新模型直接在不规则域上高效求解偏微分方程，无需坐标变换或逐步推理。
Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains
- 用图谱分解+时域傅里叶变换构建统一时空谱核
- 在不规则网格上实现高精度求解，参数少、速度快
- 零样本泛化到不同网格密度和几何类型，适合复杂物理建模
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中的核心挑战。尽管谱方法对全局相互作用建模具有强归纳偏置，但通常局限于规则域，现有神经方法常需域映射、插值或昂贵的几何嵌入。我们提出图谱神经算子（GSNO），通过统一时空谱核，将空间图谱分解与时间傅里叶变换结合，实现非笛卡尔离散化下的全局一致算子学习，无需域映射或自回归推演。通过用图拉普拉斯谱基替代学习的几何嵌入，GSNO实现了低参数复杂度的几何感知谱学习。在稳态与非稳态PDE基准测试中，其在不规则与几何依赖域上均取得高精度，运行时间更短、参数量更低，并展现出对网格分辨率和几何族的强零样本泛化能力。
原文摘要 · Abstract (English)
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.
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