# GNN-based physics solver for time-independent PDEs

arXiv: https://arxiv.org/abs/2303.15681 · HTML: https://mlsanity.com/paper/2303.15681/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 49 references that are in the map.
- First posted on arXiv 28 Mar 2023.

## Builds on (most cited first, 30 of 30)

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Graph Convolutional Networks for Text Classification](https://mlsanity.com/paper/1809.05679/) (2018)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [Structured Sequence Modeling with Graph Convolutional Recurrent Networks](https://mlsanity.com/paper/1612.07659/) (2016)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Representing Long-Range Context for Graph Neural Networks with Global Attention](https://mlsanity.com/paper/2201.08821/) (2022)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Learning continuous-time PDEs from sparse data with graph neural networks](https://mlsanity.com/paper/2006.08956/) (2020)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Graph Element Networks: adaptive, structured computation and memory](https://mlsanity.com/paper/1904.09019/) (2019)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)

## Built on by (most cited first, 11 of 11)

- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks](https://mlsanity.com/paper/2308.01602/) (2023)
- [AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics](https://mlsanity.com/paper/2407.20801/) (2024)
- [A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulations](https://mlsanity.com/paper/2402.13101/) (2024)
- [Multiscale Neural Operators for Solving Time-Independent PDEs](https://mlsanity.com/paper/2311.05964/) (2023)
- [Training Transformers for Mesh-Based Simulations](https://mlsanity.com/paper/2508.18051/) (2025)
- [ML-based identification of the interface regions for coupling local and nonlocal models](https://mlsanity.com/paper/2404.15388/) (2024)
- [Learning Deformable Body Interactions With Adaptive Spatial Tokenization](https://mlsanity.com/paper/2507.13707/) (2025)
- [Physics-Informed Coarsening for Multigrid Graph Neural Surrogates](https://mlsanity.com/paper/2605.31013/) (2026)
- [ONE CYLinder: A Benchmark for Graph-Based Surrogate Modeling of Unsteady Bluff-Body Flows](https://mlsanity.com/paper/2609.08947/) (2026)

## Abstract (arXiv)

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of requiring long-range exchange of information across the computational domain for obtaining accurate predictions. In the context of graph neural networks (GNNs), this calls for deeper networks, which, in turn, may compromise or slow down the training process. In this work, we present two GNN architectures to overcome this challenge - the Edge Augmented GNN and the Multi-GNN. We show that both these networks perform significantly better (by a factor of 1.5 to 2) than baseline methods when applied to time-independent solid mechanics problems. Furthermore, the proposed architectures generalize well to unseen domains, boundary conditions, and materials. Here, the treatment of variable domains is facilitated by a novel coordinate transformation that enables rotation and translation invariance. By broadening the range of problems that neural operators based on graph neural networks can tackle, this paper provides the groundwork for their application to complex scientific and industrial settings.