mlsanityData to 7 Oct 2026

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

arXiv 2205.11912 ↗First posted 24 May 2022cs.LG cs.CEcited by 33

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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.

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). Builds on 13 of its 36 references that are in the map. First posted on arXiv 24 May 2022.

Builds on

13 of 36 references.

References · 13

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  4. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
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  6. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  7. 2019Deep Double Descent: Where Bigger Models and More Data Hurtcited by 667
  8. 2022Message Passing Neural PDE Solverscited by 368
  9. 2017Equivariance Through Parameter-Sharingcited by 139
  10. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 123
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1 more
  1. 2020Learning to simulate and design for structural engineeringcited by 9

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Citing papers per year: 2023: 5; 2024: 9; 2025: 8; 2026: 11.

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  1. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  2. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  3. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  4. 2023Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networkscited by 17
  5. 2023Swarm Reinforcement Learning For Adaptive Mesh Refinementcited by 16
  6. 2024Graph Neural PDE Solvers with Conservation and Similarity-Equivariancecited by 15
  7. 2024SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equationscited by 13
  8. 2024Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers for non-stationary and nonlinear simulations on arbitrary meshescited by 12
  9. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  10. 2024Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular gridscited by 7
  11. 2023Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamicscited by 6
  12. 2025Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogatescited by 5
21 more
  1. 2026Imposing Boundary Conditions on Neural Operators via Learned Function Extensionscited by 3
  2. 2026ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learningcited by 3
  3. 2025A Two-Phase Deep Learning Framework for Adaptive Time-Stepping in High-Speed Flow Modelingcited by 2
  4. 2025FIGNN: Feature-Specific Interpretability for Graph Neural Network Surrogate Modelscited by 2
  5. 2025Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticitycited by 2
  6. 2025A Reward-Directed Diffusion Framework for Generative Design Optimizationcited by 2
  7. 2025Learning Equivariant Functions via Quadratic Formscited by 2
  8. 2025Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decompositioncited by 2
  9. 2024Flexible SE(2) graph neural networks with applications to PDE surrogatescited by 1
  10. 2024Physics-embedded Fourier Neural Network for Partial Differential Equationscited by 1
  11. 2026DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEscited by 1
  12. 2026Enforcing Dirichlet Boundary Conditions in Operator Learningcited by 1
  13. 2024Multi-Level GNN Preconditioner for Solving Large Scale Problemscited by 0
  14. 2025Hamiltonian Neural PDE Solvers through Functional Approximationcited by 0
  15. 2026FlowSymm: Physics Aware, Symmetry Preserving Graph Attention for Network Flow Completioncited by 0
  16. 2026FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogatescited by 0
  17. 2026An ALE-Consistent Graph Neural Operator-Transformer Framework for Fluid-Structure Interactioncited by 0
  18. 2026MC2: Monte Carlo Correction for Fast Elliptic PDE Solvingcited by 0
  19. 2026RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Correctionscited by 0
  20. 2026Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networkscited by 0
  21. 2026CraftSPH: A high-accuracy and composable differentiable SPH solver implemented in PyTorchcited by 0