A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results
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| Format: | Preprint |
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2024
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| author | Abhishek, Anuj Strauss, Thilo |
| author_facet | Abhishek, Anuj Strauss, Thilo |
| contents | In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography (EIT), where the goal is to recover the conductivity in a medium from boundary current-to-voltage measurements, i.e., the Neumann-to-Dirichlet (N--t--D) operator. We formulate this inverse problem as an operator-learning task, where the aim is to approximate the implicitly defined map from N--t--D operators to admissible conductivities. To this end, we employ a Deep Operator Network (DeepONet) architecture, thereby extending operator learning beyond the classical function-to-function setting to the more challenging operator-to-function regime. We establish a universal approximation theorem that guarantees that such operator-to-function maps can be approximated arbitrarily well by DeepONets. Furthermore, we provide a computational implementation of our approach and compare it against the iteratively regularized Gauss--Newton (IRGN) method. Our results show that the proposed framework yields accurate and robust reconstructions, outperforms the baseline, and demonstrates strong generalization. To our knowledge, this is the first work that combines rigorous approximation-theoretic guarantees with DeepONet-based inversion for EIT, thereby opening a principled and interpretable pathway for use of DeepONets in such inverse problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17182 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results Abhishek, Anuj Strauss, Thilo Machine Learning 35J15, 47H99, 41A65 In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography (EIT), where the goal is to recover the conductivity in a medium from boundary current-to-voltage measurements, i.e., the Neumann-to-Dirichlet (N--t--D) operator. We formulate this inverse problem as an operator-learning task, where the aim is to approximate the implicitly defined map from N--t--D operators to admissible conductivities. To this end, we employ a Deep Operator Network (DeepONet) architecture, thereby extending operator learning beyond the classical function-to-function setting to the more challenging operator-to-function regime. We establish a universal approximation theorem that guarantees that such operator-to-function maps can be approximated arbitrarily well by DeepONets. Furthermore, we provide a computational implementation of our approach and compare it against the iteratively regularized Gauss--Newton (IRGN) method. Our results show that the proposed framework yields accurate and robust reconstructions, outperforms the baseline, and demonstrates strong generalization. To our knowledge, this is the first work that combines rigorous approximation-theoretic guarantees with DeepONet-based inversion for EIT, thereby opening a principled and interpretable pathway for use of DeepONets in such inverse problems. |
| title | A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results |
| topic | Machine Learning 35J15, 47H99, 41A65 |
| url | https://arxiv.org/abs/2407.17182 |