Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries

• URL: http://arxiv.org/abs/2411.15178v3
• Date: Fri, 07 Feb 2025 13:53:41 GMT
• Title: Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries
• Abstract summary: This paper introduces the AMG method, a Multi-Graph neural operator approach designed for efficiently solving PDEs on Arbitrary geometries.<n> AMG leverages advanced graph-based techniques and dynamic attention mechanisms within a novel GraphFormer architecture.<n>We present a comprehensive evaluation of AMG across six benchmarks, demonstrating its consistent superiority over existing state-of-the-art models.
• Score: 9.954826518381468
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the AMG method, a Multi-Graph neural operator approach designed for efficiently solving PDEs on Arbitrary geometries. AMG leverages advanced graph-based techniques and dynamic attention mechanisms within a novel GraphFormer architecture, enabling precise management of diverse spatial domains and complex data interdependencies. By constructing multi-scale graphs to handle variable feature frequencies and a physics graph to encapsulate inherent physical properties, AMG significantly outperforms previous methods, which are typically limited to uniform grids. We present a comprehensive evaluation of AMG across six benchmarks, demonstrating its consistent superiority over existing state-of-the-art models. Our findings highlight the transformative potential of tailored graph neural operators in surmounting the challenges faced by conventional PDE solvers. Our code and datasets are available on https://github.com/lizhihao2022/AMG.

Related papers

• Beyond MMD: Evaluating Graph Generative Models with Geometric Deep Learning [0.0]
  Graph Generative Models (GGMs) have emerged as a promising solution to the problem of generating realistic graphs.<n>This paper introduces a novel methodology for evaluating GGMs that overcomes the limitations of Maximum Mean Discrepancy (MMD)<n>We present a comprehensive evaluation of two state-of-the-art Graph Generative Models: Graph Recurrent Attention Networks (GRAN) and Efficient and Degree-guided graph GEnerative model (EDGE)
  arXiv  Detail & Related papers  (2025-12-16T09:51:44Z)
• PEGNet: A Physics-Embedded Graph Network for Long-Term Stable Multiphysics Simulation [8.95344024479836]
  Physical phenomena governed by partial differential equations (PDEs) are important for scientific and engineering progress.<n>PEGNet is a physics-Embedded Graph Network that incorporates PDE-guided message passing to the redesign graph neural network architecture.<n>We show significant improvements in long-term prediction accuracy and physical consistency over existing methods.
  arXiv  Detail & Related papers  (2025-11-11T19:02:16Z)
• Scalable Graph Generative Modeling via Substructure Sequences [50.32639806800683]
  We introduce Generative Graph Pattern Machine (G$2$PM), a generative Transformer pre-training framework for graphs.<n>G$2$PM represents graph instances (nodes, edges, or entire graphs) as sequences of substructures.<n>It employs generative pre-training over the sequences to learn generalizable and transferable representations.
  arXiv  Detail & Related papers  (2025-05-22T02:16:34Z)
• PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems [31.006807854698376]
  We propose a new graph learning approach, namely, Physics-encoded Message Passing Graph Network (PhyMPGN) We incorporate a GNN into a numerical integrator to approximate the temporal marching of partialtemporal dynamics for a given PDE system. PhyMPGN is capable of accurately predicting various types of operatortemporal dynamics on coarse unstructured meshes.
  arXiv  Detail & Related papers  (2024-10-02T08:54:18Z)
• Neural P$^3$M: A Long-Range Interaction Modeling Enhancer for Geometric GNNs [66.98487644676906]
  We introduce Neural P$3$M, a versatile enhancer of geometric GNNs to expand the scope of their capabilities. It exhibits flexibility across a wide range of molecular systems and demonstrates remarkable accuracy in predicting energies and forces. It also achieves an average improvement of 22% on the OE62 dataset while integrating with various architectures.
  arXiv  Detail & Related papers  (2024-09-26T08:16:59Z)
• A hybrid numerical methodology coupling Reduced Order Modeling and Graph Neural Networks for non-parametric geometries: applications to structural dynamics problems [0.0]
  This work introduces a new approach for accelerating the numerical analysis of time-domain partial differential equations (PDEs) governing complex physical systems. The methodology is based on a combination of a classical reduced-order modeling (ROM) framework and recently-parametric Graph Neural Networks (GNNs)
  arXiv  Detail & Related papers  (2024-06-03T08:51:25Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Latent Graph Inference using Product Manifolds [0.0]
  We generalize the discrete Differentiable Graph Module (dDGM) for latent graph learning. Our novel approach is tested on a wide range of datasets, and outperforms the original dDGM model.
  arXiv  Detail & Related papers  (2022-11-26T22:13:06Z)
• MAgNet: Mesh Agnostic Neural PDE Solver [68.8204255655161]
  Climate predictions require fine-temporal resolutions to resolve all turbulent scales in the fluid simulations. Current numerical model solveers PDEs on grids that are too coarse (3km to 200km on each side) We design a novel architecture that predicts the spatially continuous solution of a PDE given a spatial position query.
  arXiv  Detail & Related papers  (2022-10-11T14:52:20Z)
• Multi-scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators [14.466945570499183]
  We study three multi-resolution schema with integral kernel operators approximated with emphMessage Passing Graph Neural Networks (MPGNNs) To validate our study, we make extensive MPGNNs experiments with well-chosen metrics considering steady and unsteady PDEs.
  arXiv  Detail & Related papers  (2022-06-29T14:42:03Z)
• EXPERT: Public Benchmarks for Dynamic Heterogeneous Academic Graphs [5.4744970832051445]
  We present a variety of large scale, dynamic heterogeneous academic graphs to test the effectiveness of models developed for graph forecasting tasks. Our novel datasets cover both context and content information extracted from scientific publications across two communities: Artificial Intelligence (AI) and Nuclear Nonproliferation (NN)
  arXiv  Detail & Related papers  (2022-04-14T19:43:34Z)
• Equivariant Graph Mechanics Networks with Constraints [83.38709956935095]
  We propose Graph Mechanics Network (GMN) which is efficient, equivariant and constraint-aware. GMN represents, by generalized coordinates, the forward kinematics information (positions and velocities) of a structural object. Extensive experiments support the advantages of GMN compared to the state-of-the-art GNNs in terms of prediction accuracy, constraint satisfaction and data efficiency.
  arXiv  Detail & Related papers  (2022-03-12T14:22:14Z)
• Multivariate Time Series Forecasting with Dynamic Graph Neural ODEs [65.18780403244178]
  We propose a continuous model to forecast Multivariate Time series with dynamic Graph neural Ordinary Differential Equations (MTGODE) Specifically, we first abstract multivariate time series into dynamic graphs with time-evolving node features and unknown graph structures. Then, we design and solve a neural ODE to complement missing graph topologies and unify both spatial and temporal message passing.
  arXiv  Detail & Related papers  (2022-02-17T02:17:31Z)
• Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations [57.15855198512551]
  We propose a novel score-based generative model for graphs with a continuous-time framework. We show that our method is able to generate molecules that lie close to the training distribution yet do not violate the chemical valency rule.
  arXiv  Detail & Related papers  (2022-02-05T08:21:04Z)
• GeoMol: Torsional Geometric Generation of Molecular 3D Conformer Ensembles [60.12186997181117]
  Prediction of a molecule's 3D conformer ensemble from the molecular graph holds a key role in areas of cheminformatics and drug discovery. Existing generative models have several drawbacks including lack of modeling important molecular geometry elements. We propose GeoMol, an end-to-end, non-autoregressive and SE(3)-invariant machine learning approach to generate 3D conformers.
  arXiv  Detail & Related papers  (2021-06-08T14:17:59Z)
• Tensor Graph Convolutional Networks for Multi-relational and Robust Learning [74.05478502080658]
  This paper introduces a tensor-graph convolutional network (TGCN) for scalable semi-supervised learning (SSL) from data associated with a collection of graphs, that are represented by a tensor. The proposed architecture achieves markedly improved performance relative to standard GCNs, copes with state-of-the-art adversarial attacks, and leads to remarkable SSL performance over protein-to-protein interaction networks.
  arXiv  Detail & Related papers  (2020-03-15T02:33:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.