On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data

2026-04-08

AI Summary

This work addresses the ill-posedness inherent in generative modeling, data-driven dynamical systems, and inverse problems for partial differential equations by investigating the unique recovery of transport maps and vector fields from limited density observations and their pushforward or divergence data under the action of mappings or vector fields. By integrating tools from differential geometry, measure theory, Whitney and Takens embedding theorems, and Koopman/Perron–Frobenius operator theory, the study establishes, for the first time, sufficient conditions for the unique identifiability of smooth diffeomorphisms or vector fields using only finitely many density measurements. A novel metric based on discrepancies between pushforward densities is introduced, and theoretical analysis confirms the well-posedness of the identification problem. Numerical experiments demonstrate the method’s effectiveness in accurately recovering underlying maps and vector fields, offering new identifiability guarantees for inverse problems involving continuity, convection, and Fokker–Planck equations.

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.

Problem

• transport maps
• vector fields
• measure-valued data
• unique recovery
• inverse problems

Innovation

• transport maps
• vector field recovery
• pushforward densities
• measure-valued data
• PDE inverse problems