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First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: Stochastic population dynamics · 0 citations (OpenAlex)

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Proposes First-Order Trajectory Matching (FTM), which learns the probability current velocity directly from SDE trajectories via a simulation-free Stratonovich-based loss, giving one-NFE-per-step ensemble surrogates that preserve marginals and capture circulations.

Problem

Surrogate models for stochastic/chaotic systems trained with MSE collapse to conditional means, while marginal-matching flows miss the probability mass transport (circulations, fluxes) and conditional generative models need multiple solver steps per physical time step.

Approach

Learn the probability current velocity v = b - (1/2) Σ∇log ρ directly from trajectories using a pathwise Stratonovich-identity loss, whose one-step form uses only adjacent states; the resulting ODE preserves time marginals and captures first-order transport, with one network evaluation per time step.

Key results

• On Duffing oscillator, FTM achieves W2 distribution error 0.075 and trajectory QoI error 0.020, the lowest among compared methods.
• On stochastic Burgers, FTM attains energy error 1.95e-3 with 1 NFE per step, versus 2.71e-3 for AR-CFM with 20 NFE and 1.24e-2 for ARDM-100.
• On stochastically forced Navier-Stokes, FTM reaches energy error 3.43e-2 with 1 NFE, versus 1.16e-1 for ARDM with 100 NFE.

Surprising findings

• Distilled MeanFlow-1 yields energy error 8.22e-1 on stochastic Burgers, far worse than FTM's one-step result.
• Marginal-only population dynamics with minimum-kinetic-energy velocity misses circulation, so matching marginals alone is insufficient for trajectory QoIs.

Limitations

• Does not reproduce martingale-dominated path statistics such as hitting-time distributions or temporal autocorrelations.
• Theory covers finite-dimensional additive-noise SDEs and does not fully apply to the high-dimensional PDE/SPDE experiments.
• Cannot capture rare transitions between wells in the two-lane example.

Experimental setup

Datasets: Duffing oscillator trajectories Rayleigh-Benard convection (nine-mode model) trajectories stochastic Burgers trajectories stochastically forced 2D Navier-Stokes trajectories

Benchmarks: Duffing oscillator Rayleigh-Benard convection stochastic Burgers stochastically forced turbulence

Metrics: W2 distribution error trajectory QoI error error in energy energy spectrum enstrophy

Baselines: Operator learning DICE Marginal matching SDE learning SDE matching ARDM AR-CFM MeanFlow

Builds on: action matching discrete inverse continuity equation (DICE) autoregressive conditional flow matching MeanFlow

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