# Deep Generalized Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2209.09893 · HTML: https://mlsanity.com/paper/2209.09893/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 72 references that are in the map.
- First posted on arXiv 20 Sep 2022.

## Builds on (most cited first, 30 of 33)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Rainbow: Combining Improvements in Deep Reinforcement Learning](https://mlsanity.com/paper/1710.02298/) (2017)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth](https://mlsanity.com/paper/2003.05508/) (2020)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networks](https://mlsanity.com/paper/1803.01299/) (2018)
- [Value Iteration in Continuous Actions, States and Time](https://mlsanity.com/paper/2105.04682/) (2021)
- [Polarization in Geometric Opinion Dynamics](https://mlsanity.com/paper/2106.12459/) (2021)
- [The Effect of Multi-step Methods on Overestimation in Deep Reinforcement Learning](https://mlsanity.com/paper/2006.12692/) (2020)
- [Density control of interacting agent systems](https://mlsanity.com/paper/2108.07342/) (2021)
- [Learning Optimal Control via Forward and Backward Stochastic Differential Equations](https://mlsanity.com/paper/1509.02195/) (2015)
- [Second-Order Neural ODE Optimizer](https://mlsanity.com/paper/2109.14158/) (2021)
- [Effective Multi-step Temporal-Difference Learning for Non-Linear Function Approximation](https://mlsanity.com/paper/1608.05151/) (2016)
- [A Geometric Model of Opinion Polarization](https://mlsanity.com/paper/1910.05274/) (2019)

## Built on by (most cited first, 30 of 47)

- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [ZigMa: A DiT-style Zigzag Mamba Diffusion Model](https://mlsanity.com/paper/2403.13802/) (2024)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Image Restoration Through Generalized Ornstein-Uhlenbeck Bridge](https://mlsanity.com/paper/2312.10299/) (2023)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Aligned Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2302.11419/) (2023)
- [RetroBridge: Modeling Retrosynthesis with Markov Bridges](https://mlsanity.com/paper/2308.16212/) (2023)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Neural Optimal Transport with Lagrangian Costs](https://mlsanity.com/paper/2406.00288/) (2024)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025)
- [Discrete Adjoint Matching](https://mlsanity.com/paper/2602.07132/) (2026)
- [Transport with Support: Data-Conditional Diffusion Bridges](https://mlsanity.com/paper/2301.13636/) (2023)
- [Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis](https://mlsanity.com/paper/2503.11347/) (2025)
- [Bidirectional Diffusion Bridge Models](https://mlsanity.com/paper/2502.09655/) (2025)
- [Non-equilibrium Annealed Adjoint Sampler](https://mlsanity.com/paper/2506.18165/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Feedback Schr"odinger Bridge Matching](https://mlsanity.com/paper/2410.14055/) (2024)
- [Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations](https://mlsanity.com/paper/2401.15482/) (2024)
- [Exploring the Design Space of Diffusion Bridge Models](https://mlsanity.com/paper/2410.21553/) (2024)
- [Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching](https://mlsanity.com/paper/2411.08314/) (2024)
- [Fractional Diffusion Bridge Models](https://mlsanity.com/paper/2511.01795/) (2025)

## Abstract (arXiv)

Mean-Field Game (MFG) serves as a crucial mathematical framework in modeling the collective behavior of individual agents interacting stochastically with a large population. In this work, we aim at solving a challenging class of MFGs in which the differentiability of these interacting preferences may not be available to the solver, and the population is urged to converge exactly to some desired distribution. These setups are, despite being well-motivated for practical purposes, complicated enough to paralyze most (deep) numerical solvers. Nevertheless, we show that Schr"odinger Bridge - as an entropy-regularized optimal transport model - can be generalized to accepting mean-field structures, hence solving these MFGs. This is achieved via the application of Forward-Backward Stochastic Differential Equations theory, which, intriguingly, leads to a computational framework with a similar structure to Temporal Difference learning. As such, it opens up novel algorithmic connections to Deep Reinforcement Learning that we leverage to facilitate practical training. We show that our proposed objective function provides necessary and sufficient conditions to the mean-field problem. Our method, named Deep Generalized Schr"odinger Bridge (DeepGSB), not only outperforms prior methods in solving classical population navigation MFGs, but is also capable of solving 1000-dimensional opinion depolarization, setting a new state-of-the-art numerical solver for high-dimensional MFGs. Our code will be made available at https://github.com/ghliu/DeepGSB.