Euclidean Fourier Neural Operators
Machine Learning
Summary
The authors explain that Fourier neural operators (FNOs) are good at learning how to solve problems on different-sized grids but have trouble when the shape or size of the area changes, because they rely on fixed wave patterns. To fix this, they created Euclidean Fourier neural operators (EFNOs), which treat wave patterns more flexibly, so the learned solutions work well on different shapes and sizes. They tested EFNOs on a simple heat problem and a materials science problem, showing EFNOs can handle new grids and domains better than traditional FNOs.
Fourier neural operatorsfunction spacesspectral kernelwavevectorperiodic domainheat equationexchange-correlation potentialsgrid resolutionoperator learningmaterials science
Authors
Nathanael Bosch, Niklas Frederik Schmitz, Michael F. Herbst
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.