Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshots

Inferring dynamics from snapshots of evolving distributions is fundamentally underdetermined: the Fokker-Planck equation constrains the drift $F$ only through its score-weighted divergence $\nabla \cdot F+F\cdot \nabla \log \rho$, leaving a $\rho$-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 through changing scores. 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 $K\ge m$ time points in intrinsic dimension $m$ eliminate every polynomial gauge direction, whereas finitely many density snapshots alone admit aliasing; we give the obstruction explicitly. Around an evolving Gaussian reference, for Sobolev smoothness $s$ and $n$ samples per time point, we derive a conditional oracle rate $(nK{)}^{-2s/(2s+m+1)}$: the lower bound requires only a first-moment information scale, while the upper bound additionally requires a near-gauge tail condition. 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.