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 {(ρj,f#ρj)}j=1m 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 {(ρj,div(ρjv))}j=1m 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.
Figures & tables
Figure 1: Flowchart of our main results.
Symbol
Meaning
M,N
Smooth, compact d -dimensional Riemannian manifolds
P(M)
Borel probability measures over M
⟨⋅,⋅⟩rx
Riemannian metric on M at x
Cℓ(M,Rn)
ℓ -times continuously differentiable functions
C+ℓ(M,Rn)
componentwise positive functions in Cℓ(M,Rn)
D+ℓ(M,Rn)
functions in C+ℓ(M,Rn) componentwise integrating to 1
Table 1: Notation used throughout Section 3 .
Figure 2: Recovering a one-dimensional function from its pushforward action on five densities.
Figure 3: Recovering the Lorenz-63 dynamics on the support of finite snapshot data {ρj}j=0m that covers the full chaotic attractor. In this example, MSE(vθ)=1.48⋅10−2 and MSE(fθ)=2.27⋅10−3.
Figure 4: Recovering the Lorenz-63 dynamics on the support of finite snapshot data {ρj}j=0m which covers only a fraction of the chaotic attractor. The dynamics are accurately identified on the support of the observed density snapshots. In this example, MSE(vθ)=8.48⋅10−2 and MSE(fθ)=3.93⋅10−3.
Figure 5: Reconstructing the vector field ( 47 ) by comparing divergence operators via the objective function ( 48 ). The reference vector field v and relative error as a function of m are shown in Figure 5(a) , while Figure 5(b) visualizes individual training results for m∈{1,2,3,4} .
We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO⋆ surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.
Francisco Andrade, Gabriel Peyré, Clarice Poon
INRIA (PreMeDICaL & HeKA) · CNRS and ENS, PSL Universit´e · Mathematics Institute, University of Warwick
Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.
Xifeng Zhang, Jin Zhao
School of Mathematical Science Capital Normal University Beijing, 100048, China · Academy for Multidisciplinary Studies Capital Normal University2026 Beijing, 100048, China
We propose an instantiation of flow matching that relies on a time-independent velocity field (an autonomous flow) to exactly map between two distributions, so long as the target is singular, i.e.\ supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.
Lee Cheuk-Kit, Florentin Coeurdoux, Yuyuan Chen +5
1Harvard University · 2Capital Fund Management · University of Pennsylvania +2