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Parallel Physics-Informed Neural Networks via Domain Decomposition

arXiv 2104.10013 ↗First posted 20 Apr 2021cs.DCcited by 112

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Abstract (arXiv)

We develop a distributed framework for the physics-informed neural networks (PINNs) based on two recent extensions, namely conservative PINNs (cPINNs) and extended PINNs (XPINNs), which employ domain decomposition in space and in time-space, respectively. This domain decomposition endows cPINNs and XPINNs with several advantages over the vanilla PINNs, such as parallelization capacity, large representation capacity, efficient hyperparameter tuning, and is particularly effective for multi-scale and multi-physics problems. Here, we present a parallel algorithm for cPINNs and XPINNs constructed with a hybrid programming model described by MPI + X, where X ∈ CPUs,~GPUs. The main advantage of cPINN and XPINN over the more classical data and model parallel approaches is the flexibility of optimizing all hyperparameters of each neural network separately in each subdomain. We compare the performance of distributed cPINNs and XPINNs for various forward problems, using both weak and strong scalings. Our results indicate that for space domain decomposition, cPINNs are more efficient in terms of communication cost but XPINNs provide greater flexibility as they can also handle time-domain decomposition for any differential equations, and can deal with any arbitrarily shaped complex subdomains. To this end, we also present an application of the parallel XPINN method for solving an inverse diffusion problem with variable conductivity on the United States map, using ten regions as subdomains.

Key facts. Cited by 112 papers in the map (data through 14 Oct 2026). First posted on arXiv 20 Apr 2021.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 11; 2022: 27; 2023: 26; 2024: 18; 2025: 18; 2026: 12.

112 papers.

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  1. 2021Neural Fields in Visual Computing and Beyondcited by 540
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  3. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  4. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  5. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  6. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 104
  7. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  8. 2021Multi-Objective Loss Balancing for Physics-Informed Deep Learningcited by 87
  9. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 87
  10. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 76
  11. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  12. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
38 more
  1. 2021Meta-learning PINN loss functionscited by 51
  2. 2021Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functionscited by 49
  3. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  4. 2021Physics informed neural networks for continuum micromechanicscited by 46
  5. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 44
  6. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  7. 2024Parameterized Physics-informed Neural Networks for Parameterized PDEscited by 38
  8. 2021A novel meta-learning initialization method for physics-informed neural networkscited by 35
  9. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  10. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  11. 2022How important are activation functions in regression and classification? A survey, performance comparison, and future directionscited by 29
  12. 2022Scalable algorithms for physics-informed neural and graph networkscited by 21
  13. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  14. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
  15. 2022Improved Training of Physics-Informed Neural Networks with Model Ensemblescited by 19
  16. 2022Variationally Mimetic Operator Networkscited by 19
  17. 2023A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Datacited by 18
  18. 2025A Practitioner's Guide to Kolmogorov-Arnold Networkscited by 18
  19. 2022RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networkscited by 14
  20. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  21. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  22. 2022State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learningcited by 12
  23. 2025Machine learning for modelling unstructured grid data in computational physics: a reviewcited by 12
  24. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  25. 2022PSO-PINN: Physics-Informed Neural Networks Trained with Particle Swarm Optimizationcited by 11
  26. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  27. 2025Lang-PINN: From Language to Physics-Informed Neural Networks via a Multi-Agent Frameworkcited by 11
  28. 2021CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometriescited by 10
  29. 2022Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)cited by 10
  30. 2021A coarse space acceleration of deep-DDMcited by 8
  31. 2022A composable machine-learning approach for steady-state simulations on high-resolution gridscited by 8
  32. 2023MyCrunchGPT: A chatGPT assisted framework for scientific machine learningcited by 8
  33. 2023Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulationscited by 8
  34. 2024Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneitycited by 8
  35. 2022An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operatorcited by 7
  36. 2022Adaptive physics-informed neural operator for coarse-grained non-equilibrium flowscited by 7
  37. 2025Complex Physics-Informed Neural Networkcited by 7
  38. 2022Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering processcited by 6

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