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.

翻译：我们提出一阶轨迹匹配（FTM），这是一种代理建模方法，通过从随机系统的轨迹中学习概率质量的一阶局部传输。通过匹配轨迹的对称一阶运动，FTM学习概率流速度，其流动通过保持时间边际来匹配集合平均值，同时捕获类似电流的轨迹量，如通量、环流和势垒穿越电流。FTM直接从轨迹学习流速度，避免漂移、扩散和分数估计。我们的稳定性分析将离散化误差与采样方差分离，并表明当时间分辨率和样本量适当平衡时，无单步仿真的FTM损失是稳定的。通过随机动力系统和偏微分方程实例，我们经验性地证明FTM以低确定性展开成本提供轨迹感知的集合预测。