Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid

TL;DR

This paper introduces Sp$^2$GNO, a novel spatio-spectral graph neural operator that effectively learns solution operators for PDEs on irregular domains and unstructured grids, overcoming limitations of existing methods.

Contribution

The paper proposes a new spatio-spectral GNN framework that combines local and global convolution to handle complex geometries efficiently.

Findings

Achieves high accuracy on PDEs in irregular domains

Handles both time-dependent and independent problems

Validated through extensive benchmarks and real-world applications

Abstract

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing in deep architectures. Conversely, spectral graph neural networks leverage global convolution to capture extensive features and long-range dependencies in domain graphs, albeit at a high computational cost due to Eigenvalue decomposition. In this paper, we introduce a novel approach, referred to as Spatio-Spectral Graph Neural Operator (Sp$^2$GNO) that integrates spatial and spectral GNNs effectively. This framework mitigates the limitations of individual methods and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems. Sp$^2$GNO demonstrates exceptional performance in solving both time-dependent and time-independent partial differential equations on regular and irregular domains. Our approach is validated through comprehensive benchmarks and practical applications drawn from computational mechanics and scientific computing literature.