# Machine learning and domain decomposition methods -- a survey

arXiv: https://arxiv.org/abs/2312.14050 · HTML: https://mlsanity.com/paper/2312.14050/

## 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 (most cited first, 30 of 41)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [A Survey on Distributed Machine Learning](https://mlsanity.com/paper/1912.09789/) (2019)
- [Conditional Computation in Neural Networks for faster models](https://mlsanity.com/paper/1511.06297/) (2015)
- [Demystifying Parallel and Distributed Deep Learning: An In-Depth Concurrency Analysis](https://mlsanity.com/paper/1802.09941/) (2018)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (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)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Theory of the Frequency Principle for General Deep Neural Networks](https://mlsanity.com/paper/1906.09235/) (2019)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [Layer-Parallel Training of Deep Residual Neural Networks](https://mlsanity.com/paper/1812.04352/) (2018)
- [Conditional physics informed neural networks](https://mlsanity.com/paper/2104.02741/) (2021)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)

## Built on by (most cited first, 14 of 14)

- [Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving](https://mlsanity.com/paper/2504.00510/) (2025)
- [AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition](https://mlsanity.com/paper/2510.08924/) (2025)
- [DDU-Net: A Domain Decomposition-Based CNN for High-Resolution Image Segmentation on Multiple GPUs](https://mlsanity.com/paper/2407.21266/) (2024)
- [PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks](https://mlsanity.com/paper/2504.19013/) (2025)
- [A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations](https://mlsanity.com/paper/2509.15900/) (2025)
- [Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEs](https://mlsanity.com/paper/2509.18110/) (2025)
- [Multi-Level GNN Preconditioner for Solving Large Scale Problems](https://mlsanity.com/paper/2402.08296/) (2024)
- [Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory](https://mlsanity.com/paper/2509.10378/) (2025)
- [Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheets](https://mlsanity.com/paper/2512.01888/) (2025)
- [Soft Partition-based KAPI-ELM for Multi-Scale PDEs](https://mlsanity.com/paper/2601.08719/) (2026)
- [How Token Influence Decays with Distance: A Green-Function View of Trained Language Models](https://mlsanity.com/paper/2606.29139/) (2026)
- [Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domains](https://mlsanity.com/paper/2608.08114/) (2026)
- [Two-level domain-decomposition AdaGrad method for scalable training of graph neural networks](https://mlsanity.com/paper/2608.22575/) (2026)
- [Parallel Training Using a CNN-DNN Architecture for Accelerated Development of Diagnostic Models](https://mlsanity.com/paper/2609.12902/) (2026)

## 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.