Characteristic Sensitivity Ensembles for Inference of Hidden Dynamics from Marginal Observations

Abstract

A framework is developed for the inference of dynamics described by a generalized system of ordinary differential equations. A stochastic gradient method is coined that infers dynamics from observed marginal probability density functions using the joint probability density function of the observable and latent variables. Diffusion and other irreversible processes observed in a low-dimensional state can be recast as deterministic, reversible flows in a sufficiently augmented state space, where the joint density satisfies the hyperbolic Liouville equation. The marginal distribution observed is the projection of these hyperbolic dynamics onto the observed coordinates, with the latent components carrying the randomness and memory. This reframing allows inference for irreversible or stochastic dynamics into the recovery of a deterministic Ordinary Differential Equation (ODE) from marginal observations. Instead of solving the high-dimensional Liouville equation for the joint density, the algorithm exploits its characteristic representation. Particles sampled from the initial distribution are transported along characteristic lines. The Eulerian sensitivity with respect to parameters is obtained by sensitivity propagation along the characteristic lines, with a crossed U-statistic producing an unbiased gradient estimator, which enables stochastic gradient descent. Four experiments validate the method: recovery of a three-mode linear system observed through the marginal of a single mode; a nonlinear Gompertz growth model with a hidden mode; a bistable system whose hidden mode turns a unimodal marginal bimodal; and Stokes--Oseen drag law recovery for particles in a cellular flow. Convergence behavior is analyzed across these settings.