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.

The optimization phase in deep learning consists in minimizing an objective function w.r.t. the set of

Section A.1 presents the lemmas used to prove the main results. Section A.2 presents the main results The first two inequalities are owing to the triangle inequality, and the third inequality is due to the definition of L-divergence Eq.(5). We complete the proof by applying Lemma A.1 to bound F ollowing the conditions of Theorem 4.1, the upper bound of null V arnull null D Based on the conditions of Theorem 4.1, we assume We complete the proof by applying Lemma A.3 and Lemma A.4 to bound the Rademacher Following the proof of Theorem 4.1, we have |D F ollowing the conditions of Proposition 4.3, as N, we have, null D Based on the result on Proposition 4.3, for any δ (0, 1), we know that 4LB ( 2 D ln 2 + 1)null We complete the proof by applying the triangle inequality. III: Samples from p and q are labeled with 0 and 1, respectively. All values are averaged over five trials.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/