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

arXiv: https://arxiv.org/abs/2508.09627 · HTML: https://mlsanity.com/paper/2508.09627/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Aug 2025.

## Built on by (most cited first, 3 of 3)

- [Geometric Laplace Neural Operator](https://mlsanity.com/paper/2512.16409/) (2025)
- [Do Neural Operators Forget Geometry? The Forgetting Hypothesis in Deep Operator Learning](https://mlsanity.com/paper/2605.05862/) (2026)
- [Local gradient neural operator](https://mlsanity.com/paper/2609.07752/) (2026)

## Abstract (arXiv)

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 (πG-Sp2GNO) for learning the solution operators of time-independent and time-dependent PDEs. The proposed approach first improves upon the recently developed Sp2GNO 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.