SNAP-Diffusion: From Unpaired Snapshots to Anisotropic Diffusion Operators

Unpaired population snapshots reveal marginal evolution but do not determine an unrestricted stochastic generator. We introduce SNAP-Diffusion, which recovers a structured anisotropic diffusion field by converting snapshot statistics and occupation quadrature into a finite weak Fokker–Planck system over a uniformly elliptic coefficient class. Restricted coercivity of the weak design yields identifiability within this class and deterministic stability to perturbations in both responses and design. Under fixed Fick drift, coefficient-recovery error further controls held-out density error and associated distributional and observable errors. In controlled two-dimensional transport, SNAP estimates one operator from short-horizon training releases and freezes it for downstream prediction. The recovered anisotropic operator improves prediction from unseen releases, remains accurate beyond the identification horizon, and transfers to first-passage functionals without refitting. A controlled FX experiment applies the same principle to state-dependent local correlation recovered from joint snapshots. On held-out tenors and contracts, SNAP reduces mean pricing error by $68\%$ and $71\%$ percent relative to constant- and time-dependent-correlation models and by $19\%$ percent relative to same-class forward calibration. Together, the results connect weak snapshot identification of structured diffusion fields to reusable operators for downstream prediction.