Core Insights - The article discusses the introduction of AlphaEvolve, a powerful new tool for mathematical discovery, co-authored by Bogdan Georgiev and Terence Tao [1][5]. Group 1: AlphaEvolve's Capabilities - AlphaEvolve was tested on 67 mathematical problems across various fields, including combinatorial mathematics, geometry, mathematical analysis, and number theory [3]. - The system outperformed traditional tools in scalability, robustness, and interpretability, and it can autonomously discover novel mathematical constructs, surpassing existing human optimal results in some cases [5][6]. Group 2: Human-AI Collaboration - In the Nikodym set problem, AlphaEvolve generated initial constructs that, while not optimal, provided valuable insights for human researchers, leading to improved upper bounds in a subsequent independent paper [6][7]. - Similarly, in the arithmetic Kakeya conjecture, AlphaEvolve played a crucial role in advancing understanding [8]. Group 3: Interpretability and Insight Generation - AlphaEvolve's ability to generate clear and interpretable program code allows human experts to analyze and extract general mathematical formulas from its outputs [10]. - For the stacking blocks problem, the system initially created a correct recursive program, which it later simplified into a more efficient explicit program, revealing the mathematical relationship with harmonic numbers [14]. Group 4: Problem-Solving Techniques - The system demonstrated its ability to navigate complex problem spaces by adapting its scoring functions to avoid local traps, ultimately converging on known theoretical optimal solutions [19]. - AlphaEvolve exhibited excellent generalization capabilities, successfully identifying universal constructs for all perfect square inputs [20][21]. Group 5: Efficiency and Expert Guidance - AlphaEvolve operates efficiently with minimal high-quality prompts, and expert guidance significantly enhances the quality of its outputs [23]. - The system supports parallelization, allowing researchers to explore multiple problem instances simultaneously, which is particularly effective for multi-parameter geometric problems [23]. Group 6: Operational Modes - AlphaEvolve functions in two primary modes: "search mode" for efficiently discovering optimal mathematical constructs and "generalizer mode" for creating universal programs applicable to various parameters [24][26]. - In search mode, the system evolves heuristic algorithms to optimize the search process, while in generalizer mode, it aims to identify patterns and develop general formulas based on observed optimal solutions [25][26]. Conclusion - Overall, AlphaEvolve exemplifies how AI-driven evolutionary search can complement human intuition, providing a robust new paradigm for mathematical research [28].

克雷西 发自 凹非寺 量子位 | 公众号 QbitAI 陶哲轩又来安利AlphaEvolve了。 在与DeepMind高级工程师Bogdan Georgiev等人合著的新论文中，陶哲轩称其为 数学发现的有力新工具 。 具体来说，他们用AlphaEvolve研究了67个数学问题，涵盖组合数学、几何、数学分析与数论等多个领域。 更关键的是，AlphaEvolve已经可以 自主发现新颖的数学构造 ，并在部分问题上超越人类已有的最优结果。 AI自主发现新数学构造 AlphaEvolve在67个问题的测试中，不仅复现了众多已知最优解，更在多个方面展现了其独特的发现能力。 一个关键的成就是AlphaEvolve 能够自主发现人类未曾一窥的新数学构造 。 例如在处理Nikodym集问题时，系统生成的初步构造虽然尚未达到最优，但它为人类研究者提供了"一个极好的人类直觉跳板" 。 基于AI提供的结构，研究人员通过人工简化和直觉推演，最终找到了一个更优的构造，改进了已知的上界，这一人机协作的成果将作为一篇独 立的数学论文发表。 结果发现，AlphaEvolve在可扩展性、鲁棒性、可解释性方面均优于传统工具。 同样地，在算术Kak ...