Graph Spectral Neural Operators: Learning Space-Time PDE Solutions on Arbitrary Geometries

Abdolmehdi Behroozi, Chaopeng Shen

Keywords: Neural Operators, Spectral Methods, Graph Learning, Irregular Geometries, PDE Solvers

TL;DR: GSNO solves PDEs on irregular domains using graph and Fourier spectral methods.

Abstract: Learning solution operators for partial differential equations (PDEs) on arbitrary geometries remains a major challenge. Traditional spectral methods are limited to regular domains, while existing neural approaches often struggle to capture global spatiotemporal structures efficiently. We introduce the Graph Spectral Neural Operator (GSNO), a geometry-adaptive framework that combines graph spectral decompositions for spatial learning with real-valued Fourier transforms for temporal modeling. By learning a joint space–time spectral kernel, GSNO enables globally coherent and mesh-invariant operator learning without domain warping or heavy graph convolutional overhead. Across a variety of steady and time-dependent PDE problems, GSNO demonstrates improved accuracy compared to well-known neural operators on irregular geometries, along with reduced runtimes. These results suggest GSNO as a scalable and resolution-robust spectral operator, capable of generalizing to higher resolutions on complex geometries and contributing to scientific machine learning for physical systems.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 12445