Mathematical exploration and discovery at scale
This provides mathematicians with a tool for exploring vast search spaces to solve complex optimization problems at scale, potentially matching or improving best-known results, though it appears incremental as it builds on existing evolutionary and LLM methods.
The authors tackled the problem of autonomously discovering novel mathematical constructions and advancing understanding of open problems by developing AlphaEvolve, an evolutionary coding agent that combines LLMs with automated evaluation, which rediscovered best-known solutions in most cases and discovered improved solutions in several out of 67 problems, with some instances generalizing results into formulas valid for all inputs.
AlphaEvolve is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time.