# MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods

arXiv: https://arxiv.org/abs/2301.11378 · HTML: https://mlsanity.com/paper/2301.11378/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 26 Jan 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations](https://mlsanity.com/paper/1911.07979/) (2019)
- [Topology Adaptive Graph Convolutional Networks](https://mlsanity.com/paper/1710.10370/) (2017)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Backpropagation-Friendly Eigendecomposition](https://mlsanity.com/paper/1906.09023/) (2019)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [Learning Interface Conditions in Domain Decomposition Solvers](https://mlsanity.com/paper/2205.09833/) (2022)
- [Optimized Sparse Matrix Operations for Reverse Mode Automatic Differentiation](https://mlsanity.com/paper/2212.05159/) (2022)

## Built on by (most cited first, 12 of 12)

- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Multi-Grid Graph Neural Networks with Self-Attention for Computational Mechanics](https://mlsanity.com/paper/2409.11899/) (2024)
- [Graph Neural Networks and Applied Linear Algebra](https://mlsanity.com/paper/2310.14084/) (2023)
- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)
- [Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations](https://mlsanity.com/paper/2502.01397/) (2025)
- [Curriculum Learning for Mesh-based simulations](https://mlsanity.com/paper/2509.13138/) (2025)
- [Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks](https://mlsanity.com/paper/2310.06717/) (2023)
- [Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheets](https://mlsanity.com/paper/2512.01888/) (2025)
- [RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections](https://mlsanity.com/paper/2605.26854/) (2026)
- [Physics-Informed Coarsening for Multigrid Graph Neural Surrogates](https://mlsanity.com/paper/2605.31013/) (2026)
- [Hierarchical GNNs for power flow: letting physics shape the hierarchy](https://mlsanity.com/paper/2609.26603/) (2026)

## Abstract (arXiv)

Domain decomposition methods (DDMs) are popular solvers for discretized systems of partial differential equations (PDEs), with one-level and multilevel variants. These solvers rely on several algorithmic and mathematical parameters, prescribing overlap, subdomain boundary conditions, and other properties of the DDM. While some work has been done on optimizing these parameters, it has mostly focused on the one-level setting or special cases such as structured-grid discretizations with regular subdomain construction. In this paper, we propose multigrid graph neural networks (MG-GNN), a novel GNN architecture for learning optimized parameters in two-level DDMs@. We train MG-GNN using a new unsupervised loss function, enabling effective training on small problems that yields robust performance on unstructured grids that are orders of magnitude larger than those in the training set. We show that MG-GNN outperforms popular hierarchical graph network architectures for this optimization and that our proposed loss function is critical to achieving this improved performance.