PINN-DG: Residual neural network methods trained with Finite Elements
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916833804681216 |
|---|---|
| author | Grekas, Georgios Makridakis, Charalambos G. Pryer, Tristan |
| author_facet | Grekas, Georgios Makridakis, Charalambos G. Pryer, Tristan |
| contents | Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical techniques such as finite elements and finite differences. In this paper, we introduce a new class of Physics-Informed Neural Networks (PINNs) trained using discontinuous Galerkin finite element methods. Unlike standard collocation-based PINNs that rely on pointwise gradient evaluations and Monte Carlo quadrature, our approach computes the loss functional using finite element interpolation and integration. This avoids costly pointwise derivative computations, particularly advantageous for elliptic PDEs requiring second-order derivatives, and inherits key stability and accuracy benefits from the finite element framework. We present a convergence analysis based on variational arguments and support our theoretical findings with numerical experiments that demonstrate improved efficiency and robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | PINN-DG: Residual neural network methods trained with Finite Elements Grekas, Georgios Makridakis, Charalambos G. Pryer, Tristan Numerical Analysis Computational Physics 65M15 (Primary), 65M12 (Secondary) Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical techniques such as finite elements and finite differences. In this paper, we introduce a new class of Physics-Informed Neural Networks (PINNs) trained using discontinuous Galerkin finite element methods. Unlike standard collocation-based PINNs that rely on pointwise gradient evaluations and Monte Carlo quadrature, our approach computes the loss functional using finite element interpolation and integration. This avoids costly pointwise derivative computations, particularly advantageous for elliptic PDEs requiring second-order derivatives, and inherits key stability and accuracy benefits from the finite element framework. We present a convergence analysis based on variational arguments and support our theoretical findings with numerical experiments that demonstrate improved efficiency and robustness. |
| title | PINN-DG: Residual neural network methods trained with Finite Elements |
| topic | Numerical Analysis Computational Physics 65M15 (Primary), 65M12 (Secondary) |
| url | https://arxiv.org/abs/2507.03521 |