发表机构
Beijing University of Technology; The University of Sydney（北京工业大学; 悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对标准图神经算子在核参数化上的高内存开销问题，提出基于函数张量分解的FTD-GNO，通过CP、Tensor-Train和Tucker分解解耦核，降低内存与训练成本，并保持算子学习性能。
AI 中文摘要
图神经算子（GNOs）为学习偏微分方程（PDEs）的解算子提供了灵活的替代模型。然而，标准GNOs通常使用一个整体神经网络对积分核进行参数化，并在图边上评估核交互，这导致在高分辨率或大邻域情况下产生大量的计算和内存开销。为了解决这些局限性，我们提出了函数张量分解图神经算子（FTD-GNO），一种内存高效的GNO框架，它将高维连续积分核解耦为低维模态函数。通过使用经典的张量分解格式（包括CP、Tensor-Train和Tucker分解）实例化核，FTD-GNO能够以代数方式重构积分算子，而无需显式地物化完整的逐边核张量。这种分解形式降低了核评估和聚合的内存占用，同时保留了GNO的连续算子学习结构。理论复杂度分析表明，FTD-GNO显著降低了与高维核构建相关的参数和激活内存成本。实验表明，与相应的未分解图积分基线相比，FTD-GNO的峰值内存更低，且记录的训练时间更短。傅里叶图实验进一步证明，FTD可以提高混合算子中图积分层的效率，并具有良好的可扩展性。
英文摘要
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.
Comments19 pages and 2 figures
