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

TL;DR

This paper provides theoretical guarantees for uniquely recovering transport maps and vector fields from finite measure data, with implications for inverse problems, dynamical systems, and generative models.

Contribution

It introduces new conditions and metrics for unique identification of diffeomorphisms and vector fields from finite measure observations, using embedding theorems.

Findings

Unique recovery of diffeomorphisms from finitely many densities.

New metric for comparing diffeomorphisms based on pushforward densities.

Guarantees for PDE inverse problems like Fokker--Planck and advection 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 $\{(\rho_j,f_\#\rho_j)\}_{j=1}^m$ 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 $\{(\rho_j,\text{div} (\rho_j v))\}_{j=1}^m$ 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.