Iteris: Agentic Research Loops for Computational Mathematics
2026-06-01 • Artificial Intelligence
Artificial IntelligenceMachine Learning
AI summaryⓘ
The authors present Iteris, an AI system designed to help solve open problems in computational mathematics by doing experiments, building examples, and drafting proofs. They tested Iteris on two challenging problems and showed it could produce useful results that experts then checked and fixed. One result compared two optimization methods, and the other found a surprising failure case in a matrix algorithm. Their work shows AI can help with math research but still needs human oversight.
large language modelsagentic AIcomputational mathematicsnumerical experimentationproof draftsconjugate gradientrandomized coordinate descentQR factorizationcolumn pivotinglow coherence
Authors
Leheng Chen, Zihao Liu, Wanyi He, Bin Dong
Abstract
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.