arXiv：cs.LG（机器学习，全量分类）· Jan Tauberschmidt, Jephte Abijuru, Samuel Okon, Naukshatro Bose, Sophie Fellenz, Marius Kloft, Jonas Latz, Sebastian Josef Vollmer·· 14 小时前AI 评分37
GObO：用格林观测算子学习子流形间的 PDE 动力学
Learning PDE Dynamics between Submanifolds Using Green's Observation Operators
AI 导读
研究者提出格林观测算子（GObO），将固定环境介质一次性映射为限制在源与观测子流形上的线性 PDE 格林核，新源只需一次低维积分、无需网络推理。在三维热传导与对流扩散任务上，GObO 仅用静态源训练即可零样本预测移动源响应，误差比黑盒代理模型低 4–8 倍，单次查询 1.4 ms。该核可跨分辨率迁移，并支持对辐射损耗、温度相关电导率等弱非线性做免重训练修正。
正文
Abstract:Many physical systems are driven and observed only on lower-dimensional submanifolds of a larger spatial domain, while their dynamics are governed by the ambient medium occupying that domain. Examples include laser-heated parts imaged by an infrared camera, and ground-level emissions measured on a sensor plane. Full-domain solvers, however, compute the entire volume for every new source although only the observation submanifold is needed, and black-box surrogates do not exploit that the ambient medium remains fixed. We introduce the \emph{Green's Observation Operator (GObO)}, which maps the ambient medium once to the Green's kernel of a linear PDE restricted to the source and observation submanifolds. New sources then cost one lower-dimensional integral and no network evaluation. Exponential rates in the kernel yield an exact finite streaming state with horizon-independent memory; we prove its stability and an approximation rate for the restricted heat kernel. On three-dimensional heat conduction and advection--diffusion with collocated and distinct source and observation geometries, GObO trained on static sources predicts responses to moving sources zero-shot with 4--8$\times$ lower error than black-box surrogates, at 1.4\,ms per query after a single conditioning pass. The same kernel transfers across resolutions and admits corrections for mild nonlinearities, including radiative losses and temperature-dependent conductivity, without retraining, at the cost of lower in-distribution accuracy.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.01697 [cs.LG] |
| (or arXiv:2610.01697v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.01697 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Jan Tauberschmidt [view email]
[v1]
Thu, 1 Oct 2026 13:47:22 UTC (1,972 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org