Adaptive Fourier Decomposition-guided Neural Operator Design for Inverse PDE Problems

Zeyuan Song, Xiaocong Zhen, Zheyu Jiang
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:6467-6491, 2026.

Abstract

Inverse problems, which are generally ill-posed, aim to identify the unknown parameters of a physical system from the observations of its output. A large class of inverse problems for partial differential equations (PDEs) are only well-defined as mappings from operators to functions. However, existing operator learning frameworks either do not explicitly account for the underlying operator space or solve the inverse problems in a {Hilbert} space. Meanwhile, it has been shown that a Banach space setting for the parameter space would be closer to reality for a wide range of problems. Driven by this, we introduce AFDONet-inv, a novel neural operator solver whose design is rigorously guided by adaptive {Fourier} decomposition (AFD) theory, to solve inverse problems for PDEs in a Banach space. Each component of AFDONet-inv’s architecture, including primal and dual nets, latent-to-RKBS (reproducing kernel Banach space) network, and dynamic convolutional kernel network (CKN), has a corresponding component in the AFD operation in Banach space. This way, AFDONet-inv is mathematically explainable and grounded in the AFD theory and possesses several desirable properties. Extensive experiments demonstrate that AFDONet-inv outperforms state-of-the-art inverse {PDE} solvers in terms of solution accuracy.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-song26a, title = {Adaptive {Fourier} Decomposition-guided Neural Operator Design for Inverse {PDE} Problems}, author = {Song, Zeyuan and Zhen, Xiaocong and Jiang, Zheyu}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {6467--6491}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/song26a/song26a.pdf}, url = {https://proceedings.mlr.press/v337/song26a.html}, abstract = {Inverse problems, which are generally ill-posed, aim to identify the unknown parameters of a physical system from the observations of its output. A large class of inverse problems for partial differential equations (PDEs) are only well-defined as mappings from operators to functions. However, existing operator learning frameworks either do not explicitly account for the underlying operator space or solve the inverse problems in a {Hilbert} space. Meanwhile, it has been shown that a Banach space setting for the parameter space would be closer to reality for a wide range of problems. Driven by this, we introduce AFDONet-inv, a novel neural operator solver whose design is rigorously guided by adaptive {Fourier} decomposition (AFD) theory, to solve inverse problems for PDEs in a Banach space. Each component of AFDONet-inv’s architecture, including primal and dual nets, latent-to-RKBS (reproducing kernel Banach space) network, and dynamic convolutional kernel network (CKN), has a corresponding component in the AFD operation in Banach space. This way, AFDONet-inv is mathematically explainable and grounded in the AFD theory and possesses several desirable properties. Extensive experiments demonstrate that AFDONet-inv outperforms state-of-the-art inverse {PDE} solvers in terms of solution accuracy.} }
Endnote
%0 Conference Paper %T Adaptive Fourier Decomposition-guided Neural Operator Design for Inverse PDE Problems %A Zeyuan Song %A Xiaocong Zhen %A Zheyu Jiang %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-song26a %I PMLR %P 6467--6491 %U https://proceedings.mlr.press/v337/song26a.html %V 337 %X Inverse problems, which are generally ill-posed, aim to identify the unknown parameters of a physical system from the observations of its output. A large class of inverse problems for partial differential equations (PDEs) are only well-defined as mappings from operators to functions. However, existing operator learning frameworks either do not explicitly account for the underlying operator space or solve the inverse problems in a {Hilbert} space. Meanwhile, it has been shown that a Banach space setting for the parameter space would be closer to reality for a wide range of problems. Driven by this, we introduce AFDONet-inv, a novel neural operator solver whose design is rigorously guided by adaptive {Fourier} decomposition (AFD) theory, to solve inverse problems for PDEs in a Banach space. Each component of AFDONet-inv’s architecture, including primal and dual nets, latent-to-RKBS (reproducing kernel Banach space) network, and dynamic convolutional kernel network (CKN), has a corresponding component in the AFD operation in Banach space. This way, AFDONet-inv is mathematically explainable and grounded in the AFD theory and possesses several desirable properties. Extensive experiments demonstrate that AFDONet-inv outperforms state-of-the-art inverse {PDE} solvers in terms of solution accuracy.
APA
Song, Z., Zhen, X. & Jiang, Z.. (2026). Adaptive Fourier Decomposition-guided Neural Operator Design for Inverse PDE Problems. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:6467-6491 Available from https://proceedings.mlr.press/v337/song26a.html.

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