Foundations of Schrödinger Bridges for Generative Modeling

by Sophia Tang
Correspondence to: sophtang@seas.upenn.edu

This comprehensive guide (220 pages, 24 figures) covers the mathematical foundation of Schrödinger bridges as a unifying principle underlying modern generative modeling techniques.

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About This Guide

The field of generative modeling has experienced rapid and transformative progress in recent years. Advances range from foundational theoretical developments—such as diffusion models and flow matching—to algorithmic improvements in sampling speed and generation quality, as well as significant applications spanning language, video, and scientific domains. While this breadth of progress has driven remarkable innovation, it has also made the field increasingly complex and difficult to navigate, particularly from a foundational perspective.

In this guide, we introduce Schrödinger bridges as a unifying theoretical framework for generative modeling. This perspective generalizes a broad class of modern approaches—including diffusion models, flow-matching methods, and stochastic control formulations—while providing a principled and flexible foundation for addressing specialized scientific problems. The body of work that has built the Schrödinger bridge framework spans multiple fields, from theoretical developments that formalize path-space optimization and entropy-regularized transport, to principled extensions in diverse problem settings, to algorithmic advances that enable scalable training and inference. Despite this progress, the literature remains fragmented, and the relationships between different formulations are often difficult to identify.

The goal of this guide is to build both intuition and a deep mathematical understanding of the core principles of Schrödinger bridges, from its origins in optimal transport to its dynamic path space formulation, which underlies modern generative modeling frameworks. At a high level, we begin from a single unifying principle: optimal stochastic bridges between distributions can be characterized as minimal-entropy deviations from a reference process subject to marginal constraints. From this perspective, we develop the mathematical tools spanning optimal transport, probability theory, and stochastic calculus required to understand the static and dynamic formulations of the Schrödinger bridge problem, before naturally extending these ideas to diverse problem settings and modern generative modeling techniques. Throughout this guide, we construct a principled toolkit for building Schrödinger bridges from first principles and show how these constructions give rise to theoretically-grounded and scalable computational methods for simulating the optimal Schrödinger bridge dynamics.

Table of Contents

This guide contains eight primary sections. See what each section covers by expanding the details below.

1 The Static Schrödinger Bridge Problem

We start our deep dive into Schrödinger bridges by tracing to its origins in optimal mass transport (OMT), which seeks the best coupling between distributions. However, classical OT can be unstable and non-unique, making it difficult to use in practice.

By introducing entropy regularization, we obtain entropic OT, leading to the static Schrödinger bridge formulation. This yields a unique, stable solution that can be efficiently computed with the classic Sinkhorn, forming the foundation of everything that follows.

  • 1.1 The Optimal Mass Transport Problem: traces back to the classical optimal mass transport (OMT) problem posed by Monge and Kantorovich, which seeks an optimal static coupling between probability distributions that minimizes total transport cost.
  • 1.2 Entropy on Probability Spaces: introduces the foundational properties of entropy on probability spaces, including the fundamental properties of the Kullback–Leibler (KL) divergence.
  • 1.3 Entropic Optimal Transport Problem: introduces the entropic optimal transport (EOT) problem, which leverages entropy as a method of regularizing the OMT problem with a pre-defined reference coupling, which yields a unique solution.
  • 1.4 Static Schrödinger Bridge Problem: extends the EOT problem to formalize the static Schrödinger bridge problem and its corresponding dual problem, which introduces a pair of Schrödinger potentials that uniquely yield a clean form of the optimal solution.
  • 1.5 Sinkhorn’s Algorithm: breaks down the classic algorithm for solving the static SB problem, which alternates between optimizing the dual Schrödinger potentials.
2 The Dynamic Schrödinger Bridge Problem

We next move to the dynamic formulation of the Schrödinger bridge problem, which lifts the problem from static couplings to continuous-time stochastic processes between distributions. Instead of directly transporting mass, we ask: among all stochastic evolutions that transform an initial distribution into a target distribution, which one is most likely relative to a given reference dynamics?

In this section, we formalize this dynamic viewpoint and develop the stochastic calculus tools—including path measures, Itô processes, and change-of-measure techniques—necessary to analyze and solve the dynamic SB problem.

  • 2.1 Dynamic Optimal Transport Problem: starts by redefining the static SB problem as learning a continuous-time deterministic flow between distributions.
  • 2.2 Dynamic Schrödinger Bridge Problem: reformulates the SB problem as an entropy minimization over stochastic processes.
  • 2.3 Path Measures and Itô Processes: provides the key definitions and theory required for understanding path measures as stochastic differential equations (SDEs) that can be steered via control drifts and transformed through functions.
  • 2.4 Fokker–Planck and Feynman–Kac Equations: derives the Fokker–Planck equation, governing how the probability density over stochastic paths evolves forward in time, and the Feynman–Kac equation, governing how functions evaluated at the end of a stochastic process evolve backward in time.
  • 2.5 Girsanov’s Theorem: derives Girsanov’s theorem from first principles, which allows us to define changes in measure and KL divergences on path space.
  • 2.6 Path Measure Radon-Nikodym Derivative and KL Divergence: explicitly breaks down the theory of Radon–Nikodym derivatives between path measures, used to define the relative-entropy between path measures.
  • 2.7 Schrödinger Bridge with Arbitrary Reference Dynamics: introduces the explicit form of the dynamic Schrödinger bridge problem as minimizing the quadratic cost of the control drift and derives its optimality conditions via the Lagrangian.
  • 2.8 Hopf–Cole Transform: transforms the system of nonlinear Hamilton–Jacobi–Bellman and Fokker–Planck PDEs into a system of linear PDEs which define a pair of time-dependent Schrödinger potentials that solve the dynamic SB problem.
  • 2.9 Schrödinger Bridges as Entropy-Regularized Dynamic Optimal Transport: reformulates the dynamic Schrödinger bridge problem as an entropy-regularized dynamic optimal transport problem by absorbing the diffusion term in the Fokker–Planck constraint into the objective functional.
3 Schrödinger Bridge Problem as Optimal Control

The key idea explored in this section is that the Schrödinger bridge problem can be solved with stochastic optimal control (SOC), which defines a path-space variational objective that minimizes a running cost and terminal constraint generated by a controlled SDE.

Crucially, the quadratic control cost is exactly a KL divergence between path measures. This reframes SB as learning minimal deviations from a reference process, without ever needing explicit couplings. We break down the dynamic programming theory and practical objectives for solving the SB problem with SOC.

  • 3.1 Stochastic Optimal Control: introduces the general framework of stochastic optimal control, including Bellman’s Principle of Optimality given a terminal constraint and deriving the value function that defines the optimal control.
  • 3.2 Schrödinger Bridges with Stochastic Optimal Control: connects the Schrödinger bridge problem to stochastic optimal control, showing that the optimal bridge corresponds to an optimal control drift.
  • 3.3 Objectives for Solving the SOC Problem: develops practical objectives for solving the stochastic optimal control problem.
4 Building Schrödinger Bridges

The theory tells us what SB is, but how do we construct it? This section covers several complementary mechanisms for building stochastic bridges between prescribed endpoint distributions, including mixtures of bridges, time-reversal, forward–backward SDEs, Doob’s $h$-transform, Markovian and reciprocal projections, and stochastic interpolants.

While each approach originates from a different mathematical viewpoint, they ultimately converge to a unified form of a Markov control drift that minimally corrects the uncontrolled reference dynamics such that they reconstruct the prescribed marginal distributions.

  • 4.1 Mixture of Conditional Bridges: introduces mixtures of conditional bridges constructed with the reference process and a pre-defined endpoint coupling.
  • 4.2 Time Reversal: derives the time-reversal formula of SDEs, which is fundamental to backward dynamics in Schrödinger bridges.
  • 4.3 Forward–Backward Stochastic Differential Equations: introduces forward–backward stochastic differential equations (FBSDEs), providing a coupled characterization of the Schrödinger bridge with respect to the time-dependent Schrödinger potentials.
  • 4.4 Doob’s $h$-Transform: presents Doob’s $h$-transform, which constructs conditioned stochastic processes by tilting the reference process using the $h$-function.
  • 4.5 Markovian and Reciprocal Projections: formalizes Markovian and reciprocal projections, which perform entropy-minimizing projections in path space that converge to the optimal Schrödinger bridge measure.
  • 4.6 Stochastic Interpolants to Schrödinger Bridges: introduces stochastic interpolants, providing a practical way to construct bridges between distributions as deterministic interpolants with Gaussian noise.
5 Variations of the Schrödinger Bridge Problem

The dynamic SB problem is just the beginning. In this section, we break down extensions of the SB framework to specialized problem settings, including Gaussian marginals, mean-field interactions, multi-marginal constraints, unbalanced marginals, branched and multi-modal marginals, and fractional Brownian motion.

Each problem adapts the dynamic SB formulation to different constraints while maintaining the same variational principles, illustrating the flexibility of the Schrödinger bridge framework in describing complex dynamical systems.

  • 5.1 Gaussian Schrödinger Bridge Problem: studies the Gaussian Schrödinger bridge problem, which admits closed-form solutions.
  • 5.2 Generalized Schrödinger Bridge Problem: introduces the generalized Schrödinger bridge problem which generalizes the dynamic formulation to model mean-field interactions.
  • 5.3 Multi-Marginal Schrödinger Bridge Problem: introduces the multi-marginal Schrödinger bridge problem, which extends the framework to settings with multiple intermediate marginal constraints.
  • 5.4 Unbalanced Schrödinger Bridge Problem: develops the unbalanced Schrödinger bridge problem, allowing mass creation and destruction along the stochastic trajectories.
  • 5.5 Branched Schrödinger Bridge Problem: introduces the branched Schrödinger bridge problem, enabling modeling of diverging trajectories to multiple distinct terminal modes.
  • 5.6 Fractional Schrödinger Bridge Problem: studies fractional Schrödinger bridge problems, incorporating long-range temporal dependencies through fractional Brownian motion.
6 Generative Modeling with Schrödinger Bridges

Modern generative models can all be interpreted as constructing Schrödinger bridges. Diffusion models, score matching, and flow matching are different parameterizations addressing the problem of learning controlled stochastic dynamics that interpolate between an initial and a target distribution while minimizing relative entropy with respect to a reference process.

This section breaks down several generative modeling techniques for solving the dynamic SB problem, from likelihood-based training to adjoint matching, highlighting how Schrödinger bridges unify these methods under a single framework.

  • 6.1 A Primer on Score-Based Generative Modeling: provides a primer on score-based generative modeling, which learns gradients of log-densities.
  • 6.2 Likelihood Training of Forward–Backward SDEs: introduces likelihood training of forward–backward SDEs, which trains the forward and backward potential drifts with likelihood objectives.
  • 6.3 Diffusion Schrödinger Bridge Matching: develops diffusion Schrödinger bridge matching, which parameterizes the Iterative Markovian Fitting procedure with a learned Markov drift.
  • 6.4 Simulation-Free Score and Flow Matching: introduces simulation-free score and flow matching, enabling training of Schrödinger bridges without simulating full trajectories.
  • 6.5 Schrödinger Bridge with Adjoint Matching: presents adjoint matching, which learns the optimal Schrödinger bridge while avoiding explicit sampling from target distributions.
7 From Continuous to Discrete State Space

We now extend SB from continuous state spaces to discrete state spaces, which can be used to model sequences of tokens, graphs, and discrete distributions. Establishing discrete state path measures not as SDEs but as continuous-time Markov chains (CTMCs) defined by their rate matrices, the discrete SB problem becomes optimizing a jump process that minimizes KL divergence from a reference process.

Despite this shift, the same foundational principles apply. We show how the discrete Schrödinger bridge problem can be solved using stochastic optimal control (SOC) and Markovian and reciprocal projections, closely mirroring the continuous setting.

  • 7.1 Continuous-Time Markov Chains: introduces continuous-time Markov chains (CTMCs) as discrete analogues of stochastic processes.
  • 7.2 Discrete Schrödinger Bridge Problem: formulates the discrete Schrödinger bridge problem, introducing Radon–Nikodym derivatives and KL divergences for CTMCs.
  • 7.3 Stochastic Optimal Control of CTMCs: develops stochastic optimal control for CTMCs as a method of learning the optimal CTMC path measure with terminal constraint.
  • 7.4 Discrete Schrödinger Bridges with Stochastic Optimal Control: connects discrete Schrödinger bridges with stochastic optimal control and introduces practical algorithms and objectives.
  • 7.5 Discrete Markov and Reciprocal Projections: introduces Markovian and reciprocal projections in discrete spaces.
  • 7.6 Discrete Diffusion Schrödinger Bridge Matching: develops discrete diffusion Schrödinger bridge matching, which parameterizes the Markovian and reciprocal generators to solve the discrete Schrödinger bridge.
8 Applications of Generative Modeling with Schrödinger Bridges

While Schrödinger bridges can be seen as a unified framework for generative modeling techniques, from denoising diffusion to flow matching, there are several applications where Schrödinger bridge frameworks are specialized to solve.

We highlight three prominent applications where advances in Schrödinger bridges have led to principled and practically effective solutions, including data translation, single-cell modeling, and sampling complex energy landscapes.

  • 8.1 Data Translation: applies SB to data translation tasks, which map between structured data distributions.
  • 8.2 Modeling Single-Cell State Dynamics: studies single-cell state dynamics, showing how Schrödinger bridge formulations can model cell population dynamics and responses to perturbation.
  • 8.3 Sampling Boltzmann Distributions: applies SB to sampling Boltzmann distributions, showing how the SB frameworks can generate from unnormalized energy distributions without explicit samples.

BibTeX

@article{tang2026foundations,
  title={Foundations of Schr\"{o}dinger Bridges for Generative Modeling},
  author={Tang, Sophia},
  journal={arXiv preprint arXiv:2603.18992},
  year={2026},
  url={https://arxiv.org/abs/2603.18992}
}