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

arXiv: https://arxiv.org/abs/2211.08939 · HTML: https://mlsanity.com/paper/2211.08939/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 40 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Generalization in Deep Learning](https://mlsanity.com/paper/1710.05468/) (2017)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depth](https://mlsanity.com/paper/2105.04550/) (2021)
- [Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions](https://mlsanity.com/paper/2105.09513/) (2021)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers](https://mlsanity.com/paper/2102.07346/) (2021)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [Robustness Implies Generalization via Data-Dependent Generalization Bounds](https://mlsanity.com/paper/2206.13497/) (2022)
- [Depth with Nonlinearity Creates No Bad Local Minima in ResNets](https://mlsanity.com/paper/1810.09038/) (2018)
- [Large-scale Neural Solvers for Partial Differential Equations](https://mlsanity.com/paper/2009.03730/) (2020)
- [Understanding Dynamics of Nonlinear Representation Learning and Its Application](https://mlsanity.com/paper/2106.14836/) (2021)
- [Learning Interface Conditions in Domain Decomposition Solvers](https://mlsanity.com/paper/2205.09833/) (2022)
- [A coarse space acceleration of deep-DDM](https://mlsanity.com/paper/2112.03732/) (2021)

## Built on by (most cited first, 30 of 44)

- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [State-space models are accurate and efficient neural operators for dynamical systems](https://mlsanity.com/paper/2409.03231/) (2024)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)
- [The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.03864/) (2024)
- [Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies](https://mlsanity.com/paper/2306.17648/) (2023)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [On the Generalization of PINNs outside the training domain and the Hyperparameters influencing it](https://mlsanity.com/paper/2302.07557/) (2023)
- [Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)](https://mlsanity.com/paper/2406.03172/) (2024)
- [Complex Physics-Informed Neural Network](https://mlsanity.com/paper/2502.04917/) (2025)
- [Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2309.02589/) (2023)
- [Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applications](https://mlsanity.com/paper/2406.10997/) (2024)
- [BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamics](https://mlsanity.com/paper/2508.03965/) (2025)
- [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024)
- [Physics-Informed Machine Learning in Biomedical Science and Engineering](https://mlsanity.com/paper/2510.05433/) (2025)
- [AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition](https://mlsanity.com/paper/2510.08924/) (2025)
- [Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks](https://mlsanity.com/paper/2409.13644/) (2024)
- [Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems](https://mlsanity.com/paper/2508.15695/) (2025)
- [M2M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations need](https://mlsanity.com/paper/2410.11617/) (2024)
- [PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks](https://mlsanity.com/paper/2504.19013/) (2025)
- [AdS/Deep-Learning made easy II: neural network-based approaches to holography and inverse problems](https://mlsanity.com/paper/2511.22522/) (2025)
- [EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networks](https://mlsanity.com/paper/2607.26490/) (2026)
- [Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradients](https://mlsanity.com/paper/2609.14841/) (2026)
- [Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics](https://mlsanity.com/paper/2402.11126/) (2024)
- [Improved physics-informed neural network in mitigating gradient related failures](https://mlsanity.com/paper/2407.19421/) (2024)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)

## 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.