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Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology

arXiv 2211.08939 ↗First posted 16 Nov 2022cs.LG cs.NA math.DScited by 43

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Abstract (arXiv)

In this paper, we propose the augmented physics-informed neural network (APINN), which adopts soft and trainable domain decomposition and flexible parameter sharing to further improve the extended PINN (XPINN) as well as the vanilla PINN methods. In particular, a trainable gate network is employed to mimic the hard decomposition of XPINN, which can be flexibly fine-tuned for discovering a potentially better partition. It weight-averages several sub-nets as the output of APINN. APINN does not require complex interface conditions, and its sub-nets can take advantage of all training samples rather than just part of the training data in their subdomains. Lastly, each sub-net shares part of the common parameters to capture the similar components in each decomposed function. Furthermore, following the PINN generalization theory in Hu et al. [2021], we show that APINN can improve generalization by proper gate network initialization and general domain & function decomposition. Extensive experiments on different types of PDEs demonstrate how APINN improves the PINN and XPINN methods. Specifically, we present examples where XPINN performs similarly to or worse than PINN, so that APINN can significantly improve both. We also show cases where XPINN is already better than PINN, so APINN can still slightly improve XPINN. Furthermore, we visualize the optimized gating networks and their optimization trajectories, and connect them with their performance, which helps discover the possibly optimal decomposition. Interestingly, if initialized by different decomposition, the performances of corresponding APINNs can differ drastically. This, in turn, shows the potential to design an optimal domain decomposition for the differential equation problem under consideration.

Key facts. Cited by 43 papers in the map (data through 7 Oct 2026). Builds on 27 of its 40 references that are in the map. First posted on arXiv 16 Nov 2022.

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27 of 40 references.

References · 27

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2017Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layercited by 3,413
  3. 2017Spectrally-normalized margin bounds for neural networkscited by 681
  4. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  5. 2016Deep Learning without Poor Local Minimacited by 233
  6. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  7. 2017Generalization in Deep Learningcited by 146
  8. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  9. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  10. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  11. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  12. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
15 more
  1. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  2. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  3. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  4. 2021Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depthcited by 55
  5. 2021Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functionscited by 49
  6. 2019D3M: A deep domain decomposition method for partial differential equationscited by 44
  7. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  8. 2021On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layerscited by 35
  9. 2019Distributed physics informed neural network for data-efficient solution to partial differential equationscited by 26
  10. 2022Robustness Implies Generalization via Data-Dependent Generalization Boundscited by 23
  11. 2018Depth with Nonlinearity Creates No Bad Local Minima in ResNetscited by 17
  12. 2020Large-scale Neural Solvers for Partial Differential Equationscited by 17
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  14. 2022Learning Interface Conditions in Domain Decomposition Solverscited by 11
  15. 2021A coarse space acceleration of deep-DDMcited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 11; 2024: 12; 2025: 13; 2026: 7.

43 papers.

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  1. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  2. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  3. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  4. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  5. 2023CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamicscited by 40
  6. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  7. 2024State-space models are accurate and efficient neural operators for dynamical systemscited by 27
  8. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  9. 2024The Challenges of the Nonlinear Regime for Physics-Informed Neural Networkscited by 21
  10. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
  11. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  12. 2023On the Generalization of PINNs outside the training domain and the Hyperparameters influencing itcited by 10
31 more
  1. 2024Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)cited by 9
  2. 2025Complex Physics-Informed Neural Networkcited by 6
  3. 2023Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networkscited by 5
  4. 2024Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applicationscited by 5
  5. 2025BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamicscited by 5
  6. 2024Resolution invariant deep operator network for PDEs with complex geometriescited by 4
  7. 2025Physics-Informed Machine Learning in Biomedical Science and Engineeringcited by 4
  8. 2024Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networkscited by 3
  9. 2025Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problemscited by 3
  10. 2024M2M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations needcited by 2
  11. 2025PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networkscited by 2
  12. 2025AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decompositioncited by 2
  13. 2025AdS/Deep-Learning made easy II: neural network-based approaches to holography and inverse problemscited by 2
  14. 2026Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradientscited by 2
  15. 2024Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metricscited by 1
  16. 2024Improved physics-informed neural network in mitigating gradient related failurescited by 1
  17. 2025A comprehensive analysis of PINNs: Variants, Applications, and Challengescited by 1
  18. 2026Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networkscited by 1
  19. 2026EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networkscited by 1
  20. 2023Dirichlet-Neumann learning algorithm for solving elliptic interface problemscited by 0
  21. 2023Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zonescited by 0
  22. 2024Hybrid data-driven and physics-informed regularized learning of cyclic plasticity with Neural Networkscited by 0
  23. 2025Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuningcited by 0
  24. 2025A Residual Guided strategy with Generative Adversarial Networks in training Physics-Informed Transformer Networkscited by 0
  25. 2025PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Samplingcited by 0
  26. 2025SA-EMO: Structure-Aligned Encoder Mixture of Operators for Generalizable Full-waveform Inversioncited by 0
  27. 2025Enhancing PINN Accuracy for the RLW Equation: Adaptive and Conservative Approachescited by 0
  28. 2026GlueNN: gluing patchwise analytic solutions with neural networkscited by 0
  29. 2026SPAMoE: Spectrum-Aware Hybrid Operator Framework for Full-Waveform Inversioncited by 0
  30. 2026Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solverscited by 0
  31. 2026Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domainscited by 0