mlsanityData to 7 Oct 2026

A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples

arXiv 2402.07156 ↗First posted 11 Feb 2024math.NA cs.LG cs.NAcited by 13

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Abstract (arXiv)

We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 19 of its 38 references that are in the map. First posted on arXiv 11 Feb 2024.

Builds on

19 of 38 references.

References · 19

  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. 2018On the Spectral Bias of Neural Networkscited by 1,321
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  7. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  8. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 360
  9. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  10. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  11. 2022U-NO: U-shaped Neural Operatorscited by 219
  12. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
7 more
  1. 2023Solving High-Dimensional PDEs with Latent Spectral Modelscited by 84
  2. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
  3. 2023Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestrationcited by 46
  4. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  5. 2023MgNO: Efficient Parameterization of Linear Operators via Multigridcited by 40
  6. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
  7. 2023Experimental observation on a low-rank tensor model for eigenvalue problemscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 4; 2025: 3; 2026: 6.

13 papers.

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  1. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 10
  2. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  3. 2024Learning solution operators of PDEs defined on varying domains via MIONetcited by 3
  4. 2025Self-composing neural operators for high-frequency and multiscale PDE surrogatescited by 3
  5. 2025Learning Singularity-Encoded Green's Functions with Application to Iterative Methodscited by 3
  6. 2024A deformation-based framework for learning solution mappings of PDEs defined on varying domainscited by 2
  7. 2025A Greedy PDE Router for Blending Neural Operators and Classical Methodscited by 0
  8. 2026Are Deep Learning Based Hybrid PDE Solvers Reliable? Why Training Paradigms and Update Strategies Mattercited by 0
  9. 2026A level-wise training scheme for learning neural multigrid smoothers with application to integral equationscited by 0
  10. 2026IV-Net: A neural network for elliptic PDEs with random and highly varying coefficientscited by 0
  11. 2026Hybrid Least Squares/Gradient Descent Methods for MIONetscited by 0
  12. 2026Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solvercited by 0
1 more
  1. 2026MENO: Memory-Efficient Neural Operatorcited by 0