# A coarse space acceleration of deep-DDM

arXiv: https://arxiv.org/abs/2112.03732 · HTML: https://mlsanity.com/paper/2112.03732/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 7 Dec 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [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)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [Physics-informed Neural Networks for Solving Nonlinear Diffusivity and Biot's equations](https://mlsanity.com/paper/2002.08235/) (2020)

## Built on by (most cited first, 8 of 8)

- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [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)
- [Learning Interface Conditions in Domain Decomposition Solvers](https://mlsanity.com/paper/2205.09833/) (2022)
- [A Nonoverlapping Domain Decomposition Method for Extreme Learning Machines: Elliptic Problems](https://mlsanity.com/paper/2406.15959/) (2024)
- [Dirichlet-Neumann learning algorithm for solving elliptic interface problems](https://mlsanity.com/paper/2301.07361/) (2023)
- [A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machines](https://mlsanity.com/paper/2503.10032/) (2025)
- [An overview of machine learning-enhanced iterative methods for systems of linear and nonlinear equations](https://mlsanity.com/paper/2610.07211/) (2026)

## Abstract (arXiv)

The use of deep learning methods for solving PDEs is a field in full expansion. In particular, Physical Informed Neural Networks, that implement a sampling of the physical domain and use a loss function that penalizes the violation of the partial differential equation, have shown their great potential. Yet, to address large scale problems encountered in real applications and compete with existing numerical methods for PDEs, it is important to design parallel algorithms with good scalability properties. In the vein of traditional domain decomposition methods (DDM), we consider the recently proposed deep-ddm approach. We present an extension of this method that relies on the use of a coarse space correction, similarly to what is done in traditional DDM solvers. Our investigations shows that the coarse correction is able to alleviate the deterioration of the convergence of the solver when the number of subdomains is increased thanks to an instantaneous information exchange between subdomains at each iteration. Experimental results demonstrate that our approach induces a remarkable acceleration of the original deep-ddm method, at a reduced additional computational cost.