j.cma.2020.113547_b49) 2006; 167 DeVore (10.1016/j.cma.2020.113547_b9) 1998; 7 Yang (10.1016/j.cma.2020.113547_b56) 2020 Hangelbroek (10.1016/j.cma.2020.113547_b18) 2010; 259 Xu (10.1016/j.cma.2020.113547_b58) 2020 Tariyal (10.1016/j.cma.2020.113547_b12) 2016 Pang (10.1016/j.cma.2020.113547_b29) 2019; 41 Mhaskar (10.1016/j.cma.2020.113547_b20) 2019 Cyr (10.1016/j.cma.2020.113547_b41) 2019 10.1016/j.cma.2020.113547_b15 Zang (10.1016/j.cma.2020.113547_b34) 2020 Karniadakis (10.1016/j.cma.2020.113547_b43) 2013 10.1016/j.cma.2020.113547_b54 Raissi (10.1016/j.cma.2020.113547_b28) 2019; 4 Xu (10.1016/j.cma.2020.113547_b57) 2020 Cao (10.1016/j.cma.2020.113547_b46) 2003; 24 Li (10.1016/j.cma.2020.113547_b8) 2019 Novak (10.1016/j.cma.2020.113547_b40) 1996; 75 E (10.1016/j.cma.2020.113547_b2) 2018; 6 Yang (10.1016/j.cma.2020.113547_b24) 2018 Khodayi-mehr (10.1016/j.cma.2020.113547_b36) 2018 Davis (10.1016/j.cma.2020.113547_b13) 1994 Al-Aradi (10.1016/j.cma.2020.113547_b33) 2019 Bao (10.1016/j.cma.2020.113547_b35) 2020 Sirignano (10.1016/j.cma.2020.113547_b23) 2018; 375 Morokoff (10.1016/j.cma.2020.113547_b38) 1995; 122 Kharazmi (10.1016/j.cma.2020.113547_b48) 2019; 80 Tromp (10.1016/j.cma.2020.113547_b47) 2005; 160 Shin (10.1016/j.cma.2020.113547_b60) 2020 DeVore (10.1016/j.cma.2020.113547_b10) 2009 Allaire (10.1016/j.cma.2020.113547_b50) 2015; 836 Daubechies (10.1016/j.cma.2020.113547_b11) 1992 Raissi (10.1016/j.cma.2020.113547_b52) 2017; 348 Meng (10.1016/j.cma.2020.113547_b55) 2020; 401 Jagtap (10.1016/j.cma.2020.113547_b30) 2019 Kharazmi (10.1016/j.cma.2020.113547_b25) 2019 Wang (10.1016/j.cma.2020.113547_b27) 2020 Bradley (10.1016/j.cma.2020.113547_b45) 2010 DeVore (10.1016/j.cma.2020.113547_b17) 2010; 362 Khodayi-Mehr (10.1016/j.cma.2020.113547_b26) 2019 Jagtap (10.1016/j.cma.2020.113547_b37) 2020; 365 Berg (10.1016/j.cma.2020.113547_b1) 2018; 317 Finlayson (10.1016/j.cma.2020.113547_b22) 1966; 19 van Leeuwen (10.1016/j.cma.2020.113547_b51) 2015; 32 Candès (10.1016/j.cma.2020.113547_b16) 2008; 25 Samaniego (10.1016/j.cma.2020.113547_b6) 2020; 362 Liao (10.1016/j.cma.2020.113547_b7) 2019 Mhaskar (10.1016/j.cma.2020.113547_b19) 1992; 13 Mojtabi (10.1016/j.cma.2020.113547_b59) 2015; 107 Raissi (10.1016/j.cma.2020.113547_b3) 2019; 378 |
| References_xml | – year: 2019 ident: b7 article-title: Deep Nitsche method: Deep Ritz method with essential boundary conditions – year: 2020 ident: b61 article-title: Error estimates of residual minimization using neural networks for linear PDEs – volume: 7 start-page: 51 year: 1998 end-page: 150 ident: b9 article-title: Nonlinear approximation publication-title: Acta Numer. – volume: 19 start-page: 735 year: 1966 end-page: 748 ident: b22 article-title: The method of weighted residuals—A review publication-title: Appl. Mech. Rev. – year: 2018 ident: b24 article-title: Physics-informed generative adversarial networks for stochastic differential equations – year: 2020 ident: b32 article-title: SciANN: A Keras wrapper for scientific computations and physics-informed deep learning using artificial neural networks – year: 2019 ident: b44 article-title: DeepXDE: A deep learning library for solving differential equations – year: 2019 ident: b8 article-title: D3M: A deep domain decomposition method for partial differential equations – volume: 360 year: 2020 ident: b4 article-title: Physics-informed neural networks for high-speed flows publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2020 ident: b56 article-title: B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data – volume: 4 start-page: 240 year: 1963 end-page: 243 ident: b39 article-title: Quadrature and interpolation formulas for tensor products of certain classes of functions publication-title: Sov. Math. Dokl. – year: 2018 ident: b36 article-title: Deep learning for robotic mass transport cloaking – volume: 378 start-page: 686 year: 2019 end-page: 707 ident: b3 article-title: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations publication-title: J. Comput. Phys. – volume: 317 start-page: 28 year: 2018 end-page: 41 ident: b1 article-title: A unified deep artificial neural network approach to partial differential equations in complex geometries publication-title: Neurocomputing – volume: 365 year: 2020 ident: b37 article-title: Conservative physics-informed neural networks on discrete domains for conservation laws publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 41 start-page: A2603 year: 2019 end-page: A2626 ident: b29 article-title: FPINNs: Fractional physics-informed neural networks publication-title: SIAM J. Sci. Comput. – year: 2020 ident: b57 article-title: Physics constrained learning for data-driven inverse modeling from sparse observations – year: 2020 ident: b58 article-title: Inverse modeling of viscoelasticity materials using physics constrained learning – volume: 75 start-page: 79 year: 1996 end-page: 97 ident: b40 article-title: High dimensional integration of smooth functions over cubes publication-title: Numer. Math. – volume: 13 start-page: 350 year: 1992 end-page: 373 ident: b19 article-title: Approximation by superposition of sigmoidal and radial basis functions publication-title: Adv. in Appl. Math. – volume: 362 year: 2020 ident: b6 article-title: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications publication-title: Comput. Methods Appl. Mech. Engrg. – reference: E.J. Candès, et al. Compressive sampling, in: Proceedings of the International Congress of Mathematicians, Vol. 3, Madrid, Spain, 2006, pp. 1433–1452. – reference: M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, et al. Tensorflow: A system for large-scale machine learning, in: 12th {USENIX} Symposium on Operating Systems Design and Implementation, {OSDI} 16, 2016, pp. 265–283. – year: 2016 ident: b12 article-title: Greedy deep dictionary learning – year: 2014 ident: b42 article-title: Adam: A method for stochastic optimization – year: 2020 ident: b34 article-title: Weak adversarial networks for high-dimensional partial differential equations publication-title: J. Comput. Phys. – year: 2019 ident: b33 article-title: Applications of the deep Galerkin method to solving partial integro-differential and Hamilton-Jacobi-Bellman equations – year: 2020 ident: b35 article-title: Numerical solution of inverse problems by weak adversarial networks – start-page: 169 year: 2009 end-page: 201 ident: b10 article-title: Nonlinear approximation and its applications publication-title: Multiscale, Nonlinear and Adaptive Approximation – volume: 348 start-page: 683 year: 2017 end-page: 693 ident: b52 article-title: Machine learning of linear differential equations using Gaussian processes publication-title: J. Comput. Phys. – volume: 6 start-page: 1 year: 2018 end-page: 12 ident: b2 article-title: The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems publication-title: Commun. Math. Stat. – year: 1992 ident: b11 article-title: Ten Lectures on Wavelets, Vol. 61 – volume: 80 start-page: 110 year: 2019 end-page: 140 ident: b48 article-title: Fractional sensitivity equation method: Application to fractional model construction publication-title: J. Sci. Comput. – volume: 836 start-page: 33 year: 2015 end-page: 36 ident: b50 article-title: A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes publication-title: Ing. Automob. – volume: 107 start-page: 189 year: 2015 end-page: 195 ident: b59 article-title: One-dimensional linear advection–diffusion equation: Analytical and finite element solutions publication-title: Comput. & Fluids – volume: 6 start-page: 1 year: 2019 ident: b5 article-title: Solving for high-dimensional committor functions using artificial neural networks publication-title: Res. Math. Sci. – year: 1994 ident: b13 article-title: Adaptive Nonlinear Approximations – year: 2013 ident: b43 article-title: Spectral/ – volume: 32 year: 2015 ident: b51 article-title: A penalty method for PDE-constrained optimization in inverse problems publication-title: Inverse Problems – volume: 160 start-page: 195 year: 2005 end-page: 216 ident: b47 article-title: Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels publication-title: Geophys. J. Int. – start-page: 109 year: 2019 end-page: 136 ident: b30 article-title: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks publication-title: J. Comput. Phys. – year: 2019 ident: b41 article-title: Robust training and initialization of deep neural networks: An adaptive basis viewpoint – volume: 375 start-page: 1339 year: 2018 end-page: 1364 ident: b23 article-title: DGM: A deep learning algorithm for solving partial differential equations publication-title: J. Comput. Phys. – volume: 122 start-page: 218 year: 1995 end-page: 230 ident: b38 article-title: Quasi-Monte Carlo integration publication-title: J. Comput. Phys. – year: 2020 ident: b60 article-title: On the convergence and generalization of physics informed neural networks – year: 2019 ident: b20 article-title: Function approximation by deep networks – volume: 4 year: 2019 ident: b28 article-title: Deep learning of turbulent scalar mixing publication-title: Phys. Rev. Fluids – volume: 357 start-page: 125 year: 2018 end-page: 141 ident: b53 article-title: Hidden physics models: Machine learning of nonlinear partial differential equations publication-title: J. Comput. Phys. – volume: 401 year: 2020 ident: b55 article-title: A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems publication-title: J. Comput. Phys. – volume: 167 start-page: 495 year: 2006 end-page: 503 ident: b49 article-title: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications publication-title: Geophys. J. Int. – year: 2020 ident: b27 article-title: Understanding and mitigating gradient pathologies in physics-informed neural networks – year: 2019 ident: b26 article-title: VarNet: Variational neural networks for the solution of partial differential equations – volume: 25 start-page: 21 year: 2008 end-page: 30 ident: b16 article-title: An introduction to compressive sampling [a sensing/sampling paradigm that goes against the common knowledge in data acquisition] publication-title: IEEE Signal Process. Mag. – volume: 259 start-page: 203 year: 2010 end-page: 219 ident: b18 article-title: Nonlinear approximation using Gaussian kernels publication-title: J. Funct. Anal. – start-page: 115 year: 2013 end-page: 119 ident: b14 article-title: Nonlinear basis pursuit publication-title: 2013 Asilomar Conference on Signals, Systems and Computers – year: 2019 ident: b21 article-title: Nonlinear approximation and (deep) ReLU networks – year: 2019 ident: b31 article-title: Locally adaptive activation functions with slope recovery term for deep and physics-informed neural networks – volume: 362 start-page: 6205 year: 2010 end-page: 6229 ident: b17 article-title: Approximation using scattered shifts of a multivariate function publication-title: Trans. Amer. Math. Soc. – year: 2010 ident: b45 article-title: PDE-Constrained Optimization and the Adjoint Method – year: 2019 ident: b25 article-title: Variational physics-informed neural networks for solving partial differential equations – volume: 24 start-page: 1076 year: 2003 end-page: 1089 ident: b46 article-title: Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution publication-title: SIAM J. Sci. Comput. – year: 2019 ident: 10.1016/j.cma.2020.113547_b21 – year: 2019 ident: 10.1016/j.cma.2020.113547_b8 – year: 2014 ident: 10.1016/j.cma.2020.113547_b42 – volume: 362 start-page: 6205 issue: 12 year: 2010 ident: 10.1016/j.cma.2020.113547_b17 article-title: Approximation using scattered shifts of a multivariate function publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-2010-05070-6 – year: 2020 ident: 10.1016/j.cma.2020.113547_b57 – year: 2020 ident: 10.1016/j.cma.2020.113547_b34 article-title: Weak adversarial networks for high-dimensional partial differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109409 – year: 2019 ident: 10.1016/j.cma.2020.113547_b20 – year: 2019 ident: 10.1016/j.cma.2020.113547_b41 – year: 2019 ident: 10.1016/j.cma.2020.113547_b44 – year: 2018 ident: 10.1016/j.cma.2020.113547_b24 – volume: 75 start-page: 79 issue: 1 year: 1996 ident: 10.1016/j.cma.2020.113547_b40 article-title: High dimensional integration of smooth functions over cubes publication-title: Numer. Math. doi: 10.1007/s002110050231 – year: 2020 ident: 10.1016/j.cma.2020.113547_b60 – year: 2019 ident: 10.1016/j.cma.2020.113547_b26 – year: 2010 ident: 10.1016/j.cma.2020.113547_b45 – volume: 167 start-page: 495 issue: 2 year: 2006 ident: 10.1016/j.cma.2020.113547_b49 article-title: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications publication-title: Geophys. J. Int. doi: 10.1111/j.1365-246X.2006.02978.x – volume: 41 start-page: A2603 issue: 4 year: 2019 ident: 10.1016/j.cma.2020.113547_b29 article-title: FPINNs: Fractional physics-informed neural networks publication-title: SIAM J. Sci. Comput. doi: 10.1137/18M1229845 – volume: 32 issue: 1 year: 2015 ident: 10.1016/j.cma.2020.113547_b51 article-title: A penalty method for PDE-constrained optimization in inverse problems publication-title: Inverse Problems – year: 2020 ident: 10.1016/j.cma.2020.113547_b32 – volume: 378 start-page: 686 year: 2019 ident: 10.1016/j.cma.2020.113547_b3 article-title: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.10.045 – year: 2020 ident: 10.1016/j.cma.2020.113547_b27 – volume: 365 year: 2020 ident: 10.1016/j.cma.2020.113547_b37 article-title: Conservative physics-informed neural networks on discrete domains for conservation laws publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2020.113028 – year: 2020 ident: 10.1016/j.cma.2020.113547_b58 – year: 2020 ident: 10.1016/j.cma.2020.113547_b35 – volume: 160 start-page: 195 issue: 1 year: 2005 ident: 10.1016/j.cma.2020.113547_b47 article-title: Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels publication-title: Geophys. J. Int. doi: 10.1111/j.1365-246X.2004.02453.x – year: 2019 ident: 10.1016/j.cma.2020.113547_b7 – year: 2018 ident: 10.1016/j.cma.2020.113547_b36 – year: 2019 ident: 10.1016/j.cma.2020.113547_b33 – volume: 259 start-page: 203 issue: 1 year: 2010 ident: 10.1016/j.cma.2020.113547_b18 article-title: Nonlinear approximation using Gaussian kernels publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2010.02.001 – year: 2019 ident: 10.1016/j.cma.2020.113547_b31 – volume: 122 start-page: 218 issue: 2 year: 1995 ident: 10.1016/j.cma.2020.113547_b38 article-title: Quasi-Monte Carlo integration publication-title: J. Comput. Phys. doi: 10.1006/jcph.1995.1209 – volume: 348 start-page: 683 year: 2017 ident: 10.1016/j.cma.2020.113547_b52 article-title: Machine learning of linear differential equations using Gaussian processes publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2017.07.050 – start-page: 115 year: 2013 ident: 10.1016/j.cma.2020.113547_b14 article-title: Nonlinear basis pursuit – volume: 360 year: 2020 ident: 10.1016/j.cma.2020.113547_b4 article-title: Physics-informed neural networks for high-speed flows publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112789 – volume: 401 year: 2020 ident: 10.1016/j.cma.2020.113547_b55 article-title: A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.109020 – volume: 13 start-page: 350 issue: 3 year: 1992 ident: 10.1016/j.cma.2020.113547_b19 article-title: Approximation by superposition of sigmoidal and radial basis functions publication-title: Adv. in Appl. Math. doi: 10.1016/0196-8858(92)90016-P – volume: 80 start-page: 110 issue: 1 year: 2019 ident: 10.1016/j.cma.2020.113547_b48 article-title: Fractional sensitivity equation method: Application to fractional model construction publication-title: J. Sci. Comput. doi: 10.1007/s10915-019-00935-0 – volume: 362 year: 2020 ident: 10.1016/j.cma.2020.113547_b6 article-title: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112790 – year: 2020 ident: 10.1016/j.cma.2020.113547_b56 – volume: 6 start-page: 1 issue: 1 year: 2018 ident: 10.1016/j.cma.2020.113547_b2 article-title: The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems publication-title: Commun. Math. Stat. doi: 10.1007/s40304-018-0127-z – ident: 10.1016/j.cma.2020.113547_b15 doi: 10.4171/022-3/69 – volume: 317 start-page: 28 year: 2018 ident: 10.1016/j.cma.2020.113547_b1 article-title: A unified deep artificial neural network approach to partial differential equations in complex geometries publication-title: Neurocomputing doi: 10.1016/j.neucom.2018.06.056 – year: 2013 ident: 10.1016/j.cma.2020.113547_b43 – volume: 24 start-page: 1076 issue: 3 year: 2003 ident: 10.1016/j.cma.2020.113547_b46 article-title: Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution publication-title: SIAM J. Sci. Comput. doi: 10.1137/S1064827501380630 – volume: 357 start-page: 125 year: 2018 ident: 10.1016/j.cma.2020.113547_b53 article-title: Hidden physics models: Machine learning of nonlinear partial differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2017.11.039 – year: 2019 ident: 10.1016/j.cma.2020.113547_b25 – volume: 19 start-page: 735 issue: 9 year: 1966 ident: 10.1016/j.cma.2020.113547_b22 article-title: The method of weighted residuals—A review publication-title: Appl. Mech. Rev. – start-page: 169 year: 2009 ident: 10.1016/j.cma.2020.113547_b10 article-title: Nonlinear approximation and its applications – volume: 375 start-page: 1339 year: 2018 ident: 10.1016/j.cma.2020.113547_b23 article-title: DGM: A deep learning algorithm for solving partial differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.08.029 – volume: 4 start-page: 240 year: 1963 ident: 10.1016/j.cma.2020.113547_b39 article-title: Quadrature and interpolation formulas for tensor products of certain classes of functions publication-title: Sov. Math. Dokl. – volume: 107 start-page: 189 year: 2015 ident: 10.1016/j.cma.2020.113547_b59 article-title: One-dimensional linear advection–diffusion equation: Analytical and finite element solutions publication-title: Comput. & Fluids doi: 10.1016/j.compfluid.2014.11.006 – year: 1992 ident: 10.1016/j.cma.2020.113547_b11 – volume: 25 start-page: 21 issue: 2 year: 2008 ident: 10.1016/j.cma.2020.113547_b16 article-title: An introduction to compressive sampling [a sensing/sampling paradigm that goes against the common knowledge in data acquisition] publication-title: IEEE Signal Process. Mag. – volume: 7 start-page: 51 year: 1998 ident: 10.1016/j.cma.2020.113547_b9 article-title: Nonlinear approximation publication-title: Acta Numer. doi: 10.1017/S0962492900002816 – start-page: 109 year: 2019 ident: 10.1016/j.cma.2020.113547_b30 article-title: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks publication-title: J. Comput. Phys. – year: 2020 ident: 10.1016/j.cma.2020.113547_b61 – year: 1994 ident: 10.1016/j.cma.2020.113547_b13 – volume: 4 issue: 12 year: 2019 ident: 10.1016/j.cma.2020.113547_b28 article-title: Deep learning of turbulent scalar mixing publication-title: Phys. Rev. Fluids doi: 10.1103/PhysRevFluids.4.124501 – volume: 836 start-page: 33 year: 2015 ident: 10.1016/j.cma.2020.113547_b50 article-title: A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes publication-title: Ing. Automob. – volume: 6 start-page: 1 issue: 1 year: 2019 ident: 10.1016/j.cma.2020.113547_b5 article-title: Solving for high-dimensional committor functions using artificial neural networks publication-title: Res. Math. Sci. doi: 10.1007/s40687-018-0160-2 – ident: 10.1016/j.cma.2020.113547_b54 – year: 2016 ident: 10.1016/j.cma.2020.113547_b12 |
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| SubjectTerms | 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 |
| Title | hp-VPINNs: Variational physics-informed neural networks with domain decomposition |
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