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Solving Schrödinger bridge problem via continuous normalizing flow

Yang Jing, Lei Li, Jingtong Zhang

TL;DR

The Schrödinger Bridge Problem (SBP) is recast as an entropy-regularized transport task and solved with a mesh-free CNF framework that learns a drift via a hypothetical velocity field, converting the Fokker–Planck equation into a transport equation. A Γ-convergence analysis guarantees that the regularized, neural-network solutions converge to the classical SBP as the regularization parameter grows, linking computational efficiency with mathematical rigor. The approach is validated through 1D and 2D experiments, including Gaussian mixtures, toy generators, and a double-well potential, demonstrating accurate density evolution, drift recovery via score matching, and barrier-aware trajectory planning. The framework offers scalable density evolution and sampling for SBP-related generative tasks and provides a principled path to convergence guarantees in CNF-based methods for probability flows and stochastic control.

Abstract

The Schrödinger Bridge Problem (SBP), which can be understood as an entropy-regularized optimal transport, seeks to compute stochastic dynamic mappings connecting two given distributions. SBP has shown significant theoretical importance and broad practical potential, with applications spanning a wide range of interdisciplinary fields. While theoretical aspects of the SBP are well-understood, practical computational solutions for general cases have remained challenging. This work introduces a computational framework that leverages continuous normalizing flows and score matching methods to approximate the drift in the dynamic formulation of the SBP. The learned drift term can be used for building generative models, opening new possibilities for applications in probability flow-based methods. We also provide a rigorous $Γ-$convergence analysis for our algorithm, demonstrating that the neuron network solutions converge to the theoretical ones as the regularization parameter tends to infinity. Lastly, we validate our algorithm through numerical experiments on fundamental cases.

Solving Schrödinger bridge problem via continuous normalizing flow

TL;DR

The Schrödinger Bridge Problem (SBP) is recast as an entropy-regularized transport task and solved with a mesh-free CNF framework that learns a drift via a hypothetical velocity field, converting the Fokker–Planck equation into a transport equation. A Γ-convergence analysis guarantees that the regularized, neural-network solutions converge to the classical SBP as the regularization parameter grows, linking computational efficiency with mathematical rigor. The approach is validated through 1D and 2D experiments, including Gaussian mixtures, toy generators, and a double-well potential, demonstrating accurate density evolution, drift recovery via score matching, and barrier-aware trajectory planning. The framework offers scalable density evolution and sampling for SBP-related generative tasks and provides a principled path to convergence guarantees in CNF-based methods for probability flows and stochastic control.

Abstract

The Schrödinger Bridge Problem (SBP), which can be understood as an entropy-regularized optimal transport, seeks to compute stochastic dynamic mappings connecting two given distributions. SBP has shown significant theoretical importance and broad practical potential, with applications spanning a wide range of interdisciplinary fields. While theoretical aspects of the SBP are well-understood, practical computational solutions for general cases have remained challenging. This work introduces a computational framework that leverages continuous normalizing flows and score matching methods to approximate the drift in the dynamic formulation of the SBP. The learned drift term can be used for building generative models, opening new possibilities for applications in probability flow-based methods. We also provide a rigorous convergence analysis for our algorithm, demonstrating that the neuron network solutions converge to the theoretical ones as the regularization parameter tends to infinity. Lastly, we validate our algorithm through numerical experiments on fundamental cases.

Paper Structure

This paper contains 20 sections, 7 theorems, 83 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continuous normalizing flows
  4. convergence
  5. Benamou-Brenier functional
  6. Governing equations in the CNF framework
  7. The CNF based machine learning framework
  8. Training hypothetical velocity field
  9. Recovering the optimal drift
  10. Convergence of the optimization problem to SBP
  11. Mathematical formulation of the problems
  12. Regularity of feasible points
  13. Convergence of the functionals
  14. Numerical experiments
  15. 1D Gaussian mixture
  16. ...and 5 more sections

Key Result

Proposition 2.1

Any cluster point of the minimizers of a $\Gamma$-convergent sequence $(f_n)$ is a minimizer of the corresponding $\Gamma$-limit functional $f$.

Figures (11)

  • Figure 1: Evolution of particle distribution transitioning from Gaussian mixture to Standard Gaussian.
  • Figure 2: ODE interpolation trajectories and SDE interpolation paths.
  • Figure 3: Comparison of score function between score matching solutions and reference ones.
  • Figure 4: Comparison of drift of solutions between our algorithm and tradition primal dual method.
  • Figure 5: Comparison of SBP hypothetical velocity field with different $\sigma$ and the velocity field of optimal transport problem.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 5.1
  • proof
  • Theorem 5.1
  • proof
  • ...and 2 more