Fourier Neural Operators with rank-1 lattice points and hyperbolic cross

2026-06-07 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

Fourier Neural Operators (FNOs), a neural network architecture for learning function space mappings, have been enhanced through a new approach called lattice-based hyperbolic-cross FNOs. This improved FNO architecture replaces traditional spatial tensor product grids with purpose-built rank-1 lattice points and incorporates a second lattice for training points in the parametric space. This modification leads to more accurate and efficient approximations, requiring fewer network parameters, spatial points, and training samples. A significant architectural simplification is achieved because the high-dimensional Fourier transform on rank-1 lattices only necessitates a one-dimensional fast Fourier transform, utilizing a hyperbolic cross frequency index set. The efficacy of these lattice-based hyperbolic-cross FNOs is demonstrated on an elliptic PDE on the torus, with the work published on 2026-06-07.

Key takeaway

For AI Scientists and Machine Learning Engineers developing or deploying Fourier Neural Operators, you should consider integrating rank-1 lattice points and hyperbolic cross techniques. This approach significantly improves FNO efficiency and accuracy, allowing you to achieve better results with fewer network parameters, spatial points, and training samples. Implementing these lattice-based hyperbolic-cross FNOs can simplify your architecture by reducing high-dimensional Fourier transforms to one-dimensional FFTs, optimizing resource usage for complex function space mappings.

Key insights

Replacing FNO's tensor product grids with rank-1 lattices and hyperbolic cross improves efficiency and accuracy using fewer resources.

Principles

• Generalization error of FNOs can be improved.
• Rank-1 lattice points enhance spatial approximation.
• Hyperbolic cross frequency sets simplify high-dimensional FFT.

Method

Replace FNO's spatial tensor product grids with rank-1 lattice points. Construct a second lattice for parametric training. Use a hyperbolic cross frequency index set for one-dimensional fast Fourier transforms.

In practice

• Achieve more accurate FNO approximations.
• Reduce FNO network parameters and spatial points.
• Apply to elliptic PDEs on a torus.

Topics

• Fourier Neural Operators
• Rank-1 Lattice Points
• Hyperbolic Cross
• Fast Fourier Transform
• Function Space Mappings
• Elliptic PDEs

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.