mlsanityData to 14 Oct 2026

D3M: A deep domain decomposition method for partial differential equations

arXiv 1909.12236 ↗First posted 24 Sep 2019cs.LG cs.NA math.NAcited by 44

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Abstract (arXiv)

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization problem, and we design a multi-fidelity neural network framework to solve this optimization problem. Our contribution is to develop a systematical computational procedure for the underlying problem in parallel with domain decomposition. Our analysis shows that the D3M approximation solution converges to the exact solution of underlying PDEs. Our proposed framework establishes a foundation to use variational deep learning in large-scale engineering problems and designs. We present a general mathematical framework of D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.

Key facts. Cited by 44 papers in the map (data through 14 Oct 2026). First posted on arXiv 24 Sep 2019.

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Citing papers per year: 2021: 9; 2022: 5; 2023: 12; 2024: 7; 2025: 8; 2026: 3.

44 papers.

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  1. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  2. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  3. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  4. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 87
  5. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  6. 2021Learning in Sinusoidal Spaces with Physics-Informed Neural Networkscited by 62
  7. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 44
  8. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  9. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 26
  10. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  11. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
  12. 2021Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolutioncited by 14
32 more
  1. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  2. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  3. 2022Learning Interface Conditions in Domain Decomposition Solverscited by 11
  4. 2023Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating methodcited by 9
  5. 2021Bayesian neural networks for weak solution of PDEs with uncertainty quantificationcited by 8
  6. 2021A coarse space acceleration of deep-DDMcited by 8
  7. 2022A composable machine-learning approach for steady-state simulations on high-resolution gridscited by 8
  8. 2025Enhancing Physics-Informed Neural Networks Through Feature Engineeringcited by 8
  9. 2023A domain-decomposed VAE method for Bayesian inverse problemscited by 6
  10. 2025Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solvingcited by 6
  11. 2025Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Methodcited by 6
  12. 2024Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating methodcited by 5
  13. 2023LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometrycited by 4
  14. 2023ParticleWNN: a Novel Neural Networks Framework for Solving Partial Differential Equationscited by 4
  15. 2023Learning from Integral Losses in Physics Informed Neural Networkscited by 4
  16. 2023A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networkscited by 4
  17. 2024Adaptive deep density approximation for stochastic dynamical systemscited by 4
  18. 2024The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order modelscited by 4
  19. 2025AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decompositioncited by 4
  20. 2025Hybrid coupling with operator inference and the overlapping Schwarz alternating methodcited by 4
  21. 2023Breaking Boundaries: Distributed Domain Decomposition with Scalable Physics-Informed Neural PDE Solverscited by 3
  22. 2023TSONN: Time-stepping-oriented neural network for solving partial differential equationscited by 3
  23. 2024Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networkscited by 3
  24. 2025DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methodscited by 3
  25. 2022SciAI4Industry -- Solving PDEs for industry-scale problems with deep learningcited by 2
  26. 2026Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradientscited by 2
  27. 2025A Variational Physics-Informed Neural Network Framework Using Petrov-Galerkin Method for Solving Singularly Perturbed Boundary Value Problemscited by 1
  28. 2024Multi-Level GNN Preconditioner for Solving Large Scale Problemscited by 0
  29. 2024Two-level deep domain decomposition methodcited by 0
  30. 2025Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Mediacited by 0
  31. 2026Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problemscited by 0
  32. 2026Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guaranteescited by 0