RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains

• URL: http://arxiv.org/abs/2501.19205v2
• Date: Tue, 14 Oct 2025 08:58:18 GMT
• Title: RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains
• Authors: Sepehr Mousavi, Shizheng Wen, Levi Lingsch, Maximilian Herde, Bogdan Raonić, Siddhartha Mishra,
• Abstract summary: We propose an end-to-end graph neuralsample (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains.<n>Our model, termed RIGNO, is tested on a challenging suite of benchmarks composed of various time-dependent and steady PDEs defined on a diverse set of domains.
• Score: 10.93287301092333
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. The approach includes novel elements aimed at ensuring spatio-temporal resolution invariance. Our model, termed RIGNO, is tested on a challenging suite of benchmarks composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen resolutions both in space and in time. Our code is publicly available at github.com/camlab-ethz/rigno.

Related papers

• Scale-Consistent Learning for Partial Differential Equations [79.48661503591943]
  We propose a data augmentation scheme based on scale-consistency properties of PDEs.<n>We then design a scale-informed neural operator that can model a wide range of scales.<n>With scale-consistency, the model trained on $Re$ of 1000 can generalize to $Re$ ranging from 250 to 10000.
  arXiv  Detail & Related papers  (2025-07-24T21:29:52Z)
• Domain Adaptation and Entanglement: an Optimal Transport Perspective [86.24617989187988]
  Current machine learning systems are brittle in the face of distribution shifts (DS), where the target distribution that the system is tested on differs from the source distribution used to train the system. For deep neural networks, a popular framework for unsupervised domain adaptation (UDA) is domain matching, in which algorithms try to align the marginal distributions in the feature or output space. In this paper, we derive new bounds based on optimal transport that analyze the UDA problem.
  arXiv  Detail & Related papers  (2025-03-11T08:10:03Z)
• Mechanistic PDE Networks for Discovery of Governing Equations [52.492158106791365]
  We present Mechanistic PDE Networks, a model for discovery of partial differential equations from data. The represented PDEs are then solved and decoded for specific tasks. We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations.
  arXiv  Detail & Related papers  (2025-02-25T17:21:44Z)
• Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids [14.93012615797081]
  GraphDeepONet is an autoregressive model based on graph neural networks (GNNs) It exhibits robust accuracy in predicting solutions compared to existing GNN-based PDE solver models. Unlike traditional DeepONet and its variants, GraphDeepONet enables time extrapolation for time-dependent PDE solutions.
  arXiv  Detail & Related papers  (2024-02-13T03:14:32Z)
• BENO: Boundary-embedded Neural Operators for Elliptic PDEs [15.18712698704595]
  We introduce Boundary-Embedded Neural Operators (BENO) for solving elliptic PDEs. BENO embeds the complex geometries and inhomogeneous boundary values into the solving of elliptic PDEs. Our model outperforms state-of-the-art neural operators and strong baselines by an average of 60.96%.
  arXiv  Detail & Related papers  (2024-01-17T16:47:39Z)
• Operator Learning with Neural Fields: Tackling PDEs on General Geometries [15.65577053925333]
  Machine learning approaches for solving partial differential equations require learning mappings between function spaces. New CORAL method leverages coordinate-based networks for PDEs on some general constraints.
  arXiv  Detail & Related papers  (2023-06-12T17:52:39Z)
• Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticity [58.44709568277582]
  Physics-informed PointNet (PIPN) is designed to fill this gap between PINNs and fully supervised learning models. We show that PIPN predicts the solution of desired partial differential equations over a few hundred domains simultaneously. Specifically, we show that PIPN predicts the solution of a plane stress problem over more than 500 domains with different geometries, simultaneously.
  arXiv  Detail & Related papers  (2023-03-22T06:49:34Z)
• RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network [4.425915683879297]
  We propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD) RBF-FD is used to construct a high-precision difference format of the differential equations to guide model training. We illustrate the generalizability, accuracy, and efficiency of the proposed algorithms on different PDE parameters.
  arXiv  Detail & Related papers  (2022-12-06T10:08:02Z)
• 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)
• PhyGNNet: Solving spatiotemporal PDEs with Physics-informed Graph Neural Network [12.385926494640932]
  We propose PhyGNNet for solving partial differential equations on the basics of a graph neural network. In particular, we divide the computing area into regular grids, define partial differential operators on the grids, then construct pde loss for the network to optimize to build PhyGNNet model.
  arXiv  Detail & Related papers  (2022-08-07T13:33:34Z)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Improved Training of Physics-Informed Neural Networks with Model Ensembles [81.38804205212425]
  We propose to expand the solution interval gradually to make the PINN converge to the correct solution. All ensemble members converge to the same solution in the vicinity of observed data. We show experimentally that the proposed method can improve the accuracy of the found solution.
  arXiv  Detail & Related papers  (2022-04-11T14:05:34Z)
• EIGNN: Efficient Infinite-Depth Graph Neural Networks [51.97361378423152]
  Graph neural networks (GNNs) are widely used for modelling graph-structured data in numerous applications. Motivated by this limitation, we propose a GNN model with infinite depth, which we call Efficient Infinite-Depth Graph Neural Networks (EIGNN) We show that EIGNN has a better ability to capture long-range dependencies than recent baselines, and consistently achieves state-of-the-art performance.
  arXiv  Detail & Related papers  (2022-02-22T08:16:58Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Handling Distribution Shifts on Graphs: An Invariance Perspective [78.31180235269035]
  We formulate the OOD problem on graphs and develop a new invariant learning approach, Explore-to-Extrapolate Risk Minimization (EERM) EERM resorts to multiple context explorers that are adversarially trained to maximize the variance of risks from multiple virtual environments. We prove the validity of our method by theoretically showing its guarantee of a valid OOD solution.
  arXiv  Detail & Related papers  (2022-02-05T02:31:01Z)
• GrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equations [0.0]
  We propose a novel framework referred to as the Graph Attention Differential Equation (GrADE) for solving time dependent nonlinear PDEs. The proposed approach couples FNN, graph neural network, and recently developed Neural ODE framework. Results obtained illustrate the capability of the proposed framework in modeling PDE and its scalability to larger domains without the need for retraining.
  arXiv  Detail & Related papers  (2021-08-24T10:49:03Z)
• PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs [8.220908558735884]
  Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (NNs) to solve PDEs as a basis for data-driven inverse analysis. We propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhCRyNet-s) for solving PDEs without any labeled data.
  arXiv  Detail & Related papers  (2021-06-26T22:22:19Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.