How AI is changing the nature of mathematical research

How AI is changing the nature of mathematical research - Amazon Science

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Machine learning

How AI is changing the nature of mathematical research

What machine learning theorists learned using AI agents to generate proofs — and what comes next.

By Michael Kearns , Aaron Roth

March 9, 2026

10 min read

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Modern AI coding tools have revolutionized software engineering, with developers now using AI assistants to write a substantial fraction of their code across a range of applications. As scientists studying the theory of machine learning, we’re already seeing a similar transformation in basic scientific methodology, especially for research of a mathematical nature. More precisely, AI tools are now able to develop and write rigorous mathematical proofs only from prompts providing high-level proof sketches. These proofs are written in longstanding “ languages ” for detailing mathematical arguments, in the same way that code is written in formal programming languages like Python. AI seems to have become proficient in both kinds of languages and their underlying logics.

Working with proof-based AI tools is akin to collaborating with a smart, broadly educated but occasionally error-prone colleague.

We came to this realization during a three-week period last summer, when we used agentic AI tools to write a mathematical paper that normally would have taken months. The 50-page paper describes and solves an optimization problem based on concepts from graph theory and machine learning. A typical prompt we would give the AI to set up the general framework for our paper looked like this: “Imagine a directed acyclic network of linear least-squares learning agents, each of which shares a common dataset but each of which sees only a different subset of the features.” A typical prompt for a theorem statement and proof went “We believe that if the network contains a sufficiently long chain of agents whose features cover the entire dataset, some agent in the chain should rapidly converge to the globally optimal linear model. The proof should use the fact that error monotonically decreases in the chain, which forces long sequences of agents to be multi-accurate with respect to each other’s features.” While incantations like these might be opaque to the casual reader, they all have precise, standard mathematical interpretations that the AI was aware of, due to its training, and it proceeded to translate informal intuitions into precise definitions and statements. This translation was imperfect (as discussed below) but resulted in a great first draft that could then be corrected and smoothed. To be clear, for this specific paper, we already knew the general outline of the proofs we had in mind. What AI did was to automate and dramatically speed up the process of filling in the missing details and writing them with formal precision. But more recently, we’ve written papers that we believe are substantially different and better than what we would have produced without...