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

• URL: http://arxiv.org/abs/2606.08871v1
• Date: Sun, 07 Jun 2026 22:55:39 GMT
• Title: Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
• Abstract summary: The emphFourier neural operator (FNO) is a neural network architecture that learns mappings between function spaces.<n>We prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with rank-1 lattice points.<n>We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples.
• Score: 40.78425934640374
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The \emph{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 \emph{one-dimensional fast Fourier transform}, and we can use a \emph{hyperbolic cross} frequency index set with lattice points. We demonstrate the benefits of our \emph{lattice-based hyperbolic-cross FNOs} for an elliptic PDE on the torus.

Related papers

• Flash EQ-Linear: Accelerating Equivariant Linear Layers via Group-wise Discrete Fourier Transform [65.89510337046077]
  Equivariant networks embed geometric symmetries as structural priors through weight sharing.<n>Existing implementations unroll the structured weights into dense matrices and dispatch them to generic dense kernels.<n>This is the first time equivariant networks strictly dominate their non-equivariant counterparts along all three axes simultaneously.
  arXiv  Detail & Related papers  (2026-07-23T12:45:21Z)
• Fourier Feature Pyramids for Physics-Informed Neural Networks [47.75068288153803]
  We present an improved neural field architecture for solving partial differential equations (PDEs)<n>We call our model beignet (Bandlimited Embedding with Interpolated Grid Network)<n>We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state-of-the-art PINN methods.
  arXiv  Detail & Related papers  (2026-05-22T23:12:56Z)
• Fermion lattices can be simulated by same-size qubit lattices with $\mathcal{O}(1)$ interaction overhead [0.0]
  Local interactions among electrons underlie many complex properties of correlated materials.<n>We show how to simulate all geometrically local interactions on an $N$-site two-dimensional fermion lattice with no overhead in the number of interactions and no space overhead.
  arXiv  Detail & Related papers  (2026-05-12T18:00:04Z)
• Adaptive Local Frequency Filtering for Fourier-Encoded Implicit Neural Representations [7.7579389797716365]
  We propose an adaptive local frequency filtering method for Fourier-encoded INRs.<n>The proposed method introduces a spatially varying parameter $(mathbfx)$ to modulate encoded Fourier components.<n>Experiments on 2D image fitting, 3D shape representation, and sparse data reconstruction demonstrate that the proposed method consistently improves reconstruction quality.
  arXiv  Detail & Related papers  (2026-04-03T08:04:30Z)
• Pseudo-differential-enhanced physics-informed neural networks [0.0]
  Pseudo-differential enhanced physics-informed neural networks (PINNs)<n>We present PINNs, an extension of enhancement but in Fourier space.<n>Our methods oftentimes achieve superior PINN versus numerical error in fewer training.
  arXiv  Detail & Related papers  (2026-02-16T11:40:58Z)
• FUTON: Fourier Tensor Network for Implicit Neural Representations [56.48739018255443]
  Implicit neural representations (INRs) have emerged as powerful tools for encoding signals, yet dominant-based designs often suffer from slow convergence, overfitting to noise, and poor extrapolation.<n>We introduce FUTON, which models signals as generalized Fourier series whose coefficients are parameterized by a low-rank tensor decomposition.
  arXiv  Detail & Related papers  (2026-02-13T19:31:44Z)
• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids [73.28614344779076]
  We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
  arXiv  Detail & Related papers  (2026-01-15T18:59:57Z)
• Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning [0.5097809301149341]
  We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series.<n>The FLM uses a simple feedforward structure with cosine activation functions to learn the frequencies, amplitudes, and phase shifts of the series as trainable parameters.<n> Computational experiments show that the performance of FLMs is comparable, and often superior, to that of established architectures like SIREN and vanilla feedforward NNs.
  arXiv  Detail & Related papers  (2025-09-10T16:49:20Z)
• Kronecker-Factored Approximate Curvature for Modern Neural Network Architectures [85.76673783330334]
  Two different settings of linear weight-sharing layers motivate two flavours of Kronecker-Factored Approximate Curvature (K-FAC) We show they are exact for deep linear networks with weight-sharing in their respective setting. We observe little difference between these two K-FAC variations when using them to train both a graph neural network and a vision transformer.
  arXiv  Detail & Related papers  (2023-11-01T16:37:00Z)
• Factorized Fourier Neural Operators [77.47313102926017]
  The Factorized Fourier Neural Operator (F-FNO) is a learning-based method for simulating partial differential equations. We show that our model maintains an error rate of 2% while still running an order of magnitude faster than a numerical solver.
  arXiv  Detail & Related papers  (2021-11-27T03:34:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.