Predictions with dynamical systems can be computationally expensive---even with reduced models---because reaching future prediction times can require many sequential time integration steps. We propose Flow-Map First-Order Trajectory Matching (FM-FTM), a nonintrusive model reduction approach that accelerates ensemble predictions of stochastic dynamical systems by learning flow maps that replace multiple time-integration steps. Our method uses first-order trajectory matching (FTM) to learn a deterministic representation of the stochastic probability-current dynamics and then reduces its evolution over time into a flow map whose evaluation directly yields predictions over long time horizons. Numerical experiments on evolution problems with stochastic forcing demonstrate that FM-FTM accelerates ensemble predictions by up to two orders of magnitude while accurately predicting time marginals and ensemble statistics.
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Figure 1: While FTM quickly loses accuracy at coarse time steps, FM-FTM maintains accurate predictions by taking far-horizon time jumps, replacing many sequential integration steps with a single flow-map evaluation and thereby accelerating ensemble prediction.
Figure 2: Duffing oscillator: FM-FTM accurately captures the evolving distributions over large time intervals, whereas FTM loses accuracy even with moderately increased time steps from the original h=0.01 to h=0.25 .
Figure 3: Duffing oscillator: The Wasserstein-2 errors of FM-FTM remain comparable to those of FTM and show little sensitivity to the jump horizon τ , demonstrating that FM-FTM preserves the accuracy of ensemble predictions while reducing the number of time-integration steps by a factor of up to 400× in this example.
Figure 4: The relative errors in predicting quantities of interest remain small across increasing jump horizons τ for both the Duffing oscillator (left) and Rayleigh-Bénard convection (right), demonstrating that FM-FTM preserves trajectory-dependent quantities beyond time-marginal statistics.
Figure 5: Rayleigh-Bénard: FTM fails to capture the multimodal distribution even already with moderately increased time-step sizes, whereas FM-FTM accurately captures the evolution of both modes at a jump horizon of τ=4 , which is 400 times larger than the time-step size used by FTM.
Figure 6: Rayleigh-Bénard: The Wasserstein-2 errors remain moderate across jump horizons up to τ=4 , corresponding to 400 times the time-step size of FTM.
Figure 7: Burgers: The relative errors in energy (left) and enstrophy (right) exhibit similar behavior for FTM and FM-FTM across different jump horizons τ , demonstrating that FM-FTM retains the accuracy of these physical ensemble statistics while reducing the number of time-integration steps.
Figure 8: Stochastically forced turbulence: Over the time interval, FM-FTM requires only 20 ( τ=1.25 ) flow-map evaluations, compared with 1000 sequential integration steps for FTM ( h=0.025 ), while achieving comparable relative errors in ensemble-mean energy (left) and enstrophy (right).
Figure 9: Stochastically forced turbulence: With a jump horizon of τ=1.25 , FM-FTM generates trajectories approximately 50× faster than FTM and about 500× faster than the physics model (left), while predicting solution fields that accurately match the decay of the kinetic-energy spectrum across wavenumbers (right).