Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs

TL;DR

This paper introduces a physics- and geometry-aware spatio-spectral graph neural operator that efficiently learns PDE solution operators for complex geometries and time-dependent problems, outperforming existing methods.

Contribution

It extends the Sp$^2$GNO framework by incorporating geometry awareness and physics-informed loss functions for improved PDE solving in complex scenarios.

Findings

Achieves accurate PDE solutions on complex geometries

Outperforms state-of-the-art physics-informed neural operators

Effective for both time-independent and time-dependent PDEs

Abstract

Solving partial differential equations (PDEs) efficiently and accurately remains a cornerstone challenge in science and engineering, especially for problems involving complex geometries and limited labeled data. We introduce a Physics- and Geometry- Aware Spatio-Spectral Graph Neural Operator ($\pi$G-Sp$^2$GNO) for learning the solution operators of time-independent and time-dependent PDEs. The proposed approach first improves upon the recently developed Sp$^2$GNO by enabling geometry awareness and subsequently exploits the governing physics to learn the underlying solution operator in a simulation-free setup. While the spatio-spectral structure present in the proposed architecture allows multiscale learning, two separate strategies for enabling geometry awareness is introduced in this paper. For time dependent problems, we also introduce a novel hybrid physics informed loss function that combines higher-order time-marching scheme with upscaled theory inspired stochastic projection scheme. This allows accurate integration of the physics-information into the loss function. The performance of the proposed approach is illustrated on number of benchmark examples involving regular and complex domains, variation in geometry during inference, and time-independent and time-dependent problems. The results obtained illustrate the efficacy of the proposed approach as compared to the state-of-the-art physics-informed neural operator algorithms in the literature.