# Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid

arXiv: https://arxiv.org/abs/2409.00604 · HTML: https://mlsanity.com/paper/2409.00604/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 45 references that are in the map.
- First posted on arXiv 1 Sep 2024.

## Builds on (most cited first, 27 of 27)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [LanczosNet: Multi-Scale Deep Graph Convolutional Networks](https://mlsanity.com/paper/1901.01484/) (2019)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [FRIGATE: Frugal Spatio-temporal Forecasting on Road Networks](https://mlsanity.com/paper/2306.08277/) (2023)
- [Assessment of DeepONet for reliability analysis of stochastic nonlinear dynamical systems](https://mlsanity.com/paper/2201.13145/) (2022)
- [Nonlinear model reduction for operator learning](https://mlsanity.com/paper/2403.18735/) (2024)
- [Koopman operator for time-dependent reliability analysis](https://mlsanity.com/paper/2203.02658/) (2022)
- [Generative adversarial wavelet neural operator: Application to fault detection and isolation of multivariate time series data](https://mlsanity.com/paper/2401.04004/) (2024)
- [SHoP: A Deep Learning Framework for Solving High-order Partial Differential Equations](https://mlsanity.com/paper/2305.10033/) (2023)
- [Waveformer for modelling dynamical systems](https://mlsanity.com/paper/2310.04990/) (2023)
- [Generative flow induced neural architecture search: Towards discovering optimal architecture in wavelet neural operator](https://mlsanity.com/paper/2405.06910/) (2024)

## Built on by (most cited first, 6 of 6)

- [Real-Time Sensing of Inaccessible Physical Fields via an Edge-Deployable Hardware-Portable Graph Neural Operator](https://mlsanity.com/paper/2604.01802/) (2026)
- [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)
- [Towards Data-Efficient Cross-Device Generalization of Grad-Shafranov Equilibria via Transfer Learning Neural Operator](https://mlsanity.com/paper/2606.15512/) (2026)
- [Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains](https://mlsanity.com/paper/2608.29892/) (2026)
- [Local gradient neural operator](https://mlsanity.com/paper/2609.07752/) (2026)

## Abstract (arXiv)

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing in deep architectures. Conversely, spectral graph neural networks leverage global convolution to capture extensive features and long-range dependencies in domain graphs, albeit at a high computational cost due to Eigenvalue decomposition. In this paper, we introduce a novel approach, referred to as Spatio-Spectral Graph Neural Operator (Sp2GNO) that integrates spatial and spectral GNNs effectively. This framework mitigates the limitations of individual methods and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems. Sp2GNO demonstrates exceptional performance in solving both time-dependent and time-independent partial differential equations on regular and irregular domains. Our approach is validated through comprehensive benchmarks and practical applications drawn from computational mechanics and scientific computing literature.