# Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshots

arXiv: https://arxiv.org/abs/2609.35060 · HTML: https://mlsanity.com/paper/2609.35060/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 47 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [The Limiting Dynamics of SGD: Modified Loss, Phase Space Oscillations, and Anomalous Diffusion](https://mlsanity.com/paper/2107.09133/) (2021)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [Multi-Marginal Stochastic Flow Matching for High-Dimensional Snapshot Data at Irregular Time Points](https://mlsanity.com/paper/2508.04351/) (2025)

## Abstract (arXiv)

Inferring dynamics from snapshots of evolving distributions is fundamentally underdetermined: the Fokker-Planck equation constrains the drift F only through its score-weighted divergence nabla· F+F·nablalogρ, leaving a ρ-solenoidal gauge invisible to any single-time constraint. Time-indexed transport formulations cannot resolve this ambiguity: every admissible marginal path admits a curl-free explanation, minimum-action reconstruction selects it, and marginal fit alone cannot distinguish dynamically inequivalent explanations. Requiring one autonomous field to explain several marginals instead makes part of the hidden circulation visible as nablalogρ changes across marginals. Separating instantaneous Fokker-Planck source constraints from the snapshot experiment, we show that the source constraints identify the field modulo the kernel of a stacked score-weighted divergence operator. For generic Gaussian shape variation, source constraints at K≥ m time points in intrinsic dimension m eliminate every polynomial gauge direction, whereas finitely many density snapshots alone admit aliasing; we give the obstruction explicitly. At a Gaussian anchor, for Sobolev smoothness s and n samples per time point, we derive a conditional lower rate (nK)-2s/(2s+m+1) for the tangent snapshot experiment, with a matching upper rate in a degreewise benchmark. Strong-form fitting is non-orthogonal to score error and cannot be repaired by spectral filtering. Instead, we estimate using smooth test functions while retaining the known diffusion term, and derive a finite-sample bound that separates sampling error from fixed-grid quadrature bias. Planted-circulation experiments confirm the predicted gauge contraction and expose a design tension between cross-slice information and covariance-aware whitening.