Identifying Stochastic Dynamics from Non-Sequential Data (DyNoSeD)

Inferring stochastic dynamics from data is central across the sciences, yet in many applications only unordered, non-sequential measurements are available-often restricted to limited regions of state space-so standard time-series methods do not apply. We introduce DyNoSeD, a first-principles framework that identifies unknown dynamical parameters from such non-sequential data by minimizing Fokker-Planck residuals. We develop two complementary routes: a local route that handles region-restricted data via locally estimated scores, and a global route that fits dynamics from globally sampled data using a kernel Stein discrepancy without explicit density or score estimation. When the dynamics are affine in the unknown parameters, we prove a necessary-and-sufficient condition for the existence and uniqueness of the inferred parameters and derive a sensitivity analysis that identifies which parameters are tightly constrained by the data and which remain effectively free under over-parameterization. For general non-affine case, both routes define differentiable losses amenable to gradient-based optimization. As demonstrations, we recover (i) the three parameters of a stochastic Lorenz system from non-sequential data (region-restricted data for the local route and full steady-state data for the global route) and (ii) a 3x7interaction matrix of a nonlinear gene-regulatory network derived from a published B-cell differentiation model, using only unordered steady-state samples and applying the global route. Finally, we show that the same Fokker-Planck residual viewpoint supports a "dynamics-to-density" complement that trains a normalized density estimator directly from known dynamics without any observations. Overall, IDyNSD provides two first-principles routes for system-identification from non-sequential data, grounded in the Fokker-Planck equation, that link data, density, and stochastic dynamics.