[ICML25] Neural Discovery in Mathematics

This repository contains the code for the ICML2025 paper "Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?".

Authors: Konrad Mundinger, Max Zimmer, Aldo Kiem, Christoph Spiegel, Sebastian Pokutta

[Paper] [Blogpost] [BibTeX]

Environment setup

We use Python 3.11.9. To install the required dependencies, run:

pip install -r requirements.txt

We use Weights & Biases (wandb) for experiment tracking and logging. To enable it:

1. Create a free account at wandb.ai.

2. Log in via the command line:

   wandb login

   This will prompt you to paste your API key, which can be found in your W&B account settings.

You can find your logs, metrics, and model checkpoints on the project dashboard linked in your Weights & Biases account.

Running the code

You can start a run by invoking

python main.py

This will use the parameters specified directly in the main.py file. For the different variants described in the paper, we provide the following config files:

Vanilla Hadwiger-Nelson problem

To run the vanilla Hadwiger-Nelson problem with seven colors in 2D, use:

python main.py --config=configs/vanilla_seven_color.yaml

Almost coloring

To minimize the occurence of the last color with a lagrangian multiplier, use:

python main.py --config=configs/lagrange_six_color.yaml

Polychromatic Number

For training on a range of distances for the last color, use:

python main.py --config=configs/polychromatic_number.yaml

Hadwiger-Nelson in three dimensions

For coloring $3$-dimensional space, use:

python main.py --config=configs/coloring_space.yaml

Note that no visualizations will be generated.

Important Notes

If you want to run the experiments for different amounts of colors or change any other parameters, you can modify the .yaml files accordingly. Please note that a single run will most likely not yield the best results. We obtained our results by running each experiment multiple times and selecting the best runs afterwards.

Discrete almost-colorings

We provide an almost-coloring of the plane with five colors and an almost-coloring of three-dimensional space with 14 colors, both obtained using Algorithm 1 from our paper. They are available in the constructions folder and can be downloaded using git-lfs.

Almost-5 coloring

The files in constructions/almost-5-coloring-2D contain a discretized almost-5-coloring.
The coloring is periodic on the parallelogram spanned by the vectors v₁ and v₂, given in parallelogram.csv.
It is constant on small parallelograms, which are spanned by scaled versions of v₁ and v₂.

The file parallelogram.csv includes both the original vectors (v₁, v₂) (columns x and y) and their scaled versions (v₁_small, v₂_small), listed under the columns x_hat and y_hat.

The rows of grid.csv specify the corner of a small parallelogram (columns x and y) and the color assigned to that parallelogram (column color).

Almost-14 coloring

Similarly, the files in constructions/almost-14-coloring-3D contain a discretized almost-14-coloring of three-dimensional space.

Contact

If you have further questions or want to discuss our work, please send an e-mail to mundinger@zib.de or spiegel@zib.de.

Citing the paper

If you find this work helpful, please consider citing our paper:

@inproceedings{mundinger2025neural,
  title = {Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?},
  shorttitle = {Neural Discovery},
  booktitle = {Forty-Second International Conference on Machine Learning},
  author = {Mundinger, Konrad and Zimmer, Max and Kiem, Aldo and Spiegel, Christoph and Pokutta, Sebastian},
  year = {2025},
  month = jul
}