Probabilistic Forecasting with Stochastic Interpolants and Föllmer
  Processes
Yifan Chen Mark Goldstein Mengjian Hua Michael S. Albergo Nicholas M. Boffi Eric Vanden-Eijnden
We propose a framework for probabilistic forecasting of dynamical systems
based on generative modeling. Given observations of the system state over time,
we formulate the forecasting problem as sampling from the conditional
distribution of the future system state given its current state. To this end,
we leverage the framework of stochastic interpolants, which facilitates the
construction of a generative model between an arbitrary base distribution and
the target. We design a fictitious, non-physical stochastic dynamics that takes
as initial condition the current system state and produces as output a sample
from the target conditional distribution in finite time and without bias. This
process therefore maps a point mass centered at the current state onto a
probabilistic ensemble of forecasts. We prove that the drift coefficient
entering the stochastic differential equation (SDE) achieving this task is
non-singular, and that it can be learned efficiently by square loss regression
over the time-series data. We show that the drift and the diffusion
coefficients of this SDE can be adjusted after training, and that a specific
choice that minimizes the impact of the estimation error gives a F\"ollmer
process. We highlight the utility of our approach on several complex,
high-dimensional forecasting problems, including stochastically forced
Navier-Stokes and video prediction on the KTH and CLEVRER datasets.