  We propose a new method for inferring the governing stochastic ordinary
differential equations (SODEs) by observing particle ensembles at discrete and
sparse time instants, i.e., multiple "snapshots". Particle coordinates at a
single time instant, possibly noisy or truncated, are recorded in each snapshot
but are unpaired across the snapshots. By training a physics-informed
generative model that generates "fake" sample paths, we aim to fit the observed
particle ensemble distributions with a curve in the probability measure space,
which is induced from the inferred particle dynamics. We employ different
metrics to quantify the differences between distributions, e.g., the sliced
Wasserstein distances and the adversarial losses in generative adversarial
networks (GANs). We refer to this method as generative "ensemble-regression"
(GER), in analogy to the classic "point-regression", where we infer the
dynamics by performing regression in the Euclidean space. We illustrate the GER
by learning the drift and diffusion terms of particle ensembles governed by
SODEs with Brownian motions and Levy processes up to 100 dimensions. We also
discuss how to treat cases with noisy or truncated observations. Apart from
systems consisting of independent particles, we also tackle nonlocal
interacting particle systems with unknown interaction potential parameters by
constructing a physics-informed loss function. Finally, we investigate
scenarios of paired observations and discuss how to reduce the dimensionality
in such cases by proving a convergence theorem that provides theoretical
support.
