The discrete inverse conductivity problem solved by the weights of an interpretable neural network
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912174676377600 |
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| author | Beretta, Elena Deng, Maolin Gandolfi, Alberto Jin, Bangti |
| author_facet | Beretta, Elena Deng, Maolin Gandolfi, Alberto Jin, Bangti |
| contents | In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The discrete inverse conductivity problem solved by the weights of an interpretable neural network Beretta, Elena Deng, Maolin Gandolfi, Alberto Jin, Bangti Numerical Analysis Analysis of PDEs In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data. |
| title | The discrete inverse conductivity problem solved by the weights of an interpretable neural network |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2501.00345 |