🤖 AI Summary
This work addresses the challenge of efficient and accurate ensemble forecasting in chaotic, turbulent, and stochastic systems by proposing a trajectory-aware surrogate modeling approach. The method uniquely learns the probability flow velocity directly from trajectory data, enabling modeling of trajectory-dependent dynamical quantities—such as fluxes and circulations—through first-order trajectory matching (FTM), without requiring estimation of conventional drift or diffusion coefficients, score functions, or explicit simulation. By integrating a simulation-free one-step training loss with stability analysis, the approach achieves high-fidelity ensemble predictions at low computational cost across diverse stochastic dynamical systems and partial differential equation benchmarks, significantly enhancing both trajectory resolution and predictive efficiency.
📝 Abstract
We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.