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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2308.16910 |
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| _version_ | 1866916147118473216 |
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| author | Rojas, Sergio Maczuga, Paweł Muñoz-Matute, Judit Pardo, David Paszynski, Maciej |
| author_facet | Rojas, Sergio Maczuga, Paweł Muñoz-Matute, Judit Pardo, David Paszynski, Maciej |
| contents | We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov-Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN's loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection-diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16910 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Robust Variational Physics-Informed Neural Networks Rojas, Sergio Maczuga, Paweł Muñoz-Matute, Judit Pardo, David Paszynski, Maciej Numerical Analysis 65N12, 65N15, 65N22, 65N30, 65N50 We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov-Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN's loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection-diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp. |
| title | Robust Variational Physics-Informed Neural Networks |
| topic | Numerical Analysis 65N12, 65N15, 65N22, 65N30, 65N50 |
| url | https://arxiv.org/abs/2308.16910 |