Trustworthy AI in numerics: On verification algorithms for neural network-based PDE solvers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908568770314240 |
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| author | Haugen, Emil Stepanenko, Alexei Hansen, Anders C. |
| author_facet | Haugen, Emil Stepanenko, Alexei Hansen, Anders C. |
| contents | We present new algorithms for a posteriori verification of neural networks (NNs) approximating solutions to PDEs. These verification algorithms compute accurate estimates of $L^p$ norms of NNs and their derivatives. When combined with residual bounds for specific PDEs, the algorithms provide guarantees of $\eps$-accuracy (in a suitable norm) with respect to the true, but unknown, solution of the PDE -- for arbitrary $\eps >0$. In particular, if the NN fails to meet the desired accuracy, our algorithms will detect that and reject it, whereas any NN that passes the verification algorithms is certified to be $\eps$-accurate. This framework enables trustworthy algorithms for NN-based PDE solvers, regardless of how the NN is initially computed. Such a posteriori verification is essential, since a priori error bounds in general cannot guarantee the accuracy of computed solutions, due to algorithmic undecidability of the optimization problems used to train NNs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trustworthy AI in numerics: On verification algorithms for neural network-based PDE solvers Haugen, Emil Stepanenko, Alexei Hansen, Anders C. Numerical Analysis 65M15, 68T07 (Primary), 26D10, 41A55, 65D32 (Secondary) We present new algorithms for a posteriori verification of neural networks (NNs) approximating solutions to PDEs. These verification algorithms compute accurate estimates of $L^p$ norms of NNs and their derivatives. When combined with residual bounds for specific PDEs, the algorithms provide guarantees of $\eps$-accuracy (in a suitable norm) with respect to the true, but unknown, solution of the PDE -- for arbitrary $\eps >0$. In particular, if the NN fails to meet the desired accuracy, our algorithms will detect that and reject it, whereas any NN that passes the verification algorithms is certified to be $\eps$-accurate. This framework enables trustworthy algorithms for NN-based PDE solvers, regardless of how the NN is initially computed. Such a posteriori verification is essential, since a priori error bounds in general cannot guarantee the accuracy of computed solutions, due to algorithmic undecidability of the optimization problems used to train NNs. |
| title | Trustworthy AI in numerics: On verification algorithms for neural network-based PDE solvers |
| topic | Numerical Analysis 65M15, 68T07 (Primary), 26D10, 41A55, 65D32 (Secondary) |
| url | https://arxiv.org/abs/2509.26122 |