mlsanityData to 7 Oct 2026

Dilated convolution neural operator for multiscale partial differential equations

arXiv 2408.00775 ↗First posted 16 Jul 2024cs.LG cs.NA math.NAcited by 3

Not on the map. Why?

Abstract (arXiv)

This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale parameterized solutions as a combination of low-rank global bases (such as low-frequency Fourier modes) and localized bases over coarse patches (analogous to dilated convolution), we propose the Dilated Convolutional Neural Operator (DCNO). The DCNO architecture effectively captures both high-frequency and low-frequency features while maintaining a low computational cost through a combination of convolution and Fourier layers. We conduct experiments to evaluate the performance of DCNO on various datasets, including the multiscale elliptic equation, its inverse problem, Navier-Stokes equation, and Helmholtz equation. We show that DCNO strikes an optimal balance between accuracy and computational cost and offers a promising solution for multiscale operator learning.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 18 of its 33 references that are in the map. First posted on arXiv 16 Jul 2024.

Builds on

18 of 33 references.

References · 18

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2021Swin Transformer: Hierarchical Vision Transformer using Shifted Windowscited by 12,082
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  5. 2018On the Spectral Bias of Neural Networkscited by 1,321
  6. 2015Multi-Scale Context Aggregation by Dilated Convolutionscited by 1,128
  7. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  8. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 360
  9. 2021Choose a Transformer: Fourier or Galerkincited by 333
  10. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  11. 2018Graph networks as learnable physics engines for inference and controlcited by 242
  12. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
6 more
  1. 2017Understanding Convolution for Semantic Segmentationcited by 168
  2. 2019The Convergence Rate of Neural Networks for Learned Functions of Different Frequenciescited by 134
  3. 2022Mitigating spectral bias for the multiscale operator learningcited by 43
  4. 2019Multi-scale Deep Neural Networks for Solving High Dimensional PDEscited by 24
  5. 2015Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Gamescited by 15
  6. 2021Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEscited by 10

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 1; 2026: 1.

3 papers.

  1. 2024PACE: Pacing Operator Learning to Accurate Optical Field Simulation for Complicated Photonic Devicescited by 5
  2. 2025FLASHμ: Fast Localizing And Sizing of Holographic Microparticlescited by 0
  3. 2026Efficient Dilated Squeeze and Excitation Neural Operator for Differential Equationscited by 0