{"ID":2882698,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.09627","arxiv_id":"2508.09627","title":"Physics- and geometry-aware spatio-spectral graph neural operator for 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 ($π$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.","short_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 ($π$G-Sp$^2$GNO) for learn...","url_abs":"https://arxiv.org/abs/2508.09627","url_pdf":"https://arxiv.org/pdf/2508.09627v1","authors":"[\"Subhankar Sarkar\",\"Souvik Chakraborty\"]","published":"2025-08-13T08:59:04Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}