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

2026-10-03

AI Summary

This study addresses the prohibitive computational and memory overheads incurred by graph neural operators (GNOs) under high-resolution discretizations or large neighborhoods due to holistic kernel parameterization. To overcome this limitation, we propose the FTD-GNO framework, which pioneers the integration of classical tensor decompositions into continuous integral operator learning. By leveraging CP, Tensor-Train, and Tucker decomposition techniques, the proposed method decouples high-dimensional integral kernels into low-dimensional modal functions, enabling algebraic reconstruction without explicitly instantiating the full edge-wise kernel tensor. Theoretical analysis and empirical evaluations demonstrate that FTD-GNO substantially reduces both model parameter counts and peak memory consumption. Furthermore, it effectively shortens training time while exhibiting superior scalability, offering a principled and efficient solution for scaling GNOs to complex, large-scale operator learning tasks.

Abstract

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.

Problem

• Graph Neural Operators
• Memory overhead
• Integral kernel
• Partial differential equations
• Computational cost

Innovation

• Graph Neural Operators
• Functional Tensor Decomposition
• Memory Efficiency
• Integral Kernel
• Partial Differential Equations