Neural Inverse Operators for Solving PDE Inverse Problems

Roberto Molinaro, Yunan Yang, Björn Engquist, Siddhartha Mishra
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:25105-25139, 2023.

Abstract

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-molinaro23a, title = {Neural Inverse Operators for Solving {PDE} Inverse Problems}, author = {Molinaro, Roberto and Yang, Yunan and Engquist, Bj\"{o}rn and Mishra, Siddhartha}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {25105--25139}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/molinaro23a/molinaro23a.pdf}, url = {https://proceedings.mlr.press/v202/molinaro23a.html}, abstract = {A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.} }
Endnote
%0 Conference Paper %T Neural Inverse Operators for Solving PDE Inverse Problems %A Roberto Molinaro %A Yunan Yang %A Björn Engquist %A Siddhartha Mishra %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-molinaro23a %I PMLR %P 25105--25139 %U https://proceedings.mlr.press/v202/molinaro23a.html %V 202 %X A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.
APA
Molinaro, R., Yang, Y., Engquist, B. & Mishra, S.. (2023). Neural Inverse Operators for Solving PDE Inverse Problems. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:25105-25139 Available from https://proceedings.mlr.press/v202/molinaro23a.html.

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