Schrödinger Bridge Over A Compact Connected Lie Group

A geometric coordinate-free solution for the isotropic Schrödinger bridge problem over Lie groups .

Hamza Mahmood, Abhishek Halder†, Adeel Akhtar

* Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology, Newark, NJ, †Department of Aerospace Engineering, Iowa State University, Ames, IA

GitHub Animations How to Run BibTeX

Snapshots of probability density function on SO(2) taken at five different time instances.

Abstract

This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$.

Animations

SBP on SO(2)

The initial PDF has three peaks while the terminal PDF has two peaks. The evolution of the intermediate PDFs illustrates how the three peaks smoothly redistribute to become two under the Schrödinger bridge dynamics on SO(2).

SBP on SO(3)

The above shows the computed $ \rho^{\mathrm{opt}}(\cdot,t) $, for $ t \in (0,1) $, supported on $ \|\omega\|_2 $. The boundary densities $ \rho_0 $ and $ \rho_1 $ are shown in dashed white and dashed red, respectively. The figure also illustrates the convergence of the Sinkhorn recursion with respect to the Hilbert metric $ d_H $. This shows the smooth evolution of optimally controlled probability density function.

Code and Reproducibility

The full source code is available in the GitHub repository: gradslab/SbpLieGroups.

After cloning the repository, install the required Python dependencies using pip install -r requirements.txt. Each experiment can then be executed directly from the command line using Python. For instance, python schrodinger_bridge_so2.py runs the SO(2) simulation, while python schrodinger_bridge_so3.py runs the SO(3) simulation. All generated figures and animation outputs are saved automatically in the assets/ directory.

Repository Structure

.
├── schrodinger_bridge_so2.py
├── schrodinger_bridge_so3.py
├── requirements.txt
├── README.md
└── assets/

How to Run

git clone https://github.com/gradslab/SbpLieGroups.git
cd SbpLieGroups

# Install required Python packages
pip install -r requirements.txt

# Run the simulations
python schrodinger_bridge_so2.py
python schrodinger_bridge_so3.py

BibTeX

@misc{sbp_lie_groups_2026,
  title        = {Schrödinger Bridge Over A Compact Connected Lie Group},
  author       = {Hamza Mahmood, Abhishek Halder, Adeel Akhtar},
  year         = {2026},
  howpublished = {GitHub repository},
  note         = {\url{https://github.com/gradslab/SbpLieGroups}}
}