A posteriori certification for neural network approximations to PDEs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911589826822144 |
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| author | Ernst, Lewin Rekatsinas, Nikolaos Urban, Karsten |
| author_facet | Ernst, Lewin Rekatsinas, Nikolaos Urban, Karsten |
| contents | We propose rigorous lower and upper error bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler domains, which are either embedded or enveloping the original domain, enabling the use of fast numerical solvers. The resulting bounds control the error in the natural norm induced by a well-posed variational formulation, require only minimal regularity assumptions, and thus remain applicable on complex geometries. The framework is detailed for elliptic as well as parabolic problems. Numerical experiments demonstrate the good quantitative behaviour of the derived upper and lower error bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20336 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A posteriori certification for neural network approximations to PDEs Ernst, Lewin Rekatsinas, Nikolaos Urban, Karsten Numerical Analysis 35J20, 65M15, 68T07 We propose rigorous lower and upper error bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler domains, which are either embedded or enveloping the original domain, enabling the use of fast numerical solvers. The resulting bounds control the error in the natural norm induced by a well-posed variational formulation, require only minimal regularity assumptions, and thus remain applicable on complex geometries. The framework is detailed for elliptic as well as parabolic problems. Numerical experiments demonstrate the good quantitative behaviour of the derived upper and lower error bounds. |
| title | A posteriori certification for neural network approximations to PDEs |
| topic | Numerical Analysis 35J20, 65M15, 68T07 |
| url | https://arxiv.org/abs/2502.20336 |