Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems

Da Long, Zhitong Xu, Qiwei Yuan, Yin Yang, Shandian Zhe
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:3043-3051, 2025.

Abstract

Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible Fourier Neural Operator (iFNO) for jointly tackling the forward and inverse problems. We developed a series of invertible Fourier blocks in the latent channel space to share the model parameters, exchange the information, and mutually regularize the learning for the bi-directional tasks. We integrated a variational auto-encoder to capture the intrinsic structures within the input space and to enable posterior inference so as to mitigate challenges of illposedness, data shortage, noises that are common in inverse problems. We proposed a three-step process to combine the invertible blocks and the VAE component for effective training. The evaluations on seven benchmark forward and inverse tasks have demonstrated the advantages of our approach. The code is available at \url{https://github.com/BayesianAIGroup/iFNO.}

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-long25a, title = {Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems}, author = {Long, Da and Xu, Zhitong and Yuan, Qiwei and Yang, Yin and Zhe, Shandian}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {3043--3051}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/long25a/long25a.pdf}, url = {https://proceedings.mlr.press/v258/long25a.html}, abstract = {Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible Fourier Neural Operator (iFNO) for jointly tackling the forward and inverse problems. We developed a series of invertible Fourier blocks in the latent channel space to share the model parameters, exchange the information, and mutually regularize the learning for the bi-directional tasks. We integrated a variational auto-encoder to capture the intrinsic structures within the input space and to enable posterior inference so as to mitigate challenges of illposedness, data shortage, noises that are common in inverse problems. We proposed a three-step process to combine the invertible blocks and the VAE component for effective training. The evaluations on seven benchmark forward and inverse tasks have demonstrated the advantages of our approach. The code is available at \url{https://github.com/BayesianAIGroup/iFNO.}} }
Endnote
%0 Conference Paper %T Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems %A Da Long %A Zhitong Xu %A Qiwei Yuan %A Yin Yang %A Shandian Zhe %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-long25a %I PMLR %P 3043--3051 %U https://proceedings.mlr.press/v258/long25a.html %V 258 %X Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible Fourier Neural Operator (iFNO) for jointly tackling the forward and inverse problems. We developed a series of invertible Fourier blocks in the latent channel space to share the model parameters, exchange the information, and mutually regularize the learning for the bi-directional tasks. We integrated a variational auto-encoder to capture the intrinsic structures within the input space and to enable posterior inference so as to mitigate challenges of illposedness, data shortage, noises that are common in inverse problems. We proposed a three-step process to combine the invertible blocks and the VAE component for effective training. The evaluations on seven benchmark forward and inverse tasks have demonstrated the advantages of our approach. The code is available at \url{https://github.com/BayesianAIGroup/iFNO.}
APA
Long, D., Xu, Z., Yuan, Q., Yang, Y. & Zhe, S.. (2025). Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:3043-3051 Available from https://proceedings.mlr.press/v258/long25a.html.

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