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.

翻译：本文提出一个基于生成建模的动力系统概率预测框架。给定系统状态随时间演化的观测数据，我们将预测问题表述为在给定当前系统状态的条件下，对未来系统状态的条件分布进行采样。为此，我们利用随机插值框架，该框架便于构建任意基分布与目标分布之间的生成模型。我们设计了一个虚构的非物理随机动力学过程：该过程以当前系统状态为初始条件，在有限时间内无偏地生成来自目标条件分布的样本。因此，该过程将集中于当前状态的质点映射为概率性的预测集合。我们证明了实现该任务的随机微分方程（SDE）中的漂移系数是非奇异的，并且可以通过时间序列数据的平方损失回归进行高效学习。研究表明，该SDE的漂移系数和扩散系数可在训练后进行调节，而通过特定选择最小化估计误差影响可得到Föllmer过程。我们在多个复杂高维预测问题上验证了本方法的有效性，包括随机强迫Navier-Stokes方程预测以及KTH和CLEVRER数据集上的视频预测。