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

• URL: http://arxiv.org/abs/2508.09627v1
• Date: Wed, 13 Aug 2025 08:59:04 GMT
• Title: Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs
• Authors: Subhankar Sarkar, Souvik Chakraborty,
• Abstract summary: We introduce a Physics- and Geometry- Aware Spatio-tral Graph Neural Operator for learning the solution operators of time-independent and time-dependent PDEs.<n>The proposed approach first improves upon recently developed Sp$2$GNO by enabling geometry awareness.<n>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 projection.
• Score: 0.9208007322096533
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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