Learning partial differential dynamics from lower dimensional observations
Learning PDE Dynamics between Submanifolds Using Green's Observation Operators
Machine Learning
Summary
Many physical systems can only be observed or affected on smaller surfaces or curves, while their actual behavior happens in a larger surrounding space. The authors propose a new method called Green's Observation Operator to efficiently learn how disturbances affect observations on these smaller parts without simulating the entire space every time. Their approach uses mathematical kernels once computed, enabling fast and accurate predictions for new situations with less computational effort. This method also adapts to different scales and mild nonlinear effects without retraining.
What this means in practice
- •For thermal engineers: Simulate heat distribution effects on sensor surfaces quickly without computing full 3D volumes for varying heat sources.
- •For environmental sensing teams: Model pollutant transport and concentration only at ground-level sensor arrays using fixed medium properties for faster predictions.
Authors
Jan Tauberschmidt, Jephte Abijuru, Samuel Okon, Naukshatro Bose, Sophie Fellenz, Marius Kloft, Jonas Latz, Sebastian Josef Vollmer
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.