First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems
2026-06-09 • Machine Learning
Machine Learning
AI summaryⓘ
The authors present First-Order Trajectory Matching (FTM), a new way to model how groups of random moving particles behave over time. Instead of estimating complicated details, FTM focuses on learning the average direction and movement of these particles directly from their paths. This approach helps capture important movements like flows and crossings without heavy computation. They also analyze how stable their method is and show it works well with the right balance of data and timing. Their tests demonstrate that FTM can accurately predict group behaviors while being efficient.
stochastic systemsprobability current velocitytrajectory matchingensemble averagefluxcirculationbarrier crossingsurrogate modelingstability analysispartial differential equations (PDEs)
Authors
Shreya Jha, Timo Schorlepp, Nicholas Geissler, Jules Berman, Benjamin Peherstorfer
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.