On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data
2026-04-09 • Machine Learning
Machine Learning
AI summaryⓘ
The authors show how to uniquely identify certain transformations (diffeomorphisms) and vector fields by looking at the effects they have on a limited number of input data distributions. They provide mathematical conditions that guarantee when these transformations can be recovered exactly from finite samples, using ideas from classical embedding theorems. Their work also connects to understanding operators that describe dynamics and gives new perspectives on solving inverse problems in certain partial differential equations. Finally, they illustrate their findings with numerical examples that confirm the unique recovery of transport maps and vector fields from finite measurements.
DiffeomorphismPushforward measureVector fieldsWhitney embedding theoremTakens embedding theoremPerron-Frobenius operatorKoopman operatorPartial differential equationsInverse problemsFokker-Planck equation
Authors
Jonah Botvinick-Greenhouse, Yunan Yang
Abstract
We establish guarantees for the unique recovery of vector fields and transport maps from finite measure-valued data, yielding new insights into generative models, data-driven dynamical systems, and PDE inverse problems. In particular, we provide general conditions under which a diffeomorphism can be uniquely identified from its pushforward action on finitely many densities, i.e., when the data $\{(ρ_j,f_\#ρ_j)\}_{j=1}^m$ uniquely determines $f$. As a corollary, we introduce a new metric which compares diffeomorphisms by measuring the discrepancy between finitely many pushforward densities in the space of probability measures. We also prove analogous results in an infinitesimal setting, where derivatives of the densities along a smooth vector field are observed, i.e., when $\{(ρ_j,\text{div} (ρ_j v))\}_{j=1}^m$ uniquely determines $v$. Our analysis makes use of the Whitney and Takens embedding theorems, which provide estimates on the required number of densities $m$, depending only on the intrinsic dimension of the problem. We additionally interpret our results through the lens of Perron--Frobenius and Koopman operators and demonstrate how our techniques lead to new guarantees for the well-posedness of certain PDE inverse problems related to continuity, advection, Fokker--Planck, and advection-diffusion-reaction equations. Finally, we present illustrative numerical experiments demonstrating the unique identification of transport maps from finitely many pushforward densities, and of vector fields from finitely many weighted divergence observations.