# Graph Element Networks: adaptive, structured computation and memory

arXiv: https://arxiv.org/abs/1904.09019 · HTML: https://mlsanity.com/paper/1904.09019/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Apr 2019.

## Built on by (most cited first, 30 of 41)

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- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
- [Reinforcement Learning for Adaptive Mesh Refinement](https://mlsanity.com/paper/2103.01342/) (2021)
- [Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions](https://mlsanity.com/paper/2205.11912/) (2022)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [HAMLET: Graph Transformer Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2402.03541/) (2024)
- [Learning Connectivity with Graph Convolutional Networks for Skeleton-based Action Recognition](https://mlsanity.com/paper/2112.03328/) (2021)
- [Bridging the Gap between Spatial and Spectral Domains: A Unified Framework for Graph Neural Networks](https://mlsanity.com/paper/2107.10234/) (2021)
- [Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks](https://mlsanity.com/paper/2206.14092/) (2022)
- [Neural Relational Inference with Fast Modular Meta-learning](https://mlsanity.com/paper/2310.07015/) (2023)
- [Grounding Graph Network Simulators using Physical Sensor Observations](https://mlsanity.com/paper/2302.11864/) (2023)
- [A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations](https://mlsanity.com/paper/2212.14545/) (2022)
- [Convergent Graph Solvers](https://mlsanity.com/paper/2106.01680/) (2021)
- [Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2205.02637/) (2022)
- [GNN-based physics solver for time-independent PDEs](https://mlsanity.com/paper/2303.15681/) (2023)
- [Better Neural PDE Solvers Through Data-Free Mesh Movers](https://mlsanity.com/paper/2312.05583/) (2023)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [Learning time-dependent PDE solver using Message Passing Graph Neural Networks](https://mlsanity.com/paper/2204.07651/) (2022)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)
- [Quasi-optimal hp-finite element refinements towards singularities via deep neural network prediction](https://mlsanity.com/paper/2209.05844/) (2022)
- [REMuS-GNN: A Rotation-Equivariant Model for Simulating Continuum Dynamics](https://mlsanity.com/paper/2205.07852/) (2022)
- [An Implicit GNN Solver for Poisson-like problems](https://mlsanity.com/paper/2302.10891/) (2023)
- [Inductive biases in deep learning models for weather prediction](https://mlsanity.com/paper/2304.04664/) (2023)
- [Learning Space-Time Continuous Neural PDEs from Partially Observed States](https://mlsanity.com/paper/2307.04110/) (2023)

## Abstract (arXiv)

We explore the use of graph neural networks (GNNs) to model spatial processes in which there is no a priori graphical structure. Similar to finite element analysis, we assign nodes of a GNN to spatial locations and use a computational process defined on the graph to model the relationship between an initial function defined over a space and a resulting function in the same space. We use GNNs as a computational substrate, and show that the locations of the nodes in space as well as their connectivity can be optimized to focus on the most complex parts of the space. Moreover, this representational strategy allows the learned input-output relationship to generalize over the size of the underlying space and run the same model at different levels of precision, trading computation for accuracy. We demonstrate this method on a traditional PDE problem, a physical prediction problem from robotics, and learning to predict scene images from novel viewpoints.