# Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift

arXiv: https://arxiv.org/abs/1912.01244 · HTML: https://mlsanity.com/paper/1912.01244/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2019.

## Built on by (most cited first, 30 of 50)

- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
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- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [Transport with Support: Data-Conditional Diffusion Bridges](https://mlsanity.com/paper/2301.13636/) (2023)
- [Variational Schr"odinger Diffusion Models](https://mlsanity.com/paper/2405.04795/) (2024)
- [Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamics](https://mlsanity.com/paper/2204.04853/) (2022)
- [Maximum entropy optimal density control of discrete-time linear systems and Schr"odinger bridges](https://mlsanity.com/paper/2204.05263/) (2022)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Simplified Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2403.14623/) (2024)
- [Conditional Lagrangian Wasserstein Flow for Time Series Imputation](https://mlsanity.com/paper/2410.07550/) (2024)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schr"odinger Bridge Matching](https://mlsanity.com/paper/2602.15396/) (2026)
- [On the Contraction Coefficient of the Schr"odinger Bridge for Stochastic Linear Systems](https://mlsanity.com/paper/2309.06622/) (2023)
- [Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr"odinger Bridge and Reaction-Diffusion PDEs](https://mlsanity.com/paper/2401.07961/) (2024)
- [Stochastic Learning of Computational Resource Usage as Graph Structured Multimarginal Schr"odinger Bridge](https://mlsanity.com/paper/2405.12463/) (2024)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [Schr"odinger Bridge Over A Compact Connected Lie Group](https://mlsanity.com/paper/2603.14049/) (2026)
- [Proximal Mean Field Learning in Shallow Neural Networks](https://mlsanity.com/paper/2210.13879/) (2022)

## Abstract (arXiv)

We study the Schr"odinger 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 time horizon, subject to a controlled stochastic differential evolution of the state vector. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the "Schr"odinger factors" -- so named since their product at any time yields the optimal controlled joint state PDF at that time. We show that if the drift is a gradient vector field, or is of mixed conservative-dissipative nature, then it is possible to transform these PDEs into a pair of initial value problems (IVPs) involving the same forward Kolmogorov operator. Combined with a recently proposed fixed point recursion that is contractive in the Hilbert metric, this opens up the possibility to numerically solve the SBPs in these cases by computing the Schr"odinger factors via a single IVP solver for the corresponding (uncontrolled) forward Kolmogorov PDE. The flows generated by such forward Kolmogorov PDEs, for the two aforementioned types of drift, in turn, enjoy gradient descent structures on the manifold of joint PDFs with respect to suitable distance functionals. We employ a proximal algorithm developed in our prior work, that exploits this geometric viewpoint, to solve these IVPs and compute the Schr"odinger factors via weighted scattered point cloud evolution in the state space. We provide the algorithmic details and illustrate the proposed framework of solving the SBPs with nonlinear prior dynamics by numerical examples.