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The Lambert problem originated in orbital mechanics. It concerns with determining the initial velocity for a boundary value problem involving the dynamical constraint due to gravitational potential with additional time horizon and endpoint…

Optimization and Control · Mathematics 2024-10-04 Alexis M. H. Teter , Iman Nodozi , Abhishek Halder

Generalized Schr\"odinger Bridges (GSBs) are a fundamental mathematical framework used to analyze the most likely particle evolution based on the principle of least action including kinetic and potential energy. In parallel to their…

Machine Learning · Statistics 2024-12-31 Guan-Horng Liu , Tianrong Chen , Evangelos A. Theodorou

Predicting the intermediate trajectories between an initial and target distribution is a central problem in generative modeling. Existing approaches, such as flow matching and Schr\"odinger bridge matching, effectively learn mappings…

Machine Learning · Computer Science 2026-03-03 Sophia Tang , Yinuo Zhang , Alexander Tong , Pranam Chatterjee

Modern distribution matching algorithms for training diffusion or flow models directly prescribe the time evolution of the marginal distributions between two boundary distributions. In this work, we consider a generalized distribution…

We study a martingale Schr\"odinger bridge problem: given two probability distributions, find their martingale coupling with minimal relative entropy. Our main result provides Schr\"odinger potentials for this coupling. Namely, under…

Probability · Mathematics 2025-09-01 Marcel Nutz , Johannes Wiesel

We study the Schr\"odinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schr\"odinger potentials via Sinkhorn iterations on empirical measures and then…

Machine Learning · Statistics 2026-02-10 Denis Belomestny , Alexey Naumov , Nikita Puchkin , Denis Suchkov

Quantum counterparts of Schrodinger's classical bridge problem have been around for the better part of half a century. During that time, several quantum approaches to this multifaceted classical problem have been introduced. In the present…

Quantum Physics · Physics 2025-03-11 Olga Movilla Miangolarra , Ralph Sabbagh , Tryphon T. Georgiou

Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual…

Machine Learning · Statistics 2025-10-21 Byoungwoo Park , Juho Lee

Simulating trajectories of multi-particle systems on complex energy landscapes is a central task in molecular dynamics (MD) and drug discovery, but remains challenging at scale due to computationally expensive and long simulations. Previous…

Machine Learning · Computer Science 2025-11-11 Sophia Tang , Yinuo Zhang , Pranam Chatterjee

Schr\"{o}dinger bridge can be viewed as a continuous-time stochastic control problem where the goal is to find an optimally controlled diffusion process whose terminal distribution coincides with a pre-specified target distribution. We…

Machine Learning · Statistics 2024-04-23 Jhanvi Garg , Xianyang Zhang , Quan Zhou

In this paper, we study the Schr\"odinger Bridge Problem (SBP), which is central to entropic optimal transport. For general reference processes and begin--endpoint distributions, we propose a forward-reverse iterative Monte Carlo procedure…

Machine Learning · Statistics 2025-07-02 Denis Belomestny , John. Schoenmakers

In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem…

Optimization and Control · Mathematics 2020-11-30 Yongxin Chen , Tryphon T. Georgiou , Michele Pavon

Score-based generative models have recently attracted significant attention for their ability to generate high-fidelity data by learning maps from simple Gaussian priors to complex data distributions. A natural generalization of this idea…

Computation · Statistics 2025-11-19 Hanwen Huang

Over the last several years, there has been significant progress in developing neural solvers for the Schr\"odinger Bridge (SB) problem and applying them to generative modelling. This new research field is justifiably fruitful as it is…

Resampling from a target measure whose density is unknown is a fundamental problem in mathematical statistics and machine learning. A setting that dominates the machine learning literature consists of learning a map from an easy-to-sample…

Machine Learning · Computer Science 2026-02-24 Leticia Mattos Da Silva , Silvia Sellán , Francisco Vargas , Justin Solomon

We characterize the Schr\"odinger bridge problems by a family of Mckean-Vlasov stochastic control problems with no terminal time distribution constraint. In doing so, we use the theory of Hilbert space embeddings of probability measures and…

Optimization and Control · Mathematics 2025-12-10 Yumiharu Nakano

We study the Schr\"{o}dinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite…

Optimization and Control · Mathematics 2021-03-17 Kenneth F. Caluya , Abhishek Halder

Motivated by modern machine learning applications where we only have access to empirical measures constructed from finite samples, we relax the marginal constraints of the classical Schr\"odinger bridge problem by penalizing the transport…

Probability · Mathematics 2026-02-10 Yifan Jiang , Renyuan Xu , Luhao Zhang

The Schr\"odinger bridge problem (SBP) aims at finding the measure $\hat{\mathbf{P}}$ on a certain path space which possesses the desired state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL…

Probability · Mathematics 2025-11-11 Andrei Zlotchevski , Linan Chen

The subject of this work has its roots in the so called Schroedginer Bridge Problem (SBP) which asks for the most likely distribution of Brownian particles in their passage between observed empirical marginal distributions at two distinct…

Systems and Control · Computer Science 2016-08-15 Yongxin Chen , Tryphon T. Georgiou , Michele Pavon