CT-OT Flow: Estimating Continuous-Time Dynamics from Discrete Temporal Snapshots

TL;DR

This work presents Continuous-Time Optimal Transport Flow (CT-OT Flow), a two-stage framework that infers high-resolution time labels by aligning neighboring intervals via partial optimal transport (POT) and reconstructs a continuous-time data distribution through temporal kernel smoothing.

Abstract

In many real-world settings--e.g., single-cell RNA sequencing, mobility sensing, and environmental monitoring--data are observed only as temporally aggregated snapshots collected over finite time windows, often with noisy or uncertain timestamps, and without access to continuous trajectories. We study the problem of estimating continuous-time dynamics from such snapshots. We present Continuous-Time Optimal Transport Flow (CT-OT Flow), a two-stage framework that (i) infers high-resolution time labels by aligning neighboring intervals via partial optimal transport (POT) and (ii) reconstructs a continuous-time data distribution through temporal kernel smoothing, from which we sample pairs of nearby times to train standard ODE/SDE models. Our formulation explicitly accounts for snapshot aggregation and time-label uncertainty and uses practical accelerations (screening and mini-batch POT), making it applicable to large datasets. Across synthetic benchmarks and two real datasets (scRNA-seq and typhoon tracks), CT-OT Flow reduces distributional and trajectory errors compared with OT-CFM, [SF]\(^{2}\)M, TrajectoryNet, MFM, and ENOT.

Figures (20)

• Figure 1: Motivating example for CT-OT Flow. (a) True dynamics (arrow) and observations (points). (b) Dynamics estimated directly from the discrete timestamps are inaccurate. (c) CT-OT Flow: Dynamics recovered accurately follow the ground truth by inferring high-resolution time labels.
• Figure 2: CT-OT Flow Pipeline: POT-based high-resolution time label estimation, kernel-based continuous-time distribution estimation, and ODE/SDE model training.
• Figure 3: Step 1: high-resolution time label estimation via POT. (a) CT-OT Flow first extracts subsets $S^-_1$ and $S^+_1$ near the boundary of two time intervals $X_{[t_{j-1}, t_j]}$ (purple) and $X_{[t_{j}, t_{j+1}]}$ (orange). (b) (Forward) CT-OT Flow iteratively identifies the subset $S_{k+1}^+$. (c) (Backward) Similarly, CT-OT Flow identifies $S_{k+1}^-$. We set $K=5$ to better visualize the subset extraction process.
• Figure 4: Steps 2&3. A kernel function produces a smoothed empirical distribution $\tilde{p}_t(\boldsymbol x)$ at any continuous time $t$. We then sample from $\tilde{p}_t$ and $\tilde{p}_{t+\delta t}$ to train a dynamics model. Unlike conventional methods that operate only on pre-specified discrete times, CT-OT Flow allows arbitrarily fine choices of $\delta t$, capturing more realistic trajectories.
• Figure 5: Estimated trajectories on the Spiral dataset. The black lines indicate the true or estimated trajectories, while the color of each point indicates its time label. For CT-OT Flow and Slingshot, the colors represent the estimated high-resolution time labels and pseudotime labels, respectively. The initial points are sampled from $p_0(\boldsymbol x)$ but not from $p_{[0,1]}(\boldsymbol x)$. The proposed method effectively reproduces the true dynamics, whereas conventional methods often yield larger deviations from the ground truth.

Theorems & Definitions (5)

• Proposition 1
• Proposition 1
• proof
• Proposition 2
• proof