The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning
Authors: Max Zimmer, Nico Pelleriti, Christophe Roux, Sebastian Pokutta
Organizations: Department for AI in Society, Science, and Technology, Zuse Institute Berlin, Germany · Institute of Mathematics, Technische Universität Berlin, Germany
AI tools and agents are reshaping how researchers work, from proving theorems to training neural networks. Yet for many, it remains unclear how these tools fit into everyday research practice. This paper is a practical guide to AI-assisted research in mathematics and machine learning: We discuss how researchers can use modern AI systems productively, where these systems help most, and what kinds of guardrails are needed to use them responsibly. It is organized into three parts: (I) a five-level taxonomy of AI integration, (II) an open-source framework that, through a set of methodological rules formulated as agent prompts, turns CLI coding agents (e.g., Claude Code, Codex CLI, OpenCode) into autonomous research assistants, and (III) case studies from deep learning and mathematics. The framework runs inside a sandboxed container, works with any frontier LLM through existing CLI agents, is simple enough to install and use within minutes, and scales from personal-laptop prototyping to multi-node, multi-GPU experimentation across compute clusters. In practice, our longest autonomous session ran for over 20 hours, dispatching independent experiments across multiple nodes without human intervention. We stress that our framework is not intended to replace the researcher in the loop, but to augment them. Our code is publicly available at https://github.com/ZIB-IOL/The-Agentic-Researcher.
Figures & tables
Figure 1
Level
Name
Tools
AI Tasks
Human Role
0
Classical
L a T e X , math. software
No AI integration
Everything
1
Consultant
LLM chatbots
Targeted queries for explanation, literature, brainstorming
Asks, evaluates
2
Typist
Editor plugins (Copilot, Cursor)
Code and text generation without execution
Thinks, reviews, decides
3
Collaborator
CLI coding agents
Human describes task, AI implements and iterates
Reviews each output, assigns next task
4
Research Assoc.
Our framework
Autonomous experiment loop following structured research plan
Steers, audits
Table 1: Five levels of AI integration in mathematical research. Each (not necessarily mutually exclusive) level represents a qualitatively different trade-off between agent autonomy and human involvement.
Figure 2: Setting up a research project. Top: the three categories of input the researcher provides, with their conceptual role (dark) and concrete realization (light). Bottom: two examples from our case studies: a deep learning project ( Section 4.1 ) and a mathematics project ( Section 4.4 ).
Figure 3: Overview of the agentic research framework. Top: The researcher writes a persistent instruction file that governs the CLI agent operating within a sandboxed environment. Bottom: Each experiment follows an eight-step loop.
Figure 4: Final validation perplexity [ sic ] from the agent’s report in Section 4.1 . Lower is better. The dashed line marks the Muon baseline; the agent’s modifications achieve ∼5% improvement over Muon and ∼8% over AdamW.
Figure 5: Training curves [ sic ] from the agent’s report in Section 4.1 . Left: full training run. Right: final 3,000 iterations (zoomed). The agent’s optimizer modifications consistently outperform both Muon and AdamW baselines throughout training, not only in the final iterations. Note that here, the agent named the new method NewMuon, which is inconsistent with the naming in Figure 4 .
Figure 6: Plots [ sic ] from the agent’s report for Section 4.2 , produced by the agent. Left: relative perplexity improvement vs. model size. Right: absolute perplexity comparison showing that the weight reconstruction method consistently outperforms the baseline across all tested model sizes.
Figure 7: A screenshot [ sic ] from the agent’s report in Section 4.3 . Before running any benchmarks, the agent audited all error propagation and refinement formulas through numeric tests on small matrices ( Commandment X ).
Figure 8: A plot [ sic ] from the agent’s report: Log-log convergence of ∥xt−e1∥2 for p∈{3,4,6} starting from x0=e2 (blue) and from x0=x0slow(10−2) (orange) where x0slow is the worst-case initialization from the proof and α is the fitted coefficient of t−α .
Figure 9: A screenshot [ sic ] from the agent’s report: The callback architecture in Section 4.5 . The conflict graph is integrated into Boscia via two standard callbacks, propagate_bounds (before each Frank-Wolfe solve) and bnb_callback (after each node), without modifying Boscia’s source code.
Figure 10: A plot [ sic ] from the agent’s report: a three-parameter slice of the 21-dimensional K7 susceptance space, obtained by varying b1 , b8 , and b9 while fixing the remaining 18 parameters at the values of the best-found instance. Each point is colored by the number of nontrivial feasible operating states. Although the color map appears nearly continuous, it represents discrete solution counts and reveals a localized high-count region around the 192-solution configuration. This suggests that the best-found parameter vector lies in a small but open region of parameter space rather than at an isolated point.
We introduce the AI co-mathematician, a workbench for mathematicians to interactively leverage AI agents to pursue open-ended research. The AI co-mathematician is optimized to provide holistic support for the exploratory and iterative reality of mathematical workflows, including ideation, literature search, computational exploration, theorem proving and theory building. By providing an asynchronous, stateful workspace that manages uncertainty, refines user intent, tracks failed hypotheses, and outputs native mathematical artifacts, the system mirrors human collaborative workflows. In early tests, the AI co-mathematician helped researchers solve open problems, identify new research directions, and uncover overlooked literature references. Besides demonstrating a highly interactive paradigm for AI-assisted mathematical discovery, the AI co-mathematician also achieves state of the art results on hard problem-solving benchmarks, including scoring 48% on FrontierMath Tier 4, a new high score among all AI systems evaluated.
AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant KG, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of KG is not known, we recently tightened the best known bounds to
116π≤KG≤2log(1+2)π−10−4.
Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.
We report new results on eight problems in mathematics and theoretical computer science, produced with the assistance of Bolzano, an open-source multi-agent LLM system. Bolzano orchestrates rounds of interaction between parallel prover agents and a verifier agent while maintaining a persistent knowledge base that is carried across rounds. Classified using the significance-autonomy taxonomy of Feng et al., six of the eight results reach the level of publishable research, and five of the eight were produced essentially autonomously by Bolzano. Our results provide evidence that LLMs can contribute meaningfully to mathematical research, complementing recent reports by Bubeck et al., Woodruff et al., and others.
Martin Balko, Jan Grebík, Pavel Hubáček +5
Department of Applied Mathematics, Charles University · Computer Science Institute, Charles University · Institute of Mathematics, Czech Academy of Sciences +1