mlsanityData to 7 Oct 2026

Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers

arXiv 2208.13273 ↗First posted 28 Aug 2022math.NA cs.LG cs.NAcited by 53

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Abstract (arXiv)

Neural networks suffer from spectral bias having difficulty in representing the high frequency components of a function while relaxation methods can resolve high frequencies efficiently but stall at moderate to low frequencies. We exploit the weaknesses of the two approaches by combining them synergistically to develop a fast numerical solver of partial differential equations (PDEs) at scale. Specifically, we propose HINTS, a hybrid, iterative, numerical, and transferable solver by integrating a Deep Operator Network (DeepONet) with standard relaxation methods, leading to parallel efficiency and algorithmic scalability for a wide class of PDEs, not tractable with existing monolithic solvers. HINTS balances the convergence behavior across the spectrum of eigenmodes by utilizing the spectral bias of DeepONet, resulting in a uniform convergence rate and hence exceptional performance of the hybrid solver overall. Moreover, HINTS applies to large-scale, multidimensional systems, it is flexible with regards to discretizations, computational domain, and boundary conditions.

Key facts. Cited by 53 papers in the map (data through 7 Oct 2026). Builds on 29 of its 102 references that are in the map. First posted on arXiv 28 Aug 2022.

Builds on

29 of 102 references.

References · 29

  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. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  5. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  7. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  8. 2021Machine learning accelerated computational fluid dynamicscited by 461
  9. 2017Neural 3D Mesh Renderercited by 437
  10. 2017PDE-Net: Learning PDEs from Datacited by 304
  11. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 201
  12. 2016Accelerating Eulerian Fluid Simulation With Convolutional Networkscited by 128
17 more
  1. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  2. 2022Learning Operators with Coupled Attentioncited by 88
  3. 2022Deep transfer operator learning for partial differential equations under conditional shiftcited by 83
  4. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  5. 2019Learning Neural PDE Solvers with Convergence Guaranteescited by 77
  6. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  7. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  8. 2020Learning Algebraic Multigrid Using Graph Neural Networkscited by 54
  9. 2022Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problemscited by 47
  10. 2019MgNet: A Unified Framework of Multigrid and Convolutional Neural Networkcited by 41
  11. 2020Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imagingcited by 40
  12. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  13. 2020MeshingNet: A New Mesh Generation Method based on Deep Learningcited by 30
  14. 2022Multigrid-augmented deep learning preconditioners for the Helmholtz equationcited by 24
  15. 2022Systems Biology: Identifiability analysis and parameter identification via systems-biology informed neural networkscited by 23
  16. 2022G2Phinet: Relating Genotype and Biomechanical Phenotype of Tissues with Deep Learningcited by 6
  17. 2022A unified analysis framework for iterative parallel-in-time algorithmscited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 5; 2024: 9; 2025: 20; 2026: 18.

53 papers.

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  1. 2022Mitigating spectral bias for the multiscale operator learningcited by 43
  2. 2025Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systemscited by 29
  3. 2023Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)cited by 16
  4. 2024Accelerating PDE Data Generation via Differential Operator Action in Solution Spacecited by 15
  5. 2024A hybrid iterative method based on MIONet for PDEs: Theory and numerical examplescited by 13
  6. 2024Accelerating Data Generation for Neural Operators via Krylov Subspace Recyclingcited by 12
  7. 2023Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layerscited by 11
  8. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  9. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  10. 2025Binned Spectral Power Loss for Improved Prediction of Chaotic Systemscited by 10
  11. 2025Enabling Local Neural Operators to perform Equation-Free System-Level Analysiscited by 10
  12. 2025NOWS: Neural Operator Warm Starts for Accelerating Iterative Solverscited by 9
38 more
  1. 2023A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimescited by 8
  2. 2023MyCrunchGPT: A chatGPT assisted framework for scientific machine learningcited by 8
  3. 2024Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problemcited by 8
  4. 2024A Pretraining-Finetuning Computational Framework for Material Homogenizationcited by 7
  5. 2025Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Methodcited by 6
  6. 2025Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learningcited by 6
  7. 2025Accurate and scalable deep Maxwell solvers using multilevel iterative methodscited by 5
  8. 2025A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solverscited by 4
  9. 2026Out-of-Distribution Generalization for Neural Physics Solverscited by 4
  10. 2024A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamicscited by 3
  11. 2025Self-composing neural operators for high-frequency and multiscale PDE surrogatescited by 3
  12. 2025Learning Singularity-Encoded Green's Functions with Application to Iterative Methodscited by 3
  13. 2025Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEscited by 3
  14. 2024Towards Automated Algebraic Multigrid Preconditioner Design Using Genetic Programming for Large-Scale Laser Beam Welding Simulationscited by 1
  15. 2025Fredholm Neural Networks for inverse problems in elliptic PDEscited by 1
  16. 2025Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operatorscited by 1
  17. 2025Learning Sparse Approximate Inverse Preconditioners for Conjugate Gradient Solvers on GPUscited by 1
  18. 2026HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEscited by 1
  19. 2026Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solverscited by 1
  20. 2024A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equationscited by 0
  21. 2025Reformulating Neural Operators in d+1 Dimensions for Embedding Evolutioncited by 0
  22. 2025Graph Lineages and Skeletal Graph Productscited by 0
  23. 2025Ultrasound Tomography of Musculoskeletal Tissues with Generative Neural Physicscited by 0
  24. 2025Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurementscited by 0
  25. 2025A Greedy PDE Router for Blending Neural Operators and Classical Methodscited by 0
  26. 2025ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marchingcited by 0
  27. 2026NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equationscited by 0
  28. 2026Are Deep Learning Based Hybrid PDE Solvers Reliable? Why Training Paradigms and Update Strategies Mattercited by 0
  29. 2026The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositionscited by 0
  30. 2026A level-wise training scheme for learning neural multigrid smoothers with application to integral equationscited by 0
  31. 2026Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithmscited by 0
  32. 2026NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspacescited by 0
  33. 2026Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approachcited by 0
  34. 2026NPSolver: Neural Poisson Solver with Iterative Physics Supervisioncited by 0
  35. 2026RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Correctionscited by 0
  36. 2026Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problemscited by 0
  37. 2026Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantificationcited by 0
  38. 2026McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equationcited by 0

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