Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN)
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929545468182528 |
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| author | Łoś, Marcin Służalec, Tomasz Maczuga, Paweł Vilkha, Askold Uriarte, Carlos Paszyński, Maciej |
| author_facet | Łoś, Marcin Służalec, Tomasz Maczuga, Paweł Vilkha, Askold Uriarte, Carlos Paszyński, Maciej |
| contents | Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs). Their loss function is founded on a strong residual minimization scheme. Variational Physics-Informed Neural Networks (VPINNs) are their natural extension to weak variational settings. In this context, the recent work of Robust Variational Physics-Informed Neural Networks (RVPINNs) highlights the importance of conveniently translating the norms of the underlying continuum-level spaces to the discrete level. Otherwise, VPINNs might become unrobust, implying that residual minimization might be highly uncorrelated with a desired minimization of the error in the energy norm. However, applying this robustness to VPINNs typically entails dealing with the inverse of a Gram matrix, usually producing slow convergence speeds during training. In this work, we accelerate the implementation of RVPINN, establishing a LU factorization of sparse Gram matrix in a kind of point-collocation scheme with the same spirit as original PINNs. We call out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN). We test our efficient CRVPINN algorithm on Laplace, advection-diffusion, and Stokes problems in two spatial dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02300 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN) Łoś, Marcin Służalec, Tomasz Maczuga, Paweł Vilkha, Askold Uriarte, Carlos Paszyński, Maciej Machine Learning Numerical Analysis 65M99, 68T07 G.1.8; I.2; I.m; G.1.10; J.2 Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs). Their loss function is founded on a strong residual minimization scheme. Variational Physics-Informed Neural Networks (VPINNs) are their natural extension to weak variational settings. In this context, the recent work of Robust Variational Physics-Informed Neural Networks (RVPINNs) highlights the importance of conveniently translating the norms of the underlying continuum-level spaces to the discrete level. Otherwise, VPINNs might become unrobust, implying that residual minimization might be highly uncorrelated with a desired minimization of the error in the energy norm. However, applying this robustness to VPINNs typically entails dealing with the inverse of a Gram matrix, usually producing slow convergence speeds during training. In this work, we accelerate the implementation of RVPINN, establishing a LU factorization of sparse Gram matrix in a kind of point-collocation scheme with the same spirit as original PINNs. We call out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN). We test our efficient CRVPINN algorithm on Laplace, advection-diffusion, and Stokes problems in two spatial dimensions. |
| title | Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN) |
| topic | Machine Learning Numerical Analysis 65M99, 68T07 G.1.8; I.2; I.m; G.1.10; J.2 |
| url | https://arxiv.org/abs/2401.02300 |