Iteris: Agentic Research Loops for Computational Ma... | AI Research

Iteris: Agentic Research Loops for Computational Mathematics
Computational mathematics research is a complex process that requires more than just solving equations; it involves running numerical experiments, building adversarial examples, and drafting formal proofs. While AI has made great strides in competition-level problem solving, it has been less effective at navigating these multi-step, open-ended research workflows. This paper introduces Iteris, an agentic AI system specifically designed to handle the diverse, iterative tasks required to make progress on open problems in computational mathematics.

A Structured Research Loop

Iteris operates through an "explore–plan–execute" cycle that mimics how a human researcher might approach a project. The system is organized into three distinct phases:

• Explore: An exploration agent probes potential research directions, reviews project history, and identifies promising paths while flagging risks.

• Plan: A planning agent reviews the project’s global state and the exploration agent’s advice to create a structured list of tasks for the next iteration.

• Execute: Specialized agents carry out specific tasks, such as running numerical experiments, constructing proofs, or auditing mathematical definitions.
  By using files as both long-term memory and a communication medium between agents, Iteris maintains a clear, checkable record of its progress. This structure prevents the system from getting stuck in "local inertia," where an AI might repeatedly refine a failing approach rather than pivoting to a more productive strategy.

Solving Open Problems

The researchers tested Iteris on two open problems from a recent Simons Workshop collection. The system successfully generated the foundational evidence and proof drafts necessary to reach verified results:

1. Conjugate Gradient vs. Randomized Coordinate Descent: The system established a phase diagram that compares the efficiency of these two methods when applied to power-law spectra. It identified the specific conditions under which one method outperforms the other. 2. QR Factorization with Column Pivoting: The system addressed a question regarding whether this algorithm reliably selects well-conditioned submatrices. By constructing a specific counterexample, the researchers proved that the algorithm can fail to do so even under low-coherence conditions.

The Role of Human Collaboration

While Iteris was instrumental in generating the exploratory artifacts and proof sketches that led to these discoveries, the authors emphasize that human involvement remains essential. The AI-generated outputs required expert review, correction, and reorganization to become rigorous, final mathematical results. For instance, in the first case study, human inspection identified an unjustified assumption in the AI’s initial analysis, which was subsequently repaired through further interaction.
These results demonstrate that agentic AI can serve as a powerful partner in the research process, provided there is a framework for human validation and oversight to ensure the accuracy and clarity of the final output.