Learning PDE Dynamics between Submanifolds Using Green's Observation Operators

2026-10-01

AI Summary

This study addresses the global solution redundancy inherent in low-dimensional manifold observations of physical systems and the neglect of medium properties by black-box surrogates. To this end, we propose the Green Observation Operator (GObO), which maps environmental media into restricted Green's kernels, enabling new source response prediction via low-dimensional integration without network re-evaluation. We theoretically establish exponential convergence and stability guarantees. Furthermore, GObO supports zero-shot generalization to moving sources, cross-resolution transfer, and nonlinearity correction without retraining. Evaluated on tasks including 3D heat conduction, our method reduces errors by a factor of 4–8 compared to black-box surrogates while achieving single-query inference in merely 1.4 ms.

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.

Problem

• Partial Differential Equations
• Submanifolds
• Green's Function
• Surrogate Models
• Observation Operators

Innovation

• Green's Observation Operator
• Submanifold learning
• Zero-shot generalization
• Finite streaming state
• Surrogate modeling