Improved Fourier Neural Operators Using Rank-1 Lattice Points and Hyperbolic Cross Indexing
A new study introduces rank-1 lattice points and hyperbolic cross frequency index sets as replacements for standard tensor product grids in Fourier Neural Operators (FNOs), reducing generalization error. The work derives regularity bounds for FNOs across both spatial and parametric variables, showing that purpose-built lattices in both domains yield more efficient approximations. The approach simplifies the architecture by reducing high-dimensional Fourier transforms to one-dimensional FFTs, potentially lowering computational costs for scientific machine learning.
Researchers have submitted a preprint to arXiv proposing modifications to the Fourier Neural Operator (FNO), a neural network architecture used to learn mappings between function spaces and commonly applied to solving partial differential equations (PDEs). The core contribution is replacing conventional spatial tensor product grids with rank-1 lattice points, and constructing a second lattice for use as training points in the parametric space. By deriving general regularity bounds for the FNO with respect to both spatial and parametric variables, the authors prove that these substitutions reduce generalization error. A key practical benefit is architectural simplification: the high-dimensional Fourier transform on rank-1 lattices reduces to a one-dimensional fast Fourier transform, and a hyperbolic cross frequency index set can be employed. The authors demonstrate their 'lattice-based hyperbolic-cross FNO' on an elliptic PDE defined on the torus, reporting improved accuracy and efficiency with fewer network parameters, spatial points, and training samples.
What's missing
The study is demonstrated only on a single benchmark problem (an elliptic PDE on the torus), leaving open questions about generalizability to more complex or higher-dimensional PDEs, time-dependent problems, and real-world applications. The work is a preprint and has not yet undergone peer review.
What different sources said
- arXiv cs.LGCenter
Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
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