# Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions

arXiv: https://arxiv.org/abs/2205.11912 · HTML: https://mlsanity.com/paper/2205.11912/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 36 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Graph Element Networks: adaptive, structured computation and memory](https://mlsanity.com/paper/1904.09019/) (2019)
- [Isometric Transformation Invariant and Equivariant Graph Convolutional Networks](https://mlsanity.com/paper/2005.06316/) (2020)
- [Learning to simulate and design for structural engineering](https://mlsanity.com/paper/2003.09103/) (2020)

## Built on by (most cited first, 30 of 33)

- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [HAMLET: Graph Transformer Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2402.03541/) (2024)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [Swarm Reinforcement Learning For Adaptive Mesh Refinement](https://mlsanity.com/paper/2304.00818/) (2023)
- [Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks](https://mlsanity.com/paper/2308.01602/) (2023)
- [Graph Neural PDE Solvers with Conservation and Similarity-Equivariance](https://mlsanity.com/paper/2405.16183/) (2024)
- [SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2403.19507/) (2024)
- [Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers for non-stationary and nonlinear simulations on arbitrary meshes](https://mlsanity.com/paper/2402.10681/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [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)
- [Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamics](https://mlsanity.com/paper/2305.14286/) (2023)
- [Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates](https://mlsanity.com/paper/2506.05513/) (2025)
- [A Two-Phase Deep Learning Framework for Adaptive Time-Stepping in High-Speed Flow Modeling](https://mlsanity.com/paper/2506.07969/) (2025)
- [Imposing Boundary Conditions on Neural Operators via Learned Function Extensions](https://mlsanity.com/paper/2602.04923/) (2026)
- [ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning](https://mlsanity.com/paper/2602.11626/) (2026)
- [FIGNN: Feature-Specific Interpretability for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2506.11398/) (2025)
- [Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity](https://mlsanity.com/paper/2507.05291/) (2025)
- [A Reward-Directed Diffusion Framework for Generative Design Optimization](https://mlsanity.com/paper/2508.01509/) (2025)
- [Learning Equivariant Functions via Quadratic Forms](https://mlsanity.com/paper/2509.22184/) (2025)
- [Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition](https://mlsanity.com/paper/2511.23037/) (2025)
- [DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs](https://mlsanity.com/paper/2603.01762/) (2026)
- [Flexible SE(2) graph neural networks with applications to PDE surrogates](https://mlsanity.com/paper/2405.20287/) (2024)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates](https://mlsanity.com/paper/2602.01941/) (2026)
- [Enforcing Dirichlet Boundary Conditions in Operator Learning](https://mlsanity.com/paper/2608.27256/) (2026)
- [Multi-Level GNN Preconditioner for Solving Large Scale Problems](https://mlsanity.com/paper/2402.08296/) (2024)
- [Hamiltonian Neural PDE Solvers through Functional Approximation](https://mlsanity.com/paper/2505.13275/) (2025)
- [FlowSymm: Physics Aware, Symmetry Preserving Graph Attention for Network Flow Completion](https://mlsanity.com/paper/2601.22317/) (2026)
- [An ALE-Consistent Graph Neural Operator-Transformer Framework for Fluid-Structure Interaction](https://mlsanity.com/paper/2605.00937/) (2026)
- [MC2: Monte Carlo Correction for Fast Elliptic PDE Solving](https://mlsanity.com/paper/2605.09288/) (2026)

## Abstract (arXiv)

Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models inadequately treat boundary conditions essential for the reliable prediction of such problems. In addition, because of the locally connected nature of GNNs, it is difficult to accurately predict the state after a long time, where interaction between vertices tends to be global. We present our approach termed physics-embedded neural networks that considers boundary conditions and predicts the state after a long time using an implicit method. It is built based on an E(n)-equivariant GNN, resulting in high generalization performance on various shapes. We demonstrate that our model learns flow phenomena in complex shapes and outperforms a well-optimized classical solver and a state-of-the-art machine learning model in speed-accuracy trade-off. Therefore, our model can be a useful standard for realizing reliable, fast, and accurate GNN-based PDE solvers. The code is available at https://github.com/yellowshippo/penn-neurips2022.