arxiv_ml 95% Match Research Paper Scientific Machine Learning Researchers,Computational Scientists,Engineers,Physicists 3 weeks ago

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

graph-neural-networks › graph-learning

📄 Abstract

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.

Authors (6)

Sepehr Mousavi

Shizheng Wen

Levi Lingsch

Maximilian Herde

Bogdan Raonić

Siddhartha Mishra

Submitted

January 31, 2025

arXiv Category

cs.LG

arXiv PDF Code

Key Contributions

RIGNO is a novel end-to-end graph neural network (GNN) based neural operator designed to learn the solution operators of Partial Differential Equations (PDEs) on arbitrary domains represented by point clouds. Its multi-scale architecture, using regional mesh downsampling and incorporating novel elements for spatio-temporal resolution invariance, allows it to generalize robustly to unseen resolutions and outperform existing neural operator baselines in accuracy.

Business Value

Accelerates scientific discovery and engineering design by enabling faster and more accurate simulations of physical phenomena, reducing the need for traditional, computationally expensive solvers.

Paper Metadata

Innovation Type

Algorithmic Framework

Deployment Feasibility

The availability of public code suggests a commitment to usability. The approach is end-to-end, potentially simplifying integration.

Limitations Addressed

The challenge of learning PDE solution operators on arbitrary domain shapes and generalizing to unseen resolutions, which are limitations of previous neural operator approaches.

Performance Gains

Significantly more accurate than neural operator baselines and robustly generalizes to unseen resolutions in space and time.

View Code on GitHub

Technical Tags

Neural OperatorsGraph Neural NetworksPartial Differential Equations (PDEs)Operator LearningArbitrary DomainsPoint CloudsMulti-scale ModelResolution Invariance

Research Topics

Scientific Machine LearningNeural OperatorsGraph Representation LearningPartial Differential Equations

Methods & Architectures

Graph Neural Network (GNN) based Neural OperatorMulti-scale modelingRegional mesh downsampling Neural OperatorGraph Neural Network (GNN)

Applications & Tasks

Physics Simulation Fluid Dynamics Weather Forecasting Materials Science Computational Engineering Learning PDE solution operatorsHandling arbitrary domain shapesGeneralization to unseen resolutions Solving PDEsPredicting PDE solutionsOperator learning

Datasets & Benchmarks

Benchmarks

Challenging suite of benchmarks (various time-dependent and steady PDEs on diverse domains)

AccuracyGeneralization to unseen resolutions (space and time)

Related Fields

Scientific Machine LearningPartial Differential EquationsGraph Neural NetworksNumerical AnalysisComputational Physics

Keywords

neural operatorsgraph neural networksPDEsoperator learningscientific machine learningarbitrary domainspoint cloudsmulti-scaleresolution invariancesimulationcomputational physicsfluid dynamicsmesh

Academic Context

ETH Zurich (implied by camlab-ethz) #Scientific Machine Learning#Neural Operators#Graph Representation Learning#Partial Differential Equations

Companies & Organizations

Research Institutions

ETH Zurich (implied by camlab-ethz)

Technology Stack

Frameworks & Libraries

Graph Neural Networks (GNNs)

Commercial Potential

Potential Products

Simulation software for scientific and engineering applicationsAccelerated PDE solversPhysics-informed ML platforms

Target Industries

AerospaceAutomotiveEnergyMaterials ScienceClimate ModelingPharmaceuticals

Use Case Examples

Simulating airflow over complex aircraft designsPredicting weather patterns on irregular geographical terrainsModeling fluid dynamics in complex microfluidic devices

Competitive Edge

Offers a more general and accurate approach to learning PDE solution operators compared to existing neural operator baselines, particularly for complex geometries and varying resolutions.

Market Opportunity

Large market for simulation software in engineering and scientific research.

Resource Requirements

Compute Needs

Training neural operators, especially on complex PDEs and domains, requires significant computational resources (GPUs).

Data Requirements

Requires datasets of PDE solutions generated from simulations or experiments across various domains and resolutions.

Deployment Constraints

Integration into existing simulation workflows might require adaptation. Real-time performance needs depend on the specific PDE and domain complexity.

Scalability

The multi-scale approach and generalization to unseen resolutions suggest good scalability properties.

Production Readiness

Maturity Level

Research

Licensing

Publicly available code suggests open-source or permissive licensing.

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