# Trajectory Inference with Smooth Schr"odinger Bridges

arXiv: https://arxiv.org/abs/2503.00530 · HTML: https://mlsanity.com/paper/2503.00530/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 1 Mar 2025.

## Builds on (most cited first, 20 of 20)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [On the Complexity of Approximating Multimarginal Optimal Transport](https://mlsanity.com/paper/1910.00152/) (2019)
- [Wasserstein barycenters are NP-hard to compute](https://mlsanity.com/paper/2101.01100/) (2021)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Measure-valued spline curves: an optimal transport viewpoint](https://mlsanity.com/paper/1801.03186/) (2018)
- [Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure](https://mlsanity.com/paper/2008.03006/) (2020)
- [Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging](https://mlsanity.com/paper/2405.19679/) (2024)
- [Wasserstein-Fisher-Rao Splines](https://mlsanity.com/paper/2203.15728/) (2022)
- [Learning Hidden Markov Models from Aggregate Observations](https://mlsanity.com/paper/2011.11236/) (2020)

## Built on by (most cited first, 11 of 11)

- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [DICE: Discrete inverse continuity equation for learning population dynamics](https://mlsanity.com/paper/2507.05107/) (2025)
- [Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr"odinger Bridge's End](https://mlsanity.com/paper/2505.16082/) (2025)
- [Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space](https://mlsanity.com/paper/2507.01709/) (2025)
- [ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space](https://mlsanity.com/paper/2604.27443/) (2026)
- [Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems](https://mlsanity.com/paper/2605.25107/) (2026)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [Trajectory inference via Acceleration Matching](https://mlsanity.com/paper/2608.03916/) (2026)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference](https://mlsanity.com/paper/2610.19878/) (2026)

## Abstract (arXiv)

Motivated by applications in trajectory inference and particle tracking, we introduce Smooth Schr"odinger Bridges. Our proposal generalizes prior work by allowing the reference process in the Schr"odinger Bridge problem to be a smooth Gaussian process, leading to more regular and interpretable trajectories in applications. Though na"ively smoothing the reference process leads to a computationally intractable problem, we identify a class of processes (including the Mat'ern processes) for which the resulting Smooth Schr"odinger Bridge problem can be lifted to a simpler problem on phase space, which can be solved in polynomial time. We develop a practical approximation of this algorithm that outperforms existing methods on numerous simulated and real single-cell RNAseq datasets. The code can be found at https://github.com/WanliHongC/SmoothSB