Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?

Konrad Mundinger, Max Zimmer, Aldo Kiem, Christoph Spiegel, Sebastian Pokutta
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:45236-45255, 2025.

Abstract

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-mundinger25a, title = {Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?}, author = {Mundinger, Konrad and Zimmer, Max and Kiem, Aldo and Spiegel, Christoph and Pokutta, Sebastian}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {45236--45255}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/mundinger25a/mundinger25a.pdf}, url = {https://proceedings.mlr.press/v267/mundinger25a.html}, abstract = {We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.} }
Endnote
%0 Conference Paper %T Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? %A Konrad Mundinger %A Max Zimmer %A Aldo Kiem %A Christoph Spiegel %A Sebastian Pokutta %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-mundinger25a %I PMLR %P 45236--45255 %U https://proceedings.mlr.press/v267/mundinger25a.html %V 267 %X We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.
APA
Mundinger, K., Zimmer, M., Kiem, A., Spiegel, C. & Pokutta, S.. (2025). Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:45236-45255 Available from https://proceedings.mlr.press/v267/mundinger25a.html.

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