GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs
Physics-Informed Neural Network (PINN) has proven itself a powerful tool to obtain the numerical solutions of nonlinear partial differential equations (PDEs) leveraging the expressivity of deep neural networks and the computing power of modern heterogeneous hardware. However, its training is still t...
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| Veröffentlicht in: | Finite elements in analysis and design Jg. 228; S. 104047 |
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01.01.2024
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| Abstract | Physics-Informed Neural Network (PINN) has proven itself a powerful tool to obtain the numerical solutions of nonlinear partial differential equations (PDEs) leveraging the expressivity of deep neural networks and the computing power of modern heterogeneous hardware. However, its training is still time-consuming, especially in the multi-query and real-time simulation settings, and its parameterization often overly excessive. In this paper, we propose the Generative Pre-Trained PINN (GPT-PINN) to mitigate both challenges in the setting of parametric PDEs. GPT-PINN represents a brand-new meta-learning paradigm for parametric systems. As a network of networks, its outer-/meta-network is hyper-reduced with only one hidden layer having significantly reduced number of neurons. Moreover, its activation function at each hidden neuron is a (full) PINN pre-trained at a judiciously selected system configuration. The meta-network adaptively “learns” the parametric dependence of the system and “grows” this hidden layer one neuron at a time. In the end, by encompassing a very small number of networks trained at this set of adaptively-selected parameter values, the meta-network is capable of generating surrogate solutions for the parametric system across the entire parameter domain accurately and efficiently.
•The use of whole pre-trained networks as the activation functions of another network•A network of networks that is adaptively generated•The adoption of the training loss of the meta-network as an error indicator•The adaptation of the classical greedy algorithm to the neural network setting |
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| AbstractList | Physics-Informed Neural Network (PINN) has proven itself a powerful tool to obtain the numerical solutions of nonlinear partial differential equations (PDEs) leveraging the expressivity of deep neural networks and the computing power of modern heterogeneous hardware. However, its training is still time-consuming, especially in the multi-query and real-time simulation settings, and its parameterization often overly excessive. In this paper, we propose the Generative Pre-Trained PINN (GPT-PINN) to mitigate both challenges in the setting of parametric PDEs. GPT-PINN represents a brand-new meta-learning paradigm for parametric systems. As a network of networks, its outer-/meta-network is hyper-reduced with only one hidden layer having significantly reduced number of neurons. Moreover, its activation function at each hidden neuron is a (full) PINN pre-trained at a judiciously selected system configuration. The meta-network adaptively “learns” the parametric dependence of the system and “grows” this hidden layer one neuron at a time. In the end, by encompassing a very small number of networks trained at this set of adaptively-selected parameter values, the meta-network is capable of generating surrogate solutions for the parametric system across the entire parameter domain accurately and efficiently.
•The use of whole pre-trained networks as the activation functions of another network•A network of networks that is adaptively generated•The adoption of the training loss of the meta-network as an error indicator•The adaptation of the classical greedy algorithm to the neural network setting |
| ArticleNumber | 104047 |
| Author | Koohy, Shawn Chen, Yanlai |
| Author_xml | – sequence: 1 givenname: Yanlai orcidid: 0000-0002-7460-8313 surname: Chen fullname: Chen, Yanlai email: yanlai.chen@umassd.edu – sequence: 2 givenname: Shawn surname: Koohy fullname: Koohy, Shawn email: skoohy@umassd.edu |
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| Cites_doi | 10.1016/j.jcp.2021.110666 10.1016/j.cma.2021.114474 10.1137/1.9781611974829.ch2 10.1038/s42256-021-00302-5 10.1016/j.jcp.2018.10.045 10.1115/1.1448332 10.1002/nme.6544 10.1016/j.jcp.2017.11.039 10.1137/0910047 10.1007/s11831-008-9019-9 10.1088/1361-6463/acb604 10.1137/100795772 10.1137/16M1059904 10.1137/130932715 10.1016/0045-7949(79)90012-9 10.1002/zamm.19950750709 10.1016/j.jcp.2022.111121 10.1016/j.camwa.2018.11.032 10.2514/3.50778 10.1016/j.jcp.2021.110545 |
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| Keywords | Physics-Informed Neural Networks Parametric PDEs Non-intrusive learning model order reduction Meta-learning Network of networks |
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| References | Patera, Rozza (b1) 2007 Raissi (b28) 2018; 19 Wight, Zhao (b48) 2020 Raissi, Karniadakis (b32) 2018; 357 Benner, Gugercin, Willcox (b7) 2015; 57 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. Wang, Sankaran, Perdikaris (b13) 2022 Rozza, Huynh, Patera (b4) 2008; 15 Nagy (b21) 1979; 10 Xu, Zhao, Zhao (b51) 2022 Quarteroni, Manzoni, Negri (b2) 2016; vol. 92 Binev, Cohen, Dahmen, Devore, Petrova, Wojtaszczyk (b15) 2011 Pinkus (b16) 1985 E, Yu (b26) 2018 Miyanawala, Jaiman (b35) 2017 Penwarden, Zhe, Narayan, Kirby (b39) 2021 Raissi, Perdikaris, Karniadakis (b8) 2019; 378 Chen, Jiang, Narayan (b22) 2019; 77 Han, Jentzen, E (b37) 2018 Lu, Jin, Karniadakis (b34) 2019 Zhang, You, Gao, Yu, Lee, Yu (b46) 2023 Raissi, Yazdani, Karniadakis (b33) 2020 Chen, Wang, Hesthaven, Zhang (b47) 2021; 446 Ryck, Mishra, Ray (b31) 2019 McClenny, Braga-Neto (b50) 2020 Baydin, Pearlmutter, Radul, Siskind (b9) 2017; 18 Flennerhag, Moreno, Lawrence, Damianou (b41) 2018 Qin, Beatson, Oktay, McGreivy, Adams (b42) 2022 Goodfellow, Bengio, Courville (b38) 2016 Peterson (b19) 1989; 10 Maday, Patera, Rovas (b5) 2002; vol. 31 Perdikaris, Raissi, Damianou, Lawrence, Karniadakis (b25) 2017 Paszke, Gross, Chintala, Chanan, Yang, DeVito, Lin, Desmaison, Antiga, Lerer (b10) 2017 Revels, Lubin, Papamarkou (b12) 2016 Finn, Abbeel, Levine (b40) 2017 Lagaris, Likas, Fotiadis (b24) 1998 Prud’homme, Rovas, Veroy, Maday, Patera, Turinici (b17) 2002; 124 Jiang, Chen (b14) 2020; 121 Mattey, Ghosh (b49) 2022; 390 Psaros, Kawaguchi, Karniadakis (b44) 2022; 458 Zhong, Wu, Wang (b45) 2023; 56 Barrett, Reddien (b20) 1995; 75 Noor, Peters (b18) 1980; 18 Khoo, Lu, Ying (b27) 2019 Cybenko (b29) 1989 Chen, Gottlieb, Ji, Maday (b23) 2021; 444 Hesthaven, Rozza, Stamm (b3) 2016 Yarotsky (b30) 2017 Lu, Jin, Pang, Zhang, Karniadakis (b43) 2021; 3 E, Han, Jentzen (b36) 2017 Welper (b52) 2017; 39 B. Haasdonk, Chapter 2: Reduced Basis Methods for Parametrized PDEsZ̃A Tutorial Introduction for Stationary and Instationary Problems, pp. 65–136. Ryck (10.1016/j.finel.2023.104047_b31) 2019 McClenny (10.1016/j.finel.2023.104047_b50) 2020 Quarteroni (10.1016/j.finel.2023.104047_b2) 2016; vol. 92 Noor (10.1016/j.finel.2023.104047_b18) 1980; 18 Chen (10.1016/j.finel.2023.104047_b22) 2019; 77 Finn (10.1016/j.finel.2023.104047_b40) 2017 E (10.1016/j.finel.2023.104047_b36) 2017 E (10.1016/j.finel.2023.104047_b26) 2018 Hesthaven (10.1016/j.finel.2023.104047_b3) 2016 Barrett (10.1016/j.finel.2023.104047_b20) 1995; 75 Raissi (10.1016/j.finel.2023.104047_b32) 2018; 357 Binev (10.1016/j.finel.2023.104047_b15) 2011 Welper (10.1016/j.finel.2023.104047_b52) 2017; 39 Prud’homme (10.1016/j.finel.2023.104047_b17) 2002; 124 Mattey (10.1016/j.finel.2023.104047_b49) 2022; 390 Wight (10.1016/j.finel.2023.104047_b48) 2020 Lagaris (10.1016/j.finel.2023.104047_b24) 1998 Han (10.1016/j.finel.2023.104047_b37) 2018 Wang (10.1016/j.finel.2023.104047_b13) 2022 Peterson (10.1016/j.finel.2023.104047_b19) 1989; 10 Psaros (10.1016/j.finel.2023.104047_b44) 2022; 458 Revels (10.1016/j.finel.2023.104047_b12) 2016 Pinkus (10.1016/j.finel.2023.104047_b16) 1985 Jiang (10.1016/j.finel.2023.104047_b14) 2020; 121 Lu (10.1016/j.finel.2023.104047_b34) 2019 Rozza (10.1016/j.finel.2023.104047_b4) 2008; 15 10.1016/j.finel.2023.104047_b11 Benner (10.1016/j.finel.2023.104047_b7) 2015; 57 Raissi (10.1016/j.finel.2023.104047_b33) 2020 Cybenko (10.1016/j.finel.2023.104047_b29) 1989 Nagy (10.1016/j.finel.2023.104047_b21) 1979; 10 Raissi (10.1016/j.finel.2023.104047_b28) 2018; 19 Paszke (10.1016/j.finel.2023.104047_b10) 2017 Goodfellow (10.1016/j.finel.2023.104047_b38) 2016 Zhang (10.1016/j.finel.2023.104047_b46) 2023 Miyanawala (10.1016/j.finel.2023.104047_b35) 2017 Maday (10.1016/j.finel.2023.104047_b5) 2002; vol. 31 Chen (10.1016/j.finel.2023.104047_b47) 2021; 446 Khoo (10.1016/j.finel.2023.104047_b27) 2019 Chen (10.1016/j.finel.2023.104047_b23) 2021; 444 Raissi (10.1016/j.finel.2023.104047_b8) 2019; 378 Patera (10.1016/j.finel.2023.104047_b1) 2007 Yarotsky (10.1016/j.finel.2023.104047_b30) 2017 Lu (10.1016/j.finel.2023.104047_b43) 2021; 3 Xu (10.1016/j.finel.2023.104047_b51) 2022 Baydin (10.1016/j.finel.2023.104047_b9) 2017; 18 10.1016/j.finel.2023.104047_b6 Perdikaris (10.1016/j.finel.2023.104047_b25) 2017 Penwarden (10.1016/j.finel.2023.104047_b39) 2021 Qin (10.1016/j.finel.2023.104047_b42) 2022 Flennerhag (10.1016/j.finel.2023.104047_b41) 2018 Zhong (10.1016/j.finel.2023.104047_b45) 2023; 56 |
| References_xml | – volume: 390 year: 2022 ident: b49 article-title: A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 57 start-page: 483 year: 2015 end-page: 531 ident: b7 article-title: A survey of projection-based model reduction methods for parametric dynamical systems publication-title: SIAM Rev. – year: 2017 ident: b35 article-title: An efficient deep learning technique for the navier-stokes equations: Application to unsteady wake flow dynamics – year: 2022 ident: b13 article-title: Respecting causality is all you need for training physics-informed neural networks – year: 2022 ident: b42 article-title: Meta-pde: Learning to solve pdes quickly without a mesh – volume: 121 start-page: 5426 year: 2020 end-page: 5445 ident: b14 article-title: Adaptive greedy algorithms based on parameter-domain decomposition and reconstruction for the reduced basis method publication-title: Internat. J. Numer. Methods Engrg. – volume: 446 year: 2021 ident: b47 article-title: Physics-informed machine learning for reduced-order modeling of nonlinear problems publication-title: J. Comput. Phys. – year: 2021 ident: b39 article-title: Physics-informed neural networks (pinns) for parameterized pdes: a metalearning approach – year: 2017 ident: b25 article-title: Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling, 473 – year: 2019 ident: b34 article-title: Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators – year: 2016 ident: b3 publication-title: Certified Reduced Basis Methods for Parametrized Partial Differential Equations – volume: 124 start-page: 70 year: 2002 end-page: 80 ident: b17 article-title: Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods publication-title: J. Fluids Eng. – year: 2018 ident: b41 article-title: Transferring knowledge across learning processes – volume: 10 start-page: 777 year: 1989 end-page: 786 ident: b19 article-title: The reduced basis method for incompressible viscous flow calculations publication-title: SIAM J. Sci. Stat. Comput. – start-page: 1 year: 2018 end-page: 12 ident: b26 article-title: The deep ritz method: A deep learning-based numerical algorithm for solving variational problems, 6 – volume: 56 year: 2023 ident: b45 article-title: Accelerating physics-informed neural network based 1d arc simulation by meta learning publication-title: J. Phys. D: Appl. Phys. – volume: 378 start-page: 686 year: 2019 end-page: 707 ident: b8 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: 444 year: 2021 ident: b23 article-title: An eim-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation publication-title: J. Comput. Phys. – start-page: 303 year: 1989 end-page: 314 ident: b29 article-title: Approximation by superpositions of a sigmoidal function, 2 – start-page: 349 year: 2017 end-page: 380 ident: b36 article-title: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, 5 – 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: 2019 ident: b27 article-title: Solving for high-dimensional committor functions using artificial neural networks, 6 – volume: 3 start-page: 218 year: 2021 end-page: 229 ident: b43 article-title: Learning nonlinear operators via deeponet based on the universal approximation theorem of operators publication-title: Nat. Mach. Intell. – start-page: 1457 year: 2011 end-page: 1472 ident: b15 article-title: Convergence rates for greedy algorithms in reduced basis methods publication-title: SIAM J. Math. Anal. – volume: 357 start-page: 125 year: 2018 end-page: 141 ident: b32 article-title: Hidden physics models: Machine learning of nonlinear partial differential equations publication-title: J. Comput. Phys. – year: 2007 ident: b1 article-title: Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations – year: 2019 ident: b31 article-title: On the approximation of rough functions with deep neural networks – start-page: 8505 year: 2018 end-page: 8510 ident: b37 article-title: Solving high-dimensional partial differential equations using deep learning, 115 – year: 2020 ident: b48 article-title: Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks – volume: 10 start-page: 683 year: 1979 end-page: 688 ident: b21 article-title: Modal representation of geometrically nonlinear behaviour by the finite element method publication-title: Comput. Struct. – start-page: 987 year: 1998 end-page: 1000 ident: b24 article-title: Artificial neural networks for solving ordinary and partial differential equations, 9 – year: 2022 ident: b51 article-title: Numerical approximations of the Allen-Cahn-Ohta-Kawasaki (ACOK) equation with modified physics informed neural networks (pinns) – year: 2023 ident: b46 article-title: Metano: How to transfer your knowledge on learning hidden physics – start-page: 1126 year: 2017 end-page: 1135 ident: b40 article-title: Model-agnostic meta-learning for fast adaptation of deep networks publication-title: International Conference on Machine Learning – volume: 18 start-page: 455 year: 1980 end-page: 462 ident: b18 article-title: Reduced basis technique for nonlinear analysis of structures publication-title: AIAA J. – volume: 77 start-page: 1963 year: 2019 end-page: 1979 ident: b22 article-title: A robust error estimator and a residual-free error indicator for reduced basis methods publication-title: Comput. Math. Appl. – volume: 75 start-page: 543 year: 1995 end-page: 549 ident: b20 article-title: On the reduced basis method publication-title: Z. Angew. Math. Mech. – year: 1985 ident: b16 article-title: N-Widths in Approximation Theory – volume: vol. 31 start-page: 533 year: 2002 end-page: 569 ident: b5 article-title: A blackbox reduced-basis output bound method for noncoercive linear problems publication-title: Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, XIV (Paris, 1997/1998) – start-page: 103 year: 2017 end-page: 114 ident: b30 article-title: Error bounds for approximations with deep relu networks, 94 – volume: 18 start-page: 5595 year: 2017 end-page: 5637 ident: b9 article-title: Automatic differentiation in machine learning: a survey publication-title: J. Mach. Learn. Res. – reference: B. Haasdonk, Chapter 2: Reduced Basis Methods for Parametrized PDEsZ̃A Tutorial Introduction for Stationary and Instationary Problems, pp. 65–136. – volume: 458 year: 2022 ident: b44 article-title: Meta-learning pinn loss functions publication-title: J. Comput. Phys. – year: 2017 ident: b10 article-title: Automatic differentiation in pytorch – year: 2020 ident: b50 article-title: Self-adaptive physics-informed neural networks using a soft attention mechanism – start-page: 1026 year: 2020 end-page: 1030 ident: b33 article-title: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, 367 – volume: 19 start-page: 25:1 year: 2018 end-page: 25:24 ident: b28 article-title: Deep hidden physics models: Deep learning of nonlinear partial differential equations publication-title: J. Mach. Learn. Res. – volume: vol. 92 year: 2016 ident: b2 publication-title: Reduced Basis Methods for Partial Differential Equations – volume: 39 start-page: A1225 year: 2017 end-page: A1250 ident: b52 article-title: Interpolation of functions with parameter dependent jumps by transformed snapshots publication-title: SIAM J. Sci. Comput. – volume: 15 start-page: 229 year: 2008 end-page: 275 ident: b4 article-title: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics publication-title: Arch. Comput. Methods Eng. – year: 2016 ident: b38 article-title: Deep Learning – year: 2016 ident: b12 article-title: Forward-mode automatic differentiation in Julia – year: 2017 ident: 10.1016/j.finel.2023.104047_b10 – volume: 446 year: 2021 ident: 10.1016/j.finel.2023.104047_b47 article-title: Physics-informed machine learning for reduced-order modeling of nonlinear problems publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2021.110666 – start-page: 349 year: 2017 ident: 10.1016/j.finel.2023.104047_b36 – volume: 390 year: 2022 ident: 10.1016/j.finel.2023.104047_b49 article-title: A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2021.114474 – year: 2016 ident: 10.1016/j.finel.2023.104047_b12 – year: 2019 ident: 10.1016/j.finel.2023.104047_b27 – start-page: 103 year: 2017 ident: 10.1016/j.finel.2023.104047_b30 – ident: 10.1016/j.finel.2023.104047_b6 doi: 10.1137/1.9781611974829.ch2 – volume: 3 start-page: 218 issue: 3 year: 2021 ident: 10.1016/j.finel.2023.104047_b43 article-title: Learning nonlinear operators via deeponet based on the universal approximation theorem of operators publication-title: Nat. Mach. Intell. doi: 10.1038/s42256-021-00302-5 – start-page: 987 year: 1998 ident: 10.1016/j.finel.2023.104047_b24 – volume: 378 start-page: 686 year: 2019 ident: 10.1016/j.finel.2023.104047_b8 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 – volume: 124 start-page: 70 issue: 1 year: 2002 ident: 10.1016/j.finel.2023.104047_b17 article-title: Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods publication-title: J. Fluids Eng. doi: 10.1115/1.1448332 – volume: 19 start-page: 25:1 year: 2018 ident: 10.1016/j.finel.2023.104047_b28 article-title: Deep hidden physics models: Deep learning of nonlinear partial differential equations publication-title: J. Mach. Learn. Res. – year: 2020 ident: 10.1016/j.finel.2023.104047_b48 – volume: 121 start-page: 5426 issue: 23 year: 2020 ident: 10.1016/j.finel.2023.104047_b14 article-title: Adaptive greedy algorithms based on parameter-domain decomposition and reconstruction for the reduced basis method publication-title: Internat. J. Numer. Methods Engrg. doi: 10.1002/nme.6544 – start-page: 303 year: 1989 ident: 10.1016/j.finel.2023.104047_b29 – volume: 357 start-page: 125 year: 2018 ident: 10.1016/j.finel.2023.104047_b32 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: 2022 ident: 10.1016/j.finel.2023.104047_b42 – volume: 10 start-page: 777 issue: 4 year: 1989 ident: 10.1016/j.finel.2023.104047_b19 article-title: The reduced basis method for incompressible viscous flow calculations publication-title: SIAM J. Sci. Stat. Comput. doi: 10.1137/0910047 – volume: 15 start-page: 229 issue: 3 year: 2008 ident: 10.1016/j.finel.2023.104047_b4 article-title: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics publication-title: Arch. Comput. Methods Eng. doi: 10.1007/s11831-008-9019-9 – volume: 56 issue: 7 year: 2023 ident: 10.1016/j.finel.2023.104047_b45 article-title: Accelerating physics-informed neural network based 1d arc simulation by meta learning publication-title: J. Phys. D: Appl. Phys. doi: 10.1088/1361-6463/acb604 – year: 2017 ident: 10.1016/j.finel.2023.104047_b35 – year: 2021 ident: 10.1016/j.finel.2023.104047_b39 – year: 2018 ident: 10.1016/j.finel.2023.104047_b41 – year: 2019 ident: 10.1016/j.finel.2023.104047_b31 – start-page: 1 year: 2018 ident: 10.1016/j.finel.2023.104047_b26 – year: 2022 ident: 10.1016/j.finel.2023.104047_b13 – start-page: 1457 year: 2011 ident: 10.1016/j.finel.2023.104047_b15 article-title: Convergence rates for greedy algorithms in reduced basis methods publication-title: SIAM J. Math. Anal. doi: 10.1137/100795772 – year: 2023 ident: 10.1016/j.finel.2023.104047_b46 – year: 2022 ident: 10.1016/j.finel.2023.104047_b51 – start-page: 1126 year: 2017 ident: 10.1016/j.finel.2023.104047_b40 article-title: Model-agnostic meta-learning for fast adaptation of deep networks – year: 2020 ident: 10.1016/j.finel.2023.104047_b50 – volume: 39 start-page: A1225 issue: 4 year: 2017 ident: 10.1016/j.finel.2023.104047_b52 article-title: Interpolation of functions with parameter dependent jumps by transformed snapshots publication-title: SIAM J. Sci. Comput. doi: 10.1137/16M1059904 – year: 2016 ident: 10.1016/j.finel.2023.104047_b3 – year: 2019 ident: 10.1016/j.finel.2023.104047_b34 – volume: 57 start-page: 483 issue: 4 year: 2015 ident: 10.1016/j.finel.2023.104047_b7 article-title: A survey of projection-based model reduction methods for parametric dynamical systems publication-title: SIAM Rev. doi: 10.1137/130932715 – volume: vol. 31 start-page: 533 year: 2002 ident: 10.1016/j.finel.2023.104047_b5 article-title: A blackbox reduced-basis output bound method for noncoercive linear problems – volume: 10 start-page: 683 year: 1979 ident: 10.1016/j.finel.2023.104047_b21 article-title: Modal representation of geometrically nonlinear behaviour by the finite element method publication-title: Comput. Struct. doi: 10.1016/0045-7949(79)90012-9 – start-page: 8505 year: 2018 ident: 10.1016/j.finel.2023.104047_b37 – volume: 75 start-page: 543 issue: 7 year: 1995 ident: 10.1016/j.finel.2023.104047_b20 article-title: On the reduced basis method publication-title: Z. Angew. Math. Mech. doi: 10.1002/zamm.19950750709 – year: 2007 ident: 10.1016/j.finel.2023.104047_b1 – year: 2017 ident: 10.1016/j.finel.2023.104047_b25 – year: 2016 ident: 10.1016/j.finel.2023.104047_b38 – volume: 458 year: 2022 ident: 10.1016/j.finel.2023.104047_b44 article-title: Meta-learning pinn loss functions publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2022.111121 – volume: vol. 92 year: 2016 ident: 10.1016/j.finel.2023.104047_b2 – ident: 10.1016/j.finel.2023.104047_b11 – volume: 18 start-page: 5595 issue: 1 year: 2017 ident: 10.1016/j.finel.2023.104047_b9 article-title: Automatic differentiation in machine learning: a survey publication-title: J. Mach. Learn. Res. – year: 1985 ident: 10.1016/j.finel.2023.104047_b16 – volume: 77 start-page: 1963 year: 2019 ident: 10.1016/j.finel.2023.104047_b22 article-title: A robust error estimator and a residual-free error indicator for reduced basis methods publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2018.11.032 – volume: 18 start-page: 455 issue: 4 year: 1980 ident: 10.1016/j.finel.2023.104047_b18 article-title: Reduced basis technique for nonlinear analysis of structures publication-title: AIAA J. doi: 10.2514/3.50778 – start-page: 1026 year: 2020 ident: 10.1016/j.finel.2023.104047_b33 – volume: 444 year: 2021 ident: 10.1016/j.finel.2023.104047_b23 article-title: An eim-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2021.110545 |
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