cs.LGMar 16, 2026

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

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Abstract

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
LevelNameToolsAI TasksHuman Role
0ClassicalL a T e X , math. softwareNo AI integrationEverything
1ConsultantLLM chatbotsTargeted queries for explanation, literature, brainstormingAsks, evaluates
2TypistEditor plugins (Copilot, Cursor)Code and text generation without executionThinks, reviews, decides
3CollaboratorCLI coding agentsHuman describes task, AI implements and iteratesReviews each output, assigns next task
4Research Assoc.Our frameworkAutonomous experiment loop following structured research planSteers, 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%{\sim}5\% improvement over Muon and ∼8%{\sim}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\|x_{t}-e_{1}\|^{2} for p∈{3,4,6}p\in\{3,4,6\} starting from x0=e2x_{0}=e_{2} (blue) and from x0=x0slow(10−2)x_{0}=x_{0}^{\mathrm{slow}}(10^{-2}) (orange) where x0slowx_{0}^{\mathrm{slow}} is the worst-case initialization from the proof and α\alpha is the fitted coefficient of t−αt^{-\alpha} .
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 K7K_{7} susceptance space, obtained by varying b1b_{1} , b8b_{8} , and b9b_{9} 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.

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