Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916896032423936 |
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| author | Sarkar, Subhankar Chakraborty, Souvik |
| author_facet | Sarkar, Subhankar Chakraborty, Souvik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09627 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs Sarkar, Subhankar Chakraborty, Souvik Machine Learning 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. |
| title | Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2508.09627 |