GEORCE: A Fast Control Algorithm for Constructing Geodesics

Efficient computations of geodesic algorithms formulating optimization of the energy functional as a control problem.

Installation and Requirements

The implementations in the GitHub is Python using JAX. To clone the GitHub reporsitory and install packages type the following in the terminal

git clone https://github.com/FrederikMR/georce.git
cd georce
pip install -r requirements.txt

The first line clones the repository, the second line moves you to the location of the files, while the last line install the packages used in repository.

Code Structure

The following shows the structure of the code. All general implementations of geometry and optimization algorithms can be found in the "geometry" folder for both the Riemannian and Finsler case.

.
├── load_manifold.py                   # Load manifolds and points for connecting geodesic
├── runtime.py                         # Times length and runtime for different optimization algorithms to consturct geodesic
├── train_vae.py                       # Training of the VAE
├── finsler_geodesic.ipynb             # Finsler geometry figures and plots
├── riemannian_geodesics.ipynb         # Riemannian geometry figures and plots
├── vae_geodesics.ipynb                # Geodesics for learned manifolds using vae
├── runtime_estimates.ipynb            # Runtime tables and figures
├── georce.ipynb                       # An example of how to use GEORCE for Riemannian and Finsler manifolds
├── timing_cpu                         # Contains all timing results for cpu
├── timing_gpu                         # Contains all timing results for gpu
├── vae                                # Contains implementation of the VAE
├── geometry                           # Contains implementation of Finsler and Riemannian manifolds as well as geodesic optimization algorithms, inlcuding GEORCE
├── georce                             # A folder containing the GEORCE algorithm for Finsler and Riemannian manifolds that can be directly used in your application
└── README.md

Reproducing Experiments

All experiments can be re-produced by running the notebooks and the runtime.py package for the given manifold, hyper-parameters and optimization method.

Logging

All experimental results for the runtime and length estimates are saved as .pkl files in the folder "timing".

How to use GEORCE to compute your geodesics

If you want to clone this repository and use GEORCE to compute geodesics for your manifolds, then the folder "georce" contains implementations of GEORCE in JAX for Riemannian and Finsler manifolds, which takes any given metric as input. The notebook, georce_example.ipynb, illustrates how to compute geodesics using GEORCE.

Reference

If you want to use GEORCE for scientific purposes, please cite:

@misc{rygaard2025georcefastnewcontrol,
      title={GEORCE: A Fast New Control Algorithm for Computing Geodesics}, 
      author={Frederik Möbius Rygaard and Søren Hauberg},
      year={2025},
      eprint={2505.05961},
      archivePrefix={arXiv},
      primaryClass={math.DG},
      url={https://arxiv.org/abs/2505.05961}, 
}