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Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity

Ruifeng Shang, Shu Liu, Yuhua Zhu

arXiv:2609.32026v1stat.MLcs.LGmath.DS

TL;DR

Physical systems are often observed only at discrete times despite continuous and potentially stochastic evolution, making transition-only learning an incomplete account of their dynamics. PhiBE-Flow learns the SDE-induced probability velocity from those observations without known SDE coefficients or score estimation. It recovers probability flows and physical statistics across settings and outperforms representative baselines on Navier–Stokes and KTH videos.

  • Problem

    Physical systems evolve continuously and may be stochastic, but are typically observed discretely, while transition-based approaches do not explicitly capture their physical-time evolution.

    Source: §1 · p. 1

  • Method

    PhiBE-Flow uses local short-time transition information to learn an SDE-induced probability velocity through a model-free, score-free objective without known SDE coefficients.

    Source: §1 · p. 2, §3.3 · p. 4

  • Results

    PhiBE-Flow recovers stochastic probability flows, preserves multiscale physical statistics, and improves generation accuracy, physical consistency, and video prediction metrics over representative baselines on Navier–Stokes and KTH videos.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The method generates deterministic continuous-time trajectories whose time-dependent marginal distributions approximate those of the underlying stochastic system.

    Source: §1 · p. 2

    PhiBE-Flow assumes an underlying SDE representation, and high-dimensional probability-velocity learning can be computationally expensive, making performance dependent on whether latent representations preserve essential data geometry and dynamics.

    Source: §6 · p. 9

Abstract

from arXiv · show

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.

1 Introduction

Physical systems are observed discretely but often evolve continuously and stochastically, so learning their dynamics calls for more than transitions between sampled states. PhiBE-Flow learns the SDE-induced probability velocity from discrete observations and outperforms representative baselines on Navier–Stokes and KTH video datasets.

1 Introduction

  • Continuous-time learning is needed because physical systems may be stochastic or chaotic, while frame-level visual realism does not ensure coherent motion or physical consistency.The introduction situates this challenge in mechanics, fluid dynamics, and video modeling.Source: §1 · p. 1
  • Autoregressive and bridge-based approaches succeed at generation but do not explicitly model the underlying evolution in physical time.Neural ODEs and SDEs address continuous-time dynamics, but stochastic probability velocity depends on unknown drift, diffusion, and density score.Source: §1 · pp. 1–2
  • PhiBE-Flow learns SDE-induced probability velocity fields from discrete observations through a model-free framework for continuous-time trajectory generation.Its score-free variational objective does not assume known SDE coefficients, and the method has convergence guarantees accounting for time-discretization and finite-sample errors.Source: §1 · p. 2
  • PhiBE-Flow improves generation accuracy, physical consistency, and video prediction metrics over representative baselines on Navier–Stokes and KTH video datasets.The paper also reports recovery of probability flows and preservation of multiscale physical statistics across systems of increasing complexity.Source: §1 · p. 2

2 Related Work

Prior approaches model discrete transitions, auxiliary paths, or stochastic dynamics explicitly, but each leaves a gap for learning physical-time evolution from discrete observations. PhiBE-Flow instead uses local transition information in a continuous-time PDE formulation without first fitting a complete SDE model.

2.1 Autoregressive Generation

  • Autoregressive models generate stepwise predictions and can support arbitrary horizons, but learn discrete-time transitions rather than the continuous-time dynamics underlying observations.Recurrent models predict the next state or frame, while Transformers use attention for longer-range dependencies.Source: §2.1 · p. 2

2.2 Bridge-Based Generation

  • Bridge-based methods use auxiliary paths, including diffusion, flow matching, and stochastic interpolants, that need not match the observed system’s physical-time evolution.These approaches model forecast uncertainty through transport between source and target distributions.Source: §2.2 · p. 3

2.3 Learning Stochastic Dynamics

  • Neural and latent SDE methods learn unknown drift and diffusion, but explicit SDE learning can pose scalability challenges in high-dimensional systems.PhiBE instead incorporates local transition information into a continuous-time PDE formulation without first fitting a complete SDE model, while retaining an underlying-SDE assumption.Source: §2.3 · p. 3

3 Method

The method learns a deterministic probability flow whose time-dependent marginals represent the evolution of an observed stochastic system. It derives a score-free objective and makes it estimable from discrete observations using local drift and diffusion approximations from short-time transitions.

3.1 Problem Setting

  • The target is a deterministic vector field that evolves the marginal distribution of an SDE from discrete observations, without identifying the complete drift and diffusion functions.The continuous trajectory is inaccessible; observations comprise independent trajectories sampled at discrete times.Source: §3.1 · p. 3

3.2 Probability-Flow Formulation

  • The Fokker–Planck equation expresses marginal evolution as a continuity equation, motivating a probability velocity learned by minimizing its population discrepancy from the true velocity.Although integration by parts removes the density score, the resulting objective still depends on unknown drift and diffusion.Source: §3.2 · pp. 3–4

3.3 PhiBE-Flow Objective

  • For sufficiently small sampling intervals, PhiBE approximates drift and diffusion from short-time transitions, making the score-free objective accessible from discrete observations.In practice, observed increments replace conditional expectations; the covariance-weighted Jacobian term can be costly in high dimensions.Source: §3.3 · p. 4

4 Theoretical Guarantees

The guarantees separate error from using discrete observations to approximate the population dynamics from error due to finitely many transitions. They cover population approximation, finite-sample convergence for linear-basis models, and overall velocity error at observation times.

4 Theoretical Guarantees

  • The analysis separates population approximation error induced by discrete observations from finite-sample error arising from finitely many transitions.The finite-sample analysis is for fixed linear-basis models.Source: §4 · p. 4

4.1 Population Approximation

  • First-order PhiBE estimates of drift and diffusion covariance support population approximation guarantees for the induced probability velocity, with constants independent of the sampling interval.The OU process also admits explicit approximate-velocity and error-bound expressions.Source: §4.1 · p. 5

4.2 Finite-Sample Convergence

  • For fixed linear-basis models satisfying Assumption 1, including bv ∈ Vp, the theorem gives finite-sample convergence; fixed-feature networks qualify, while out-of-class fields need an approximation term.The basis fields may be nonlinear in state and time, and transitions within a trajectory may be dependent.Source: §4.2 · p. 5

4.3 Overall Convergence

  • Under the assumptions of Theorems 1 and 2, Corollary 1 gives an overall convergence guarantee with probability at least 1 − δ.The stated bound concerns velocity error at observation times and does not by itself bound trajectories of the learned field.Source: §4.3 · p. 6

5 Experiments

Experiments span four problems, from OU and double-pendulum systems to stochastic Navier–Stokes and KTH video. PhiBE-Flow closely tracks ground-truth behavior, better preserves Navier–Stokes multiscale statistics, and achieves the best KTH results across four metrics.

5.1 Learning from Stochastic Dynamical Systems

  • For non-reversible OU dynamics, PhiBE-Flow reproduces circulating probability-flow trajectories despite stationary Gaussian marginals; it also closely recovers analytical velocities and paths for reversible OU.Source: Figure 2 · p. 6, §5.1 · p. 6
  • In state space, the learned rollout closely follows nonlinear joint-angle and angular-velocity evolution over the generation horizon.Source: Figure 3 · p. 7, §5.1 · p. 7
  • From rendered frames, a geometry-preserving latent encoding lets PhiBE-Flow roll out motion while preserving the two-link system’s geometry.Source: §5.1 · p. 7, Figure 4 · p. 7

5.2 2D Navier–Stokes Equations Generation

  • On Navier–Stokes, PhiBE-Flow more accurately reproduces vorticity evolution, with smaller ground-truth error-map deviations and closer enstrophy spectra across wavenumbers than MCVD and PFI.Source: Figure 5 · p. 8, Figure 6 · p. 8, §5.2 · p. 8

5.3 Video Generation

  • PhiBE-Flow achieves the best KTH performance across MSE, FVD, PSNR, and SSIM, while better preserving body structure and motion with less accumulated drift than MCVD and PFI.Source: Table 1 · p. 8, §5.3 · p. 9

6 Conclusion

PhiBE-Flow learns SDE-induced probability velocity fields from discrete observations, with convergence guarantees and demonstrations across stochastic systems, Navier–Stokes dynamics, and video. Its theory also identifies an integration-by-parts objective whose unique minimizer is the population field under the stated conditions.

6 Conclusion

  • PhiBE-Flow generates continuous-time trajectories consistent with density evolution and is demonstrated on low-dimensional stochastic systems, Navier–Stokes dynamics, and video generation.The framework is model-free and comes with convergence guarantees.Source: §6 · p. 9
  • The framework assumes an underlying SDE; high-dimensional probability-velocity learning can be expensive, and latent-space performance depends on preserving essential geometry and dynamics.The authors identify more general stochastic dynamics and real-world systems as future work.Source: §6 · p. 9

A.1 Population Conditions and Objective Identity

  • Under the stated integrability and integration-by-parts conditions, the population objective extends to L2(ρt) and has the unique minimizer bv.Here bv uses the true density ρt.Source: §A.1 · p. 13

A.2 Proof of the PhiBE Coefficient Approximation

The proof uses the generator’s action on coordinate and centered quadratic functions to establish first-order PhiBE coefficient approximation bounds.

A.2 Proof of the PhiBE Coefficient Approximation

  • Applying the generator to coordinate functions gives the drift term, while applying it to centered quadratic functions gives the diffusion covariance.For the quadratic functions, the generator at x equals Σjk(x).Source: §A.2 · p. 13
  • The assumed semigroup expansion and boundedness on Ω yield the remaining coefficient inequalities for sufficiently small time increments.The proof states that the expansion and its first spatial derivative hold uniformly on Ω.Source: §A.2 · pp. 13–14

A.3 Proof of the General Population Approximation

The appendix derives population trajectory bounds from coefficient approximation and Grönwall’s inequality, then treats constant isotropic diffusion and an Ornstein–Uhlenbeck example.

A.3 Proof of the General Population Approximation

  • Subtracting the population objectives and applying the coefficient approximation bounds leads to a trajectory error bound via Grönwall, while both flows remain in Ω.The bound is uniform over the stated times t ≤ T.Source: §A.3 · p. 14

A.4 Constant Isotropic Diffusion

  • For known constant isotropic diffusion, the score-free objective simplifies, but this does not automatically simplify the empirical objective because the raw increment second moment may vary with x.The same flow bound applies under the stated containment condition.Source: §A.4 · p. 14

A.5 Ornstein–Uhlenbeck Population Approximation

  • For an Ornstein–Uhlenbeck system, the first-order PhiBE velocity has a uniform population approximation bound as h decreases, with a trajectory bound while paths remain in Ω.The example assumes linear drift, positive-definite diffusion covariance, and Gaussian initial state.Source: §A.5 · pp. 14–15

A.6 Linear-Basis Finite-Sample Setting and Proof

The finite-sample analysis fits fixed linear-basis fields using independent SDE trajectories, allowing dependence among observations within each trajectory. Under sampling and realizability conditions, it bounds estimator error, while nonrealizable bases target a projection.

A.6 Linear-Basis Finite-Sample Setting and Proof

  • The empirical objective averages contributions over observation times, and its identity requires no time-quadrature claim.The independent sampling units are trajectories; transitions within each trajectory need not be independent.Source: §A.6 · pp. 15–16
  • The fixed basis fields may be nonlinear in state and time, but only their coefficients are fitted, with the basis chosen independently of the training sample.The fitting class is the span of the selected basis fields.Source: §A.6 · p. 16
  • The theorem’s moment conditions depend on the chosen basis and trajectory distribution, and freezing neural features alone establishes neither realizability nor those bounds.The constants may depend on the fixed basis and its conditioning; no uniform claim over growing basis families is made without uniform bounds.Source: §A.6 · pp. 16–17
  • Under the stated conditions, the estimator is bounded relative to the population projection with probability at least 1−δ; when the field is realizable, that projection equals bv.For a nonrealizable basis, the target is the L2(µL) projection onto the fitted class.Source: §A.6 · pp. 16–17

A.7 Ornstein–Uhlenbeck Finite-Sample Convergence

The section establishes finite-sample convergence for affine probability-velocity estimates in an Ornstein–Uhlenbeck setting and connects it to the overall convergence proof. The guarantee concerns compact evaluation regions and does not require independent transitions within a trajectory.

A.7 Ornstein–Uhlenbeck Finite-Sample Convergence

  • The fitted affine field is an L2(ρtj) projection of the unrestricted probability velocity and need not equal that field.The compact set Ω is only an evaluation region; Gaussian samples are not truncated to it.Source: §A.7 · p. 17
  • The affine OU convergence corollary gives a high-probability guarantee at observation times when the sample size meets a dimension- and confidence-dependent threshold.Its stated threshold is N ≥ cOU paff ℓδ, with paff = d(d + 1) and ℓδ = log(2paffL/δ).Source: §A.7 · p. 18
  • Concentration across independent trajectories yields simultaneous control across entries and times, without assuming independent transitions within each trajectory.The proof uses scalar Bernstein inequalities and a union bound.Source: §A.7 · p. 18

A.8 Proof of the Overall Convergence

  • The overall convergence proof separates the error into a restricted-domain term and a velocity-approximation term bounded by Theorem 1.The second term is bounded by ∥bv − v∥∞,Ω×[0,T].Source: §A.8 · p. 19

B.1 Ornstein–Uhlenbeck Process

The section derives and tests probability flows for Ornstein–Uhlenbeck dynamics, then describes their use in double-pendulum state-space and image experiments. The OU flows reproduce exact marginal statistics, while the pendulum experiments use observed transitions and learned image representations.

B.1 Ornstein–Uhlenbeck Process

  • For Gaussian initial conditions, OU marginals remain Gaussian, and the process has a unique stationary Gaussian distribution.The stationary covariance is defined by a Lyapunov equation.Source: §B.1 · p. 19
  • In the non-reversible case, the limiting flow preserves the stationary Gaussian while allowing persistent probability circulation along closed elliptic orbits.The limiting velocity matrix is antisymmetric, and the trajectories are periodic.Source: §B.1 · pp. 20–21
  • An affine velocity field is fit from adjacent observed data pairs using the empirical least-squares objective.When the augmented sample matrix has full row rank, the minimizer is unique.Source: §B.1 · pp. 21–22
  • Computed probability flows agree well with exact mean and covariance evolution for both reversible and non-reversible OU processes.Figure 8 compares selected marginal mean and covariance components with exact solutions.Source: §B.1 · p. 22, Figure 8 · p. 22

B.2.1 State-Space Implementation Details

  • The double-pendulum state-space experiment uses stochastic perturbations to angular accelerations and trains on consecutive observations of the four-dimensional state.The described dataset contains 1,000 trajectories with 240 observations over eight seconds.Source: §B.2 · p. 23, §B.2.1 · p. 23

B.2.2 Image Data Generation

  • For image experiments, joint angles are rendered as grayscale pendulum frames, with 1,000 trajectories of 240 frames used for representation learning and latent-space generation.The links are unit length and are rendered against a black background.Source: §B.2.2 · p. 23

B.2.3 Geometry-Preserving Encoder and Decoder

  • The image pipeline uses a geometry-preserving encoder to map frames into a three-dimensional latent representation and a decoder to reconstruct images.Gaussian perturbations are added to latent variables during decoder training to improve robustness to autoregressive prediction errors.Source: §B.2.3 · p. 24

B.2.4 Encoder and Decoder Training

The section describes training a geometry-preserving image representation, learning probability velocities in latent dynamical space, and applying the workflow to stochastic Navier–Stokes fields and KTH videos. It also specifies the video metrics and evaluation protocol.

B.2.4 Encoder and Decoder Training

  • The encoder is trained to preserve pairwise latent geometry while keeping adjacent observations close, then fixed while the decoder learns reconstruction from noisy latent representations.The described procedure has separate encoder and decoder training stages.Source: §B.2.4 · p. 25

B.2.5 Latent-Space Dynamics

  • Latent positions are augmented with finite-difference velocity to form a six-dimensional dynamical state, whose probability velocity is learned for autoregressive prediction.The method applies the same probability-velocity objective as the explicit state-space experiment.Source: §B.2.5 · pp. 25–26, §B.2.6 · p. 26

B.2.6 Latent-Space Training and Prediction

  • Latent-space prediction advances the learned state autoregressively and decodes each predicted latent state into an image.The algorithm uses forward Euler updates with time input.Source: §B.2.5 · p. 26, §B.2.6 · p. 26

B.3 2D Navier–Stokes Equations Generation

  • The stochastic Navier–Stokes dataset contains 1,000 trajectories of two-dimensional vorticity fields, recorded every 0.5 time units and downsampled to 128 × 128.The simulation discards the initial transient and uses stochastic forcing on selected Fourier modes.Source: §B.3 · p. 26

B.4 KTH Video Generation

  • For KTH video generation, a pretrained VQGAN maps 64 × 64 frames to 4 × 8 × 8 latents, which PhiBE-Flow evolves before decoding back to frames.Evaluation uses frame-level MSE, PSNR, and SSIM, plus FVD in a learned spatiotemporal feature space.Source: §B.4 · pp. 26–27

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