Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? (2501.18527v3)

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. Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

• The paper formulates the Hadwiger-Nelson coloring problem as a continuous optimization task using neural networks as function approximators.
• It discovers two novel six-colorings of the Euclidean plane, significantly advancing results in discrete geometry.
• The study demonstrates that probabilistic loss functions and gradient-based methods can effectively bridge discrete and continuous mathematical challenges.

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

The paper "Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?" investigates a novel approach for addressing mathematical problems using neural networks (NNs), focusing specifically on the Hadwiger-Nelson (HN) problem. This problem, a long-standing question in discrete geometry and combinatorics, involves determining the minimum number of colors needed to color the Euclidean plane such that no two points at unit distance share the same color. The paper employs NNs as function approximators to reformulate this mixed discrete-continuous problem as an optimization problem with a probabilistic, differentiable loss function.

Methodological Framework

The authors introduce a framework where the geometric coloring problem is transformed into a continuous optimization task. By using probabilistic coloring and differentiable loss functions, they enable the application of gradient-based methods to explore the solution space. Neural networks are utilized as universal function approximators within this probabilistic context, providing the flexibility necessary to investigate solutions without relying heavily on symmetry or periodicity assumptions.

The continuous loss function measures how much the probabilistic coloring violates the constraints of the HN problem. By minimizing this loss function, the neural networks generate new color configurations. Through this process, two novel six-colorings of the plane were discovered, providing significant improvements to variants of the HN problem.

Contributions and Results

The paper makes several key contributions:

1. Optimization Framework: The authors propose a framework for solving geometric coloring problems using continuous optimization approaches and neural networks, bridging discrete and continuous domains effectively.
2. Novel Six-Colorings: They present two newly discovered six-colorings of the plane, extending the range of realizable distances and improving the known results for a variant of the HN problem for the first time in three decades.
3. Numerical Evidence on Other Variants: The research provides numerical results that illuminate potential solutions to other forms of coloring problems, such as avoiding monochromatic triangles in specific configurations.

Implications

These findings are indicative of how machine learning, particularly neural networks, can assist in mathematical discovery by providing intuition and possible solutions to classical problems. While the primary focus is on the HN problem, the methodology is broadly applicable to various other mathematical challenges involving mixed discrete and continuous components.

The research opens up possibilities for further exploration into using AI for other unsolved problems in discrete mathematics. The paper's success in discovering new configurations implies that similar methods could potentially resolve or provide insights into other longstanding mathematical conjectures.

Future Directions

The success of applying neural networks in this manner suggests multiple avenues for future research:

• Scalability and Generalization: Extending the approach to more complex and higher-dimensional problems, such as coloring in three-dimensional spaces or other metric spaces, offers a fertile ground for exploration.
• Broader Combinatorial Applications: Beyond geometric coloring, adapting these methods to other combinatorial structures could yield new results in extremal combinatorics and related fields.
• Enhanced Neural Architectures: Investigating other neural network architectures, such as graph neural networks or physics-informed neural networks, could refine the efficacy and applicability of this framework to a broader class of problems.

The paper contributes to a growing body of work illustrating how artificial intelligence can function not merely as a tool for automation but as an active participant in mathematical inquiry. Through continuous optimization frameworks and neural function approximation, new solutions to classical mathematical dilemmas become accessible, demonstrating the ever-expanding potential of AI in theoretical research.