mlsanityData to 7 Oct 2026

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

arXiv 2606.08871 ↗First posted 7 Jun 2026math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

The Fourier neural operator (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a one-dimensional fast Fourier transform, and we can use a hyperbolic cross frequency index set with lattice points. We demonstrate the benefits of our lattice-based hyperbolic-cross FNOs for an elliptic PDE on the torus.

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References · 22

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  5. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  6. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 225
  7. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  8. 2025FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scalecited by 56
  9. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  10. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 43
  11. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  12. 2019Deep learning observables in computational fluid dynamicscited by 38
10 more
  1. 2024A Library for Learning Neural Operatorscited by 31
  2. 2020The universal approximation theorem for complex-valued neural networkscited by 28
  3. 2020Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequencescited by 20
  4. 2024Discretization Error of Fourier Neural Operatorscited by 20
  5. 2021Physics-enhanced deep surrogates for partial differential equationscited by 18
  6. 2023Physics-Informed Polynomial Chaos Expansionscited by 15
  7. 2020Higher-order Quasi-Monte Carlo Training of Deep Neural Networkscited by 12
  8. 2022Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integrationcited by 2
  9. 2026Lattice-based Deep Neural Networks: Regularity and Tailored Regularizationcited by 2
  10. 2025Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equationcited by 1

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