mlsanityData to 7 Oct 2026

Graph Neural PDE Solvers with Conservation and Similarity-Equivariance

arXiv 2405.16183 ↗First posted 25 May 2024cs.LG cs.AI cs.CEcited by 15

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Abstract (arXiv)

Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all potential scenarios through data-driven methodologies alone. Moreover, there are legitimate concerns regarding the generalization and reliability of such approaches, as they often overlook inherent physical constraints. In response to these challenges, this study introduces a novel machine-learning architecture that is highly generalizable and adheres to conservation laws and physical symmetries, thereby ensuring greater reliability. The foundation of this architecture is graph neural networks (GNNs), which are adept at accommodating a variety of shapes and forms. Additionally, we explore the parallels between GNNs and traditional numerical solvers, facilitating a seamless integration of conservative principles and symmetries into machine learning models. Our findings from experiments demonstrate that the model's inclusion of physical laws significantly enhances its generalizability, i.e., no significant accuracy degradation for unseen spatial domains while other models degrade. The code is available at https://github.com/yellowshippo/fluxgnn-icml2024.

Key facts. Cited by 15 papers in the map (data through 7 Oct 2026). Builds on 22 of its 38 references that are in the map. First posted on arXiv 25 May 2024.

Builds on

22 of 38 references.

References · 22

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10 more
  1. 2019Scale-Equivariant Steerable Networkscited by 110
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 4; 2026: 8.

15 papers.

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  1. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  2. 2024Conservation-informed Graph Learning for Spatiotemporal Dynamics Predictioncited by 4
  3. 2026Simple yet Effective: Low-Rank Spatial Attention for Neural Operatorscited by 4
  4. 2025Dynami-CAL GraphNet: A Physics-Informed Graph Neural Network Conserving Linear and Angular Momentum for Dynamical Systemscited by 3
  5. 2024Spatiotemporal Graph Learning with Direct Volumetric Information Passing and Feature Enhancementcited by 1
  6. 2025Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flowscited by 1
  7. 2024A nudge to the truth: atom conservation as a hard constraint in models of atmospheric composition using an uncertainty-weighted correctioncited by 0
  8. 2025SUPRA: Subspace Parameterized Attention for Neural Operator on General Domainscited by 0
  9. 2025OmniFluids: Physics Pre-trained Modeling of Fluid Dynamicscited by 0
  10. 2026Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEscited by 0
  11. 2026FlowSymm: Physics Aware, Symmetry Preserving Graph Attention for Network Flow Completioncited by 0
  12. 2026FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogatescited by 0
3 more
  1. 2026A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Lawscited by 0
  2. 2026Conservative Hybrid Graph Networks for Process Systems with Learned Routingcited by 0
  3. 2026Neural Constitutive Learning for Generalized Reaction-Diffusion Systemscited by 0