发表机构
Penn State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对不规则域上PDE解算子学习的挑战，提出GSNO，通过统一时空谱核实现全局一致的算子学习，在PDE基准测试中精度优异且零样本泛化能力强。
AI 中文摘要
在不规则且依赖几何的域上学习偏微分方程（PDE）的解算子，仍是科学机器学习领域的核心挑战。尽管谱方法为建模全局交互提供了强归纳偏置，但通常局限于规则域，而现有神经方法往往需要域变形、插值或高代价的几何嵌入。我们提出图谱神经算子（Graph Spectral Neural Operator, GSNO），这是一种通过统一时空谱核将空间图谱分解与时间傅里叶变换相结合的神经算子。该公式可在非笛卡尔离散化上实现全局一致的算子学习，无需域变形或自回归滚动。通过用图拉普拉斯谱基替代学习到的几何嵌入，GSNO以低参数复杂度实现了感知几何的谱学习。在不规则及依赖几何域上的稳态与非稳态PDE基准测试中，GSNO在降低运行时间和参数数量的同时达到了优异精度，还展现出跨网格分辨率和几何族的零样本泛化能力。
英文摘要
Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the \textbf{Graph Spectral Neural Operator (GSNO)}, a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.
