# Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free

arXiv: https://arxiv.org/abs/2505.16644 · HTML: https://mlsanity.com/paper/2505.16644/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 61 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 27 of 27)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (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)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Inverse optimal transport](https://mlsanity.com/paper/1905.03950/) (2019)
- [Optimal steering of a linear stochastic system to a final probability distribution](https://mlsanity.com/paper/1408.2222/) (2014)
- [Optimal steering of a linear stochastic system to a final probability distribution, part II](https://mlsanity.com/paper/1410.3447/) (2014)
- [Tsallis Regularized Optimal Transport and Ecological Inference](https://mlsanity.com/paper/1609.04495/) (2016)
- [Stochastic bridges of linear systems](https://mlsanity.com/paper/1407.3421/) (2014)
- [Optimal transport over a linear dynamical system](https://mlsanity.com/paper/1502.01265/) (2015)
- [Inferring biological processes with intrinsic noise from cross-sectional data](https://mlsanity.com/paper/2410.07501/) (2024)
- [Joint trajectory and network inference via reference fitting](https://mlsanity.com/paper/2409.06879/) (2024)
- [Flow matching for stochastic linear control systems](https://mlsanity.com/paper/2412.00617/) (2024)
- [Entropy production of Multivariate Ornstein-Uhlenbeck processes correlates with consciousness levels in the human brain](https://mlsanity.com/paper/2207.05197/) (2022)

## Built on by (most cited first, 3 of 3)

- [Simulation-free Structure Learning for Stochastic Population Dynamics](https://mlsanity.com/paper/2510.16656/) (2025)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [PRISM: Principled Reference Identification for Schrodinger Bridge Model](https://mlsanity.com/paper/2608.06893/) (2026)

## Abstract (arXiv)

We consider the Schr"odinger bridge problem which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the "most likely" evolution of the system compatible with the data. Most existing literature assume Brownian reference dynamics, and are implicitly limited to modelling systems driven by the gradient of a potential energy. We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix A ∈ ℝd × d. When A is asymmetric, this corresponds to a non-equilibrium system in which non-gradient forces are at play: this is important for applications to biological systems, which naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise exactly the solution of both the static and dynamic Schr"odinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning an approximation to the Schr"odinger bridge. In application to a range of problems based on synthetic and real single cell data, we demonstrate that mvOU-OTFM achieves higher accuracy compared to competing methods, whilst being significantly faster to train.