cs.LGAug 28, 2026

Euclidean Fourier Neural Operators

Authors: Nathanael Bosch, Niklas Frederik Schmitz, Michael F. Herbst

Organizations: Mathematics for Materials Modelling (MatMat), Institute of Mathematics & Institute of Materials, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

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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.

Figures & tables

Figure 1: Domain transfer test for a learned convolution. a.\textbf{a}. A periodic array of Gaussian sources, smoothed by the heat kernel. Both models are fitted in the small rhombic cell (dashed), and evaluated on the equivalent rectangular cell (solid). The FNO ( b.\textbf{b}. ) distorts visibly, while the EFNO ( c. ) transfers perfectly. d. Relative L2L_{2} error of the FNO and EFNO on n×nn\times n supercells of the fitting cell. The FNO error grows with the size of the supercell, while the EFNO error remains constant.
Figure 2: Overview and results of the exchange-correlation potential learning task. a. Example system, showing a diamond unit cell with 16 atoms. b.–c. PBE electron density and RPA exchange-correlation potential on the slice plane highlighted in ( a. ); these are the inputs and outputs for the operator learning task. d.–e. Error (WRMSE) against cell volume for both tasks. The FNO extrapolates poorly from the smaller training cells to the larger test cells; the EFNO generalizes well and achieves errors that are an order of magnitude smaller than the PBE baseline.
 DiamondWater
ModelTrainTestTrainTest
PBE826.67825.171350.331348.37
FNO81.961815.42108.031675.35
EFNO47.8856.1157.95103.54
Table 1: Density-weighted root mean square error (WRMSE); median value within each split.
Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

MethodSpectral weightKernel argument
FNO [ Li et al., 2021 ]lookup tablemode number
AM-FNO [ Xiao et al., 2024 ]learned functionmode number
SirenFNO [ Shi et al., 2026 ]learned functionmode number
PDNO [ Shin et al., 2024 ]learned functionmode number & position
KANO [ Lee et al., 2026 ]learned functionmode number & position
EFNO (ours)learned functionphysical wavevector
Appendix
Table 2: Parameterization of the spectral weight across FNO variants. FNO stores a discrete table over modes; the others learn it as a function. Its argument is the mode number, with PDNO and KANO adding the position (a pseudo-differential symbol); EFNO alone uses the physical wavevector, the quantity that changes with the domain.
Real-space kernel κ\kappaSpectral kernel κ^\widehat{\kappa}
Real-valuednessHermitian symmetry
κ\kappa real-valuedκ^(−k)=κ^(k)‾\widehat{\kappa}(-k)=\overline{\widehat{\kappa}(k)}
SmoothnessDecay at large \absolutevaluek\absolutevalue{k}
κ\kappa has ss derivatives in L2L^{2}\absolutevalueks κ^∈L2\absolutevalue{k}^{s}\,\widehat{\kappa}\in L^{2}
RangeSmoothness of κ^\widehat{\kappa}
κ\kappa supported in \absolutevaluex≤R\absolutevalue{x}\leq Rκ^\widehat{\kappa} entire, \absolutevalueκ^(k)≲eR\absolutevalueIm⁡k\absolutevalue{\widehat{\kappa}(k)}\lesssim e^{R\absolutevalue{\operatorname{Im} k}}
Appendix
Table 3: Correspondence between real-space and spectral properties of the kernel. Each row is an equivalence.
 Single celln×nn\times n supercell
ModelFittingRectangularn=2n=2n=3n=3n=4n=4n=5n=5n=6n=6
FNO∼ ⁣0\sim\!049.7049.7066.0966.0985.1685.1689.6089.6090.1290.1290.1690.16
EFNO3.5×10−63.5{\times}10^{-6}3.5×10−63.5{\times}10^{-6}3.5×10−63.5{\times}10^{-6}3.5×10−63.5{\times}10^{-6}3.5×10−63.5{\times}10^{-6}3.5×10−63.5{\times}10^{-6}3.5×10−63.5{\times}10^{-6}
Appendix
Table 4: Relative L2L_{2} error (%) of the FNO and EFNO. The FNO is near-exact on the fitting cell but fails to transfer; the EFNO stays at its basis-approximation residual on every cell.
Figure 3: Predicted potentials for a representative diamond train and test structure. a. Unit cell of a training system, showing the plane along which the density and potential are visualized. b. RPA reference potential. c.–e. PBE, FNO, and EFNO predictions. f. Input electron density (PBE). g.–i. Difference of each prediction to the reference. j.–r. Corresponding panels showing the cell, potentials, density, and errors for a test system. On the training system, both the FNO and the EFNO are able to reproduce the RPA potential well and show small errors ( h.–i. ). On the test system, the FNO shows substantial errors ( q. ) that are larger than the PBE baseline ( p. ), while the EFNO stays close to the reference ( r. ).
Figure 4: Predicted potentials for a representative liquid-water train and test structure. a. Unit cell of a training system, showing the plane along which the density and potential are visualized. b. RPA reference potential. c.–e. PBE, FNO, and EFNO predictions. f. Input electron density (PBE). g.–i. Difference of each prediction to the reference. j.–r. Corresponding panels showing the cell, potentials, density, and errors for a test system. On the training system, both the FNO and the EFNO are able to reproduce the RPA potential well and show small errors ( h.–i. ). On the test system, the FNO shows substantial errors ( q. ) that are larger than the PBE baseline ( p. ), while the EFNO stays close to the reference ( r. ).
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