mlsanityData to 14 Oct 2026

Machine learning and domain decomposition methods -- a survey

arXiv 2312.14050 ↗First posted 21 Dec 2023math.NA cs.LG cs.NAcited by 14

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Abstract (arXiv)

Hybrid algorithms, which combine black-box machine learning methods with experience from traditional numerical methods and domain expertise from diverse application areas, are progressively gaining importance in scientific machine learning and various industrial domains, especially in computational science and engineering. In the present survey, several promising avenues of research will be examined which focus on the combination of machine learning (ML) and domain decomposition methods (DDMs). The aim of this survey is to provide an overview of existing work within this field and to structure it into domain decomposition for machine learning and machine learning-enhanced domain decomposition, including: domain decomposition for classical machine learning, domain decomposition to accelerate the training of physics-aware neural networks, machine learning to enhance the convergence properties or computational efficiency of DDMs, and machine learning as a discretization method in a DDM for the solution of PDEs. In each of these fields, we summarize existing work and key advances within a common framework and, finally, disuss ongoing challenges and opportunities for future research.

Key facts. Cited by 14 papers in the map (data through 14 Oct 2026). Builds on 41 of its 122 references that are in the map. First posted on arXiv 21 Dec 2023.

Builds on

41 of 122 references.

References · 41

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
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29 more
  1. 2018Demystifying Parallel and Distributed Deep Learning: An In-Depth Concurrency Analysiscited by 156
  2. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  3. 2019The Convergence Rate of Neural Networks for Learned Functions of Different Frequenciescited by 135
  4. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  5. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 81
  6. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  7. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 58
  8. 2020EikoNet: Solving the Eikonal equation with Deep Neural Networkscited by 50
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  10. 2019Theory of the Frequency Principle for General Deep Neural Networkscited by 46
  11. 2019D3M: A deep domain decomposition method for partial differential equationscited by 44
  12. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 44
  13. 2020PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometriescited by 38
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  17. 2021Conditional physics informed neural networkscited by 23
  18. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  19. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
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  22. 2023MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methodscited by 12
  23. 2022Learning Interface Conditions in Domain Decomposition Solverscited by 11
  24. 2023Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating methodcited by 9
  25. 2021A coarse space acceleration of deep-DDMcited by 8
  26. 2023Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networkscited by 6
  27. 2023A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networkscited by 4
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  29. 2023A Domain Decomposition-Based CNN-DNN Architecture for Model Parallel Training Applied to Image Recognition Problemscited by 1

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Citing papers per year: 2024: 2; 2025: 7; 2026: 5.

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  1. 2025Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solvingcited by 6
  2. 2025AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decompositioncited by 4
  3. 2024DDU-Net: A Domain Decomposition-Based CNN for High-Resolution Image Segmentation on Multiple GPUscited by 2
  4. 2025PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networkscited by 2
  5. 2025A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulationscited by 1
  6. 2025Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEscited by 1
  7. 2024Multi-Level GNN Preconditioner for Solving Large Scale Problemscited by 0
  8. 2025Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theorycited by 0
  9. 2025Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheetscited by 0
  10. 2026Soft Partition-based KAPI-ELM for Multi-Scale PDEscited by 0
  11. 2026How Token Influence Decays with Distance: A Green-Function View of Trained Language Modelscited by 0
  12. 2026Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domainscited by 0
2 more
  1. 2026Two-level domain-decomposition AdaGrad method for scalable training of graph neural networkscited by 0
  2. 2026Parallel Training Using a CNN-DNN Architecture for Accelerated Development of Diagnostic Modelscited by 0