Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity

Ruifeng Shang; Shu Liu; Yuhua Zhu

stat.MLSep 25, 2026

Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity

Authors: Ruifeng Shang, Shu Liu, Yuhua Zhu

Organizations: Department of Mathematics Florida State University Tallahassee, FL 32306, USA · Department of Statistics and Data Science University of California, Los Angeles Los Angeles, CA 90095, USA

Abstract

Physical systems evolve continuously in time, yet their states are typically observed only at discrete times. Generating trajectories consistent with their probability densities from such observations therefore requires capturing the continuous-time evolution rather than only learning transition mappings between consecutive observations. We propose PhiBE-Flow, a framework that directly estimates the probability velocity field induced by the stochastic differential equation (SDE) which governs this continuous-time distributional evolution. PhiBE-Flow learns from discrete observations using a model-free approach requiring neither known SDE coefficients nor score estimation. We establish convergence guarantees for the method, accounting for both time-discretization and finite-sample errors. We evaluate PhiBE-Flow on systems of increasing complexity, from controlled stochastic numerical systems to Navier--Stokes dynamics and real-world videos. Our results show that PhiBE-Flow accurately recovers probability flows of stochastic dynamics, preserves multiscale physical statistics, and improves video generation performance over representative baselines. The code is available at https://github.com/R1fe/PhiBE-Flow.

Figures & tables

Figure 1: Overview of PhiBE-Flow. Top: frame-to-frame methods learn relations between sampled states. Bottom: PhiBE-Flow directly learns the SDE-induced probability velocity.

Panel of Figure 2 — Figure 2: PhiBE-Flow on reversible and non-reversible OU processes. (a) PhiBE-Flow (green) and analytical (orange) probability-flow trajectories for the non-reversible case. (b) PhiBE-Flow (blue) and OU-process (red) samples at $t=15$ . (c) PhiBE-Flow (green) and analytical (orange) trajectories for the reversible case. (d) Global $L^{2}$ velocity error versus temporal observations $N_{t}$ . (e) Global $L^{2}$ velocity error versus observed trajectories $N$ .

Figure 3: State-space learning on the stochastic double-pendulum system. Ground-truth and PhiBE-Flow predictions for two joint angles $\theta_{1},\theta_{2}$ and angular velocities $\dot{\theta}_{1},\dot{\theta}_{2}$ over physical time.

Figure 4: Image-based Learning on the stochastic double-pendulum system. The top row shows the ground-truth image sequence. The bottom row shows results from PhiBE-Flow.

Figure 5: Temporal generation results on Navier–Stokes. Ground-truth vorticity fields and results from MCVD, PFI, and PhiBE-Flow, together with their corresponding error maps.

	Standard metrics				Adjusted metrics
Method	MSE( $\times 10^{-2}$ ) $\downarrow$	FVD $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	MSE( $\times 10^{-2}$ ) $\downarrow$	FVD $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$
MCVD	0.8550	307.3	20.68	0.6986	–	–	–	–
PFI	0.6637	280.1	21.78	0.7254	0.6576	247.1	21.82	0.7372
PhiBE-Flow	0.4405	227.5	23.56	0.7663	0.4305	192.7	23.66	0.7800

Table 1: Metric results on KTH. Standard metrics compare generated results with the original ground truth. Adjusted metrics compare generated results with the reconstructed ground truth.

Figure 7: Video generation on the KTH dataset. The top row shows the ground-truth trajectory. The following three rows show generation results from MCVD, PFI, and PhiBE-Flow, respectively.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 8 — Figure 8: Time evolution of selected mean and covariance components of the marginal distributions for reversible (left two panels) and non-reversible (right two panels) OU processes. Results obtained from the computed probability flows are compared with the exact solutions.

Model	Layer	Output Shape
Encoder	Input image $x$ (resized)	$1\times 32\times 32$
	Conv2d( $1\rightarrow 32$ ), LeakyReLU, BatchNorm	$32\times 16\times 16$
	Conv2d( $32\rightarrow 64$ ), LeakyReLU	$64\times 8\times 8$
	Conv2d( $64\rightarrow 128$ ), LeakyReLU, BatchNorm	$128\times 4\times 4$
	Flatten, Linear( $2048\rightarrow 3$ )	$3$
Decoder	Input latent $z$	$3\times 1\times 1$

Appendix

Table 2: Architectures of the geometry-preserving encoder

T_{\phi}

and decoder

S_{\psi}

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