Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes
This work addresses probabilistic forecasting for complex dynamical systems, offering a method to generate unbiased forecasts in finite time, though it appears incremental as it builds on existing stochastic interpolant frameworks.
The authors tackled probabilistic forecasting of dynamical systems by developing a generative modeling framework that maps a point mass at the current state to a probabilistic ensemble of forecasts, achieving efficient learning via square loss regression and demonstrating utility on high-dimensional problems like Navier-Stokes and video prediction.
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 Föllmer 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.