Euclidean Fourier Neural Operators
Abstract
Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces as they are, by construction, independent of the grid resolution at which they are trained and evaluated. However, FNOs are not independent of the periodic domain they are applied to: their discrete spectral weights are indexed by integer Fourier mode numbers, which correspond to physical wavevectors. When applied to a different domain, the same trained weights act at different wavevectors, and the FNO silently represents a different operator. This makes FNOs unsuitable for tasks where transfer across domains is crucial. We propose Euclidean Fourier neural operators~(EFNOs) as a domain-independent alternative to FNOs. By parameterizing the spectral kernel as a continuous function of the physical wavevector, the EFNO can learn operators that act consistently across periodic domains of varying shape and size. We evaluate the EFNO on a simple heat equation and on a practically relevant materials science task of learning exchange-correlation potentials across different crystal structures, and demonstrate that the EFNO is able to generalize to unseen grid sizes and domains.
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BibTeX
@misc{bosch2026euclidean,
title = {{Euclidean Fourier Neural Operators}},
author = {Nathanael Bosch and Niklas Frederik Schmitz and Michael F. Herbst},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.28425},
eprint = {2608.28425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28425},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Euclidean Fourier Neural Operators
AU - Nathanael Bosch
AU - Niklas Frederik Schmitz
AU - Michael F. Herbst
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2608.28425
UR - https://arxiv.org/abs/2608.28425
AB - Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces as they are, by construction, independent of the grid resolution at which they are trained and evaluated. However, FNOs are not independent of the periodic domain they are applied to: their discrete spectral weights are indexed by integer Fourier mode numbers, which correspond to physical wavevectors. When applied to a different domain, the same trained weights act at different wavevectors, and the FNO silently represents a different operator. This makes FNOs unsuitable for tasks where transfer across domains is crucial. We propose Euclidean Fourier neural operators~(EFNOs) as a domain-independent alternative to FNOs. By parameterizing the spectral kernel as a continuous function of the physical wavevector, the EFNO can learn operators that act consistently across periodic domains of varying shape and size. We evaluate the EFNO on a simple heat equation and on a practically relevant materials science task of learning exchange-correlation potentials across different crystal structures, and demonstrate that the EFNO is able to generalize to unseen grid sizes and domains.
ER -