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：我们提出了一种基于生成模型的动力学系统概率预测框架。通过观测系统状态随时间演化的数据，我们将预测问题转化为在给定当前系统状态条件下对其未来状态的条件分布进行采样。为此，我们采用随机插值框架，该框架可在任意基础分布与目标分布之间构建生成模型。我们设计了一种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输出目标条件分布的样本。因此，该过程将当前状态处的点质量映射为预测的概率集合。我们证明了实现这一目标的随机微分方程（SDE）中的漂移系数是非奇异的，并且可以通过时间序列数据上的平方损失回归高效学习。研究表明，该SDE的漂移系数和扩散系数可在训练后进行调整，其中通过特定选择最小化估计误差影响可得到Föllmer过程。我们通过多个复杂高维预测问题验证了该方法的应用价值，包括随机强迫Navier-Stokes方程以及KTH和CLEVRER数据集上的视频预测任务。