AI系统Iteris助力计算数学难题，生成可验证的数值证据与反例。
Iteris: Agentic Research Loops for Computational Mathematics
- 构建自研智能体系统Iteris，自动执行数学研究中的实验与推导
- 在两个开放问题上生成了经专家验证的相图与反例结果
- 适合数学研究者与AI辅助探索方向的开发者参考
大型语言模型与智能体系统的发展推动了数学发现的进步，从竞赛题到高阶猜想均有突破。然而，计算数学领域的开放问题仍较少受到关注：这类研究不仅需要证明，还需数值实验、反例构造与算法设计。本文提出一个面向计算数学开放问题的智能体研究系统Iteris。将其应用于近期西蒙斯研讨会论文集（arXiv:2602.05394）中的两个问题。在案例研究中，Iteris生成了数值证据、构造方案与证明草稿，经专家评审修正后得到可验证结果。第一项成果为幂律谱上共轭梯度与随机坐标下降渐近比较的相图；第二项为反例，证明列主元QR分解在低相干性下仍可能选出病态子矩阵。这些案例表明，智能体系统可在计算数学研究中发挥实质性作用，但人类验证仍是关键环节。
原文摘要 · Abstract (English)
Recent advances in large language models and agentic AI systems have enabled significant progress in mathematical discovery, from solving competition problems to tackling research-level conjectures. However, open problems in computational mathematics have received comparatively less attention: research in this area often requires not only proofs but also numerical experimentation, adversarial constructions, and algorithm design. In this paper, we introduce an agentic research system, Iteris, designed for open problems in computational mathematics. We apply Iteris to two open problems from a recent Simons Workshop collection (arXiv:2602.05394). In these case studies, Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results. The first result is a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra; the second is a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. These case studies suggest that agentic AI systems can participate meaningfully in research workflows for open problems in computational mathematics, while human validation remains essential.
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