  We develop an algorithm to approximate the time evolution of a probability
measure without explicitly learning an operator that governs the evolution. A
particular application of interest is discrete measures $\mu_t^N$ that arise
from particle systems. In many such situations, the individual particles move
chaotically on short time scales, making it difficult to learn the dynamics of
a governing operator, but the bulk distribution $\mu_t^N$ approximates an
absolutely continuous measure $\mu_t$ that evolves ``smoothly.'' If $\mu_t$ is
known on some time interval, then linearized optimal transport theory provides
an Euler-like scheme for approximating the evolution of $\mu_t$ using its
``tangent vector field'' (represented as a time-dependent vector field on
$\mathbb R^d$), which can be computed as a limit of optimal transport maps. We
propose an analog of this Euler approximation to predict the evolution of the
discrete measure $\mu_t^N$ (without knowing $\mu_t$). To approximate the
analogous tangent vector field, we use a finite difference over a time step
that sits between the two time scales of the system -- long enough for the
large-$N$ evolution ($\mu_t$) to emerge but short enough to satisfactorily
approximate the derivative object used in the Euler scheme. By allowing the
limiting behavior to emerge, the optimal transport maps closely approximate the
vector field describing the bulk distribution's smooth evolution instead of the
individual particles' more chaotic movements. We demonstrate the efficacy of
this approach with two illustrative examples, Gaussian diffusion and a cell
chemotaxis model, and show that our method succeeds in predicting the bulk
behavior over relatively large steps.
