Advancing mathematics by guiding human intuition with AI

By A. Davies, Petar Velickovic, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanburn, Peter W. Battaglia, Charles Blundell, András Juhász and 4 more. Nature, 2021

574 citations recorded by the source corpus.

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning. A framework through which machine learning can guide mathematicians in discovering new conjectures and theorems is presented and shown to yield mathematical insight on important open problems in different areas of pure mathematics.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c0cd4b4844c31a27a7900a754d0a91b160b00e55

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Citations in the corpus

KAN: Kolmogorov-Arnold Networks (2024, unknown): “A Google Deepmind team [29] manually chose signature to be the target variable, otherwise they would face this combinatorial problem described above.”
Mathematical Capabilities of ChatGPT (2023, background): “[17] Alex Davies, Petar Veličković, Lars Buesing, Sam Blackwell, and Daniel Zheng et al.”
Empowering Biomedical Discovery with AI Agents (2024, unknown): “…methods to rank order hypotheses considering their potential biomedical value with experimental cost, characterize skepticism via uncertainty quantiﬁcation and analysis of experiments in reference to the original hypothesis, and reﬁne hypotheses using data and counterexamples from experiments [75].”
On scientific understanding with artificial intelligence (2022, background): “hinted on relations between previously unconnected invariants in knot theory, which allowed mathematicians to conjecture and prove new theorems [59].”
Everything is Connected: Graph Neural Networks (2023, background): “Further, GNNbased systems have helped mathematicians uncover the hidden structure of mathematical objects (Davies et al., 2021), leading to new top-tier conjectures in the area of representation theory (Blundell et al.”
Machine learning and deep learning—A review for ecologists (2022, unknown): “For example, Davies et al. ( 2021 ) demonstrated that DL can aid researcher by generating new hypotheses which were tested afterwards, or it was shown that modern DL models can achieve human-like performance in text generation (Brown et al., 2020 ).”
MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning (2023, methodology): “For the out-of-domain datasets, we choose SVAMP (Patel et al., 2021), Mathematics (Davies et al., 2021), SimulEq (Koncel-Kedziorski et al., 2016), SAT-Math (Zhong et al., 2023), and MMLU-Math (Hendrycks et al., 2021a).”
Combinatorial optimization and reasoning with graph neural networks (2021, background): “Further, it offers a framework for building neural networks in the presence of an algorithmic prior, which demonstrated significant returns for applications in pure mathematics (Davies et al., 2021; Blundell et al., 2022).”
Artificial intelligence and illusions of understanding in scientific research (2024, unknown): “There is excitement about the potential of AI to open up new knowledge “frontiers” 35 ; for example, in mathematics 55 and in “uncovering new cognitive and behavioural phenomena” 56 .”
AlphaEvolve: A coding agent for scientific and algorithmic discovery (2025, unknown): “AI systems have also contributed to advances in pure mathematics [23].”