发表机构
German Research Center for Artificial Intelligence (DFKI); Rheinland-Pfälzische Technische Universität (RPTU); University of Manchester（德国人工智能研究中心; 莱茵兰-普法尔茨工业大学; 曼彻斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出格林观测算子（GObO），通过将周围介质映射为受限格林核，实现子流形间PDE动力学的零样本预测，误差较黑盒替代模型降低4-8倍，且支持跨分辨率迁移与非线性修正。
AI 中文摘要
许多物理系统仅在大空间域的低维子流形上被驱动和观测，而其动力学由占据该域的周围介质控制。例如，红外相机成像的激光加热部件，以及传感器平面上测量的地面排放。然而，全域求解器为每个新源计算整个体积，尽管只需要观测子流形，而黑盒替代模型没有利用周围介质保持不变的事实。我们引入了格林观测算子（GObO），它将周围介质一次映射到限制在源和观测子流形上的线性PDE的格林核。新的源随后只需一次低维积分，无需网络评估。核中的指数速率产生精确的有限流式状态，具有与视界无关的记忆；我们证明了其稳定性以及受限热核的逼近速率。在三维热传导和具有共置及不同源与观测几何的对流-扩散中，在静态源上训练的GObO以零样本预测移动源的响应，误差比黑盒替代模型低4-8倍，在单次条件化传递后每次查询仅需1.4毫秒。同一核可跨分辨率迁移，并允许对轻微非线性进行修正，包括辐射损失和温度相关的电导率，无需重新训练，但代价是分布内精度较低。
英文摘要
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.
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