# FTD-GNO: Memory-Efficient Graph Neural Operators through Functional Tensor Decomposition of the Kernel

arXiv: https://arxiv.org/abs/2610.04212 · HTML: https://mlsanity.com/paper/2610.04212/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 33 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 16 of 16)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [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)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Functional Tensor Decompositions for Physics-Informed Neural Networks](https://mlsanity.com/paper/2408.13101/) (2024)
- [F-INR: Functional Tensor Decomposition for Implicit Neural Representations](https://mlsanity.com/paper/2503.21507/) (2025)
- [TensorGRaD: Tensor Gradient Robust Decomposition for Memory-Efficient Neural Operator Training](https://mlsanity.com/paper/2501.02379/) (2025)
- [Robust Tensor Decomposition for Image Representation Based on Generalized Correntropy](https://mlsanity.com/paper/2005.04605/) (2020)
- [Comparison of Neural FEM and Neural Operator Methods for applications in Solid Mechanics](https://mlsanity.com/paper/2307.02494/) (2023)
- [Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Families](https://mlsanity.com/paper/2602.11630/) (2026)

## Abstract (arXiv)

Graph Neural Operators (GNOs) provide flexible surrogate models for learning solution operators of partial differential equations (PDEs). However, standard GNOs typically parameterize the integral kernel with a monolithic neural network and evaluate kernel interactions over graph edges, leading to substantial computational and memory overhead at high resolutions or with large neighborhoods. To address these limitations, we propose Functional Tensor Decomposition Graph Neural Operator (FTD-GNO), a memory-efficient GNO framework that decouples the high-dimensional continuous integral kernel into low-dimensional mode-wise functions. By instantiating the kernel with classical tensor decomposition formats, including CP, Tensor-Train, and Tucker decompositions, FTD-GNO enables algebraic reconstruction of the integral operator without explicitly materializing full edge-wise kernel tensors. This factorized formulation reduces the memory footprint of kernel evaluation and aggregation while retaining the continuous operator-learning structure of GNOs. Theoretical complexity analysis shows that FTD-GNO substantially lowers parameter and activation-memory costs associated with high-dimensional kernel construction. Experiments show lower peak memory than the corresponding unfactorized graph-integral baselines, with shorter recorded training times. Fourier-graph experiments further demonstrate that FTD can improve the efficiency of a graph-integral layer within a hybrid operator and has good scalability.