mlsanityData to 7 Oct 2026

Component Fourier Neural Operator for Singularly Perturbed Differential Equations

arXiv 2409.04779 ↗First posted 7 Sep 2024cs.LG cs.NA math.NAcited by 7

Not on the map. Why?

Abstract (arXiv)

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations motivates us to employ these methods for solving SPDEs. In this manuscript, we introduce Component Fourier Neural Operator (ComFNO), an innovative operator learning method that builds upon Fourier Neural Operator (FNO), while simultaneously incorporating valuable prior knowledge obtained from asymptotic analysis. Our approach is not limited to FNO and can be applied to other neural network frameworks, such as Deep Operator Network (DeepONet), leading to potential similar SPDEs solvers. Experimental results across diverse classes of SPDEs demonstrate that ComFNO significantly improves accuracy compared to vanilla FNO. Furthermore, ComFNO exhibits natural adaptability to diverse data distributions and performs well in few-shot scenarios, showcasing its excellent generalization ability in practical situations.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 16 of its 31 references that are in the map. First posted on arXiv 7 Sep 2024.

Builds on

16 of 31 references.

References · 16

  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. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
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  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  8. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  9. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  10. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  11. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  12. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
4 more
  1. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61
  2. 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
  3. 2022Applications of physics informed neural operatorscited by 33
  4. 2020A nonlocal physics-informed deep learning framework using the peridynamic differential operatorcited by 21

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 4; 2026: 3.

7 papers.

  1. 2025Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradientscited by 7
  2. 2025SAOT: An Enhanced Locality-Aware Spectral Transformer for Solving PDEscited by 6
  3. 2025Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experimentscited by 1
  4. 2025Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEscited by 1
  5. 2026Discontinuous Galerkin finite element operator network for solving non-smooth PDEscited by 1
  6. 2026DiffPDE: Masked Diffusion Language Models as PDE Solvercited by 0
  7. 2026Selective boundary condition reduction via learned error gatingcited by 0