A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909389820002304 |
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| author | Makridakis, Charalambos G. Pim, Aaron Pryer, Tristan |
| author_facet | Makridakis, Charalambos G. Pim, Aaron Pryer, Tristan |
| contents | We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems.
We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08702 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods Makridakis, Charalambos G. Pim, Aaron Pryer, Tristan Numerical Analysis 65N12, 65K10 G.1.8 We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems. We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains. |
| title | A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods |
| topic | Numerical Analysis 65N12, 65K10 G.1.8 |
| url | https://arxiv.org/abs/2411.08702 |