# Optimal Scheduling of Dynamic Transport

arXiv: https://arxiv.org/abs/2504.14425 · HTML: https://mlsanity.com/paper/2504.14425/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2025.

## Built on by (most cited first, 13 of 13)

- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Self-Consistent Flow: Unifying Velocity and Endpoint Prediction for Rectified Flow Models](https://mlsanity.com/paper/2607.12171/) (2026)
- [Sharpen Your Flow: Sharpness-Aware Sampling for Flow Matching](https://mlsanity.com/paper/2605.11547/) (2026)
- [Learning Paths for Dynamic Measure Transport: A Control Perspective](https://mlsanity.com/paper/2511.03797/) (2025)
- [Adaptive Path Integral Diffusion: AdaPID](https://mlsanity.com/paper/2512.11858/) (2025)
- [Entropy Across the Bridge: Conditional-Marginal Discretization for Flow and Schr"odinger Samplers](https://mlsanity.com/paper/2605.16126/) (2026)
- [Gaussian Flow-Matching Schedules: Implications for Sampling and Training](https://mlsanity.com/paper/2609.25839/) (2026)
- [Time-dependent density estimation using binary classifiers](https://mlsanity.com/paper/2506.15505/) (2025)
- [Population-Level Generative Modeling for Ranking Data](https://mlsanity.com/paper/2608.08422/) (2026)
- [Model-Aware Schedules Improve Generation via Fiberwise Optimal Transport](https://mlsanity.com/paper/2609.11842/) (2026)
- [Geometry-Aware Time Reparameterization for Flow-Map Distillation](https://mlsanity.com/paper/2610.02427/) (2026)

## Abstract (arXiv)

Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a transport map that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable design freedom, and a central question is how to exploit this freedom. Though many popular methods seek straight line (i.e., zero acceleration) trajectories, we show here that a specific class of ``curved'' trajectories can significantly improve approximation and learning. In particular, we consider the unit-time interpolation of any given transport map T and seek the schedule τ: [0,1] → [0,1] that minimizes the spatial Lipschitz constant of the corresponding velocity field over all times t ∈ [0,1]. This quantity is crucial as it allows for control of the approximation error when the velocity field is learned from data. We show that, for a broad class of source/target measures and transport maps T, the optimal schedule can be computed in closed form, and that the resulting optimal Lipschitz constant is exponentially smaller than that induced by an identity schedule (corresponding to, for instance, the Wasserstein geodesic). Our proof technique relies on the calculus of variations and Gamma-convergence, allowing us to approximate the aforementioned degenerate objective by a family of smooth, tractable problems.