Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity
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.
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BibTeX
@misc{shang2026continuoustime,
title = {{Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity}},
author = {Ruifeng Shang and Shu Liu and Yuhua Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32026},
eprint = {2609.32026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32026},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity
AU - Ruifeng Shang
AU - Shu Liu
AU - Yuhua Zhu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.32026
UR - https://arxiv.org/abs/2609.32026
AB - 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.
ER -