Identifying stochastic dynamics from non-sequential data (DyNoSeD).

Inferring stochastic dynamics from data is central; yet, in many applications, only unordered, non-sequential measurements are available-often restricted to limited regions of state space-so standard time-series methods fail. We introduce DyNoSeD (Identifying Dynamics from Non-Sequential Data), 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 local score estimation, and a global route that fits dynamics from globally sampled data using a kernel Stein discrepancy without density- or score estimation. When the dynamics are affine-in-the-unknown-parameters (while remaining nonlinear-in-the-state), we prove necessary-and-sufficient conditions for the existence and uniqueness of the inferred parameter vector 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 parameterizations, 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 observations (region-restricted data for the local route and full steady-state data for the global route) and (ii) a 3×7 interaction 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. Overall, DyNoSeD 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.