DaviespersName, VeličkovićpersName, BuesingpersName, BlackwellpersName, ZhengpersName, TomaševpersName, TanburnpersName, BattagliapersName (2021date). “Advancing mathematics by guiding human intuition with AI.” Naturetitle·j.
Object of study — The conjecture-making stage of mathematical practice, and whether a supervised model plus attribution can guide it.
Research question — Computers have generated mathematical data for decades - can a learned model do the harder thing and help a mathematician see which relationships are worth conjecturing?
Contribution — A framework - hypothesise a relation, learn it, attribute it, conjectureterm·dom a closed form - demonstrated by two genuinely new results in different areas of pure mathematics.
phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | machine-learning-guided mathematical conjecture discovery — “One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjectures: statements that are suspected to be true but have not been proven to hold in all cases.” | |
| claim-type | method — “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.” | |
| method — evidence | — “We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory.” | |
| object of study — evidence | — “One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjectures: statements that are suspected to be true but have not been proven to hold in all cases.” | |
| research question — evidence | — “Prior systems for generating conjectures have either contributed genuinely useful research conjectures 9 via methods that do not easily generalize to other mathematical areas 10 , or have demonstrated novel, general methods for finding conjectures 11 that have not yet yielded mathematically valuable results.” | |
| contribution — evidence | — “Our contribution shows how mature machine learning methodologies can be adapted and integrated into existing mathematical workflows to achieve novel results.” | |
| transfer | from machine learning / artificial intelligence: pattern-recognition and model-interpretation methods are imported wholesale into the working practice of pure mathematics, as an addition to the mathematician's toolkit — “We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory.” | transfers[0] |
| transfer | from classical statistics: the gradient-based attribution method used to interpret the trained network is identified as the machine-learning counterpart of sensitivity analysis — “In this work we use gradient-based techniques 40 , broadly similar to sensitivity analysis in classical statistics and sometimes referred to as saliency maps.” | transfers[1] |
| transfer | from machine learning: supervised regression and gradient-based attribution, imported into pure mathematics not to prove anything but to indicate where a relationship might exist — “Our contribution shows how mature machine learning methodologies can be adapted and integrated into existing mathematical workflows to achieve novel results.” | transfers[2] |
| transfer | from machine learning (pattern recognition and attribution): the mathematician's toolkit is augmented with pattern-recognition and attribution methods taken from machine learning, used to propose conjectures rather than to prove them — “We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory.” | transfers[3] |
| mechanism | Supervised learning first certifies that a pattern exists at all (better-than-chance prediction), then attribution techniques localise which input features carry it, so the mathematician's search space is narrowed before any conjecture is formed — “The framework helps guide the intuition of mathematicians in two ways: by verifying the hypothesized existence of structure/patterns in mathematical objects through the use of supervised machine learning; and by helping in the understanding of these patterns through the use of attribution techniques.” | mechanisms[0] |
| mechanism | Gradient saliency assigns importance by measuring how much the learned function's prediction shifts under small perturbations of each input component — “These techniques attribute importance to the elements of X ( z ), by calculating how much f ˆ changes in predictions of Y (z) given small changes in X ( z ).” | mechanisms[1] |
| mechanism | accuracy above chance is used as evidence that a relationship EXISTS, before anyone knows what it is - the model detects presence, not content — “If f ˆ is more accurate than would be expected by chance, it indicates that there may be such a relationship to explore.” | mechanisms[2] |
| mechanism | attribution turns an opaque learned function into something a person can reason about, by showing which inputs it is sensitive to — “Attribution techniques can be used to understand which aspects of f ˆ are relevant for predictions of Y ( z ).” | mechanisms[3] |
| mechanism | the human keeps the selection step: conjectures are chosen for being interesting and suggestive of a proof, not merely for fitting — “In this process, the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.” | mechanisms[4] |
| mechanism | the loop is iterative rather than a pipeline - the framework expects several passes before a conjecture is worth stating — “This iterative process might need to be repeated several times before a viable conjecture is settled on.” | mechanisms[5] |
| extends | prior work using supervised learning to find patterns in mathematics (the paper's refs 20-24) — “This extends work using supervised learning to find patterns 20 – 24 by focusing on enabling mathematicians to understand the learned functions and derive useful mathematical insight.” | claim_relations[0] |
| extends | the combinatorial invariance conjecture for symmetric groups — “This conjectured solution, if proven true, would settle the combinatorial invariance conjecture for symmetric groups.” | claim_relations[1] |
| builds-on | prior work on subgraphs of Bruhat intervals (the paper's ref 31) — “In particular, we found that a subgraph inspired by prior work 31 may be sufficient to calculate the KL polynomial, and this was supported by a much more accurate estimated function.” | claim_relations[2] |
| extends | prior supervised-learning work on finding patterns in mathematical data — “This extends work using supervised learning to find patterns 20 – 24 by focusing on enabling mathematicians to understand the learned functions and derive useful mathematical insight.” | claim_relations[3] |
| builds-on | the long tradition of computer-generated data in conjecture formation, from Gauss's prime tables to Birch and Swinnerton-Dyer — “Mathematicians have always used data to help in this process—from the early hand-calculated prime tables used by Gauss and others that led to the prime number theorem 5 , to modern computer-generated data 1 , 5 in cases such as the Birch and Swinnerton-Dyer conjecture 2 .” | claim_relations[4] |
| contradicts | prior conjecture-generation systems, as either non-generalisable or not yet mathematically useful — “Prior systems for generating conjectures have either contributed genuinely useful research conjectures 9 via methods that do not easily generalize to other mathematical areas 10 , or have demonstrated novel, general methods for finding conjectures 11 that have not yet yielded mathematically valuable results.” | claim_relations[5] |
| sense | intuition - as the faculty being ASSISTED: a hypothesis the mathematician brings, which the model then confirms or fails to confirm — “In the supervised learning stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). By” | senses[0] |
| sense | intuition - as the faculty being TESTED: the framework as a bench on which a hunch can be checked cheaply before it is pursued — “Conceptually, this framework provides a ‘test bed for intuition’—quickly verifying whether an intuition about the relationship between two quantities may be worth pursuing and, if so, guidance as to how they may be related.” | senses[1] |
| sense | discovery - as detecting that a relation exists, which the model does — “If f ˆ is more accurate than would be expected by chance, it indicates that there may be such a relationship to explore.” | senses[2] |
| sense | discovery - as deciding which relation is worth stating, which stays with the mathematician — “In this process, the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.” | senses[3] |
| control | a new model trained from scratch to predict the signature from ONLY the three measurements the attribution had identified, and shown to reach the same performance as the original — held — rules out an attribution method that names features which are not in fact carrying the relationship - retraining on those three alone is the demonstration the attribution cannot give — “This was confirmed by training a new model to predict the signature from only these three measurements, which was able to achieve the same level of performance as the original model.” | controls[0] |
| edge | gradient saliency —is-a→ attribution techniques — “The attribution technique we use in our work, gradient saliency, does this by calculating the derivative of outputs of f ˆ , with respect to the inputs.” | concept_edges[0] |
| audit | supervised learning —part-of→ machine learning — Audit 2026-09-28: RESTORED - endpoint by anaphora: 'this regression process' = the supervised learning of f described in the previous sentence; part-of is loose. The earlier withdrawal is kept in the record. — “The key contributions of machine learning in this regression process are the broad set of possible nonlinear functions that can be learned given a sufficient amount of data.” | concept_edges[1] |
| audit | machine learning —causes→ intuition — Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the authors PROPOSE a process in which machine learning observations guide intuition; causes is loose. — “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.” | concept_edges[2] |
| evidence status note | concept — This quote ends with the stray word 'By', the first word of the next sentence. It is NOT trimmed, because the trimmed sentence occurs twice in the staged body and an anchor that matches twice is not an anchor. The claim is carried by the sentence before 'By'. — “In the supervised learning stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). By” | concepts[2] |
| outcome (stated by the tagger, no quotation) | control — held | controls[0] |
| for which use | limitation — the applicability of the whole framework: IT REQUIRES THE ABILITY TO GENERATE LARGE DATASETS OF THE REPRESENTATIONS OF OBJECTS AND FOR THE PATTERNS TO BE DETECTABLE — “There are limitations to where this framework will be useful—it requires the ability to generate large datasets of the representations of objects and for the patterns to be detectable in examples that are calculable.” | limitations[0] |
| for which use | limitation — domains where the target function resists learning: IN SOME DOMAINS THE FUNCTIONS OF INTEREST MAY BE DIFFICULT TO LEARN IN THIS PARADIGM — “Further, in some domains the functions of interest may be difficult to learn in this paradigm.” | limitations[1] |
| for which use | limitation — every result the pipeline produces: THE RESULTS OF THE SUBSEQUENT STAGES WILL HOLD TRUE ONLY FOR THE DISTRIBUTION P_Z, AND NOT THE WHOLE SPACE Z — “The results of the subsequent stages will hold true only for the distribution P Z , and not the whole space Z .” | limitations[2] |
| for which use | limitation — any negative finding from the method: IF THE MODEL CANNOT PREDICT THE RELATIONSHIP BETTER THAN CHANCE, IT MAY MEAN THAT A PATTERN EXISTS, BUT IS SUFFICIENTLY COMPLICATED THAT IT CANNOT [be found] - a null is uninformative by construction — “The converse is not true; namely, if the model cannot predict the relationship better than chance, it may mean that a pattern exists, but is sufficiently complicated that it cannot be captured by the given model and training procedure.” | limitations[3] |
| for which use | limitation — the generality of the framework: it is argued from two demonstrations in two areas rather than established — “We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory.” | limitations[4] |
| for which use | limitation — the older data-driven route specifically: FOR X(z) AND Y(z) IN HIGHER-DIMENSIONAL SPACES, OR OF MORE COMPLEX TYPES, SUCH AS GRAPHS ... THIS APPROACH IS EITHER LESS USEFUL — “However, for X ( z ) and Y ( z ) in higher-dimensional spaces, or of more complex types, such as graphs, and for more complicated, nonlinear f ˆ , this approach is either less useful or entirely infeasible.” | limitations[5] |
| for which use | limitation — any claim that the framework automates discovery: THE MATHEMATICIAN CAN GUIDE THE CHOICE OF CONJECTURES TO THOSE THAT NOT JUST FIT THE DATA BUT ALSO SEEM INTERESTING, PLAUSIBLY TRUE - the judgement stays human — “In this process, the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.” | limitations[6] |
| evidence status note | sense — This quote ends with the stray word 'By', the first word of the next sentence. It is NOT trimmed, because the trimmed sentence occurs twice in the staged body and an anchor that matches twice is not an anchor. The claim is carried by the sentence before 'By'. — “In the supervised learning stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). By” | senses[0] |
| replication | untested | |
| limitation | The framework needs large generatable datasets and patterns visible in computable examples — “There are limitations to where this framework will be useful—it requires the ability to generate large datasets of the representations of objects and for the patterns to be detectable in examples that are calculable.” | limitations[0] |
| limitation | In some domains the target functions may simply not be learnable this way — “Further, in some domains the functions of interest may be difficult to learn in this paradigm.” | limitations[1] |
| limitation | Conclusions hold only for the sampling distribution used, not the whole space — “The results of the subsequent stages will hold true only for the distribution P Z , and not the whole space Z .” | limitations[2] |
| limitation | A negative result is uninformative: failure to predict does not mean no pattern exists — “The converse is not true; namely, if the model cannot predict the relationship better than chance, it may mean that a pattern exists, but is sufficiently complicated that it cannot be captured by the given model and training procedure.” | limitations[3] |
| limitation | Two demonstrations in two areas; the generality of the framework is argued from them rather than established. — “We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory.” | limitations[4] |
| limitation | The paper is explicit that the older data-driven route fails as objects get higher-dimensional or more structured - which is the regime the framework is for, and also the regime where its own outputs are hardest to check. — “However, for X ( z ) and Y ( z ) in higher-dimensional spaces, or of more complex types, such as graphs, and for more complicated, nonlinear f ˆ , this approach is either less useful or entirely infeasible.” | limitations[5] |
| limitation | Every step that decides whether a pattern is worth pursuing is human; the framework does not claim to automate judgement. — “In this process, the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.” | limitations[6] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | The proposed machine-learning framework produced genuinely new mathematics in two unrelated fields: a knot-theoretic connection between algebraic and geometric invariants, and a candidate resolution of a long-open conjecture in representation theory. — “In this work we have demonstrated a framework for mathematicians to use machine learning that has led to mathematical insight across two distinct disciplines: one of the first connections between the algebraic and geometric structure of knots and a proposed resolution to a long-standing open conjecture in representation theory.” | content_tags.findings[0] |
| finding | The model reached 78% accuracy on a held-out test set, with no error larger than 2. — “The model was able to achieve an accuracy of 78% on a held-out test set, with no errors larger than ±2.” | content_tags.findings[1] |
| finding | The natural slope linearly captures all the information about the signature that the original model drew from the geometry. — “This implies that the slope linearly captures all of the information about the signature that the original model had extracted from the geometry.” | content_tags.findings[2] |
| finding | The theorem is one of the first results connecting algebraic and geometric knot invariants, with various applications. — “The above theorem is one of the first results that connect the algebraic and geometric invariants of knots and has various interesting applications.” | content_tags.findings[3] |
| finding | The first conjecture, though supported by several large sampled datasets, had counterexamples built from braids of a specific form. — “While this conjecture was supported by an analysis of several large datasets sampled from different distributions, we were able to construct counterexamples using braids of a specific form.” | content_tags.findings[4] |
| finding | Salient-subgraph analysis in the representation-theory case led to the discovery that a Bruhat interval decomposes into a hypercube plus a graph isomorphic to an interval in the smaller symmetric group. — “Considering this observation led us to the discovery that there is a natural decomposition of an interval into two parts—a hypercube induced by one set of extremal edges and a graph isomorphic to an interval in S N −1 .” | content_tags.findings[5] |
| finding | This decomposition yields a proved theorem: every Bruhat interval admits a canonical hypercube decomposition from which the Kazhdan-Lusztig polynomial can be computed directly. — “Every Bruhat interval admits a canonical hypercube decomposition along its extremal reflections, from which the KL polynomial is directly computable.” | content_tags.findings[6] |
| finding | Computational testing suggested the stronger claim that any hypercube decomposition, not just the canonical one, recovers the correct KL polynomial. — “Remarkably, further tests suggested that all hypercube decompositions correctly determine the KL polynomial.” | content_tags.findings[7] |
| finding | The edge labels are neither given to the network nor recoverable from the unlabelled Bruhat interval. — “This is notable because the edge labels are not given to the network and are not recoverable from the unlabelled Bruhat interval.” | content_tags.findings[8] |
| finding | The paper presents new fundamental results in pure mathematics found with machine learning's help, as a method for aiding the discovery of conjectures and theorems. — “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.” | content_tags.findings[9] |
| finding | Earlier conjecture-generating systems split into two failure modes: useful but not generalisable, or general but not yet useful. — “Prior systems for generating conjectures have either contributed genuinely useful research conjectures 9 via methods that do not easily generalize to other mathematical areas 10 , or have demonstrated novel, general methods for finding conjectures 11 that have not yet yielded mathematically valuable results.” | content_tags.findings[10] |
| finding | The models needed can be trained within several hours on a single-GPU machine. — “In each of these cases, the necessary models can be trained within several hours on a machine with a single graphics processing unit.” | content_tags.findings[11] |
| finding | AI had already served mathematics by finding counterexamples, accelerating calculations, generating symbolic solutions and detecting structure. — “In mathematics, it has been shown that AI can be used as a valuable tool by finding counterexamples to existing conjectures 16 , accelerating calculations 17 , generating symbolic solutions 18 and detecting the existence of structure in mathematical objects 19 .” | content_tags.findings[12] |
| quantity | 78% — held-out accuracy of the network predicting knot signature from geometric invariants — “The model was able to achieve an accuracy of 78% on a held-out test set, with no errors larger than ±2.” | content_tags.quantities[0] |
| quantity | 25% — chance baseline accuracy against which the 78% signature-prediction result was judged — “This is substantially higher than chance (a baseline accuracy of 25%), which gave us strong confidence that a relationship may exist.” | content_tags.quantities[1] |
| quantity | 0.23392 — empirical lower bound on the constant c in the slope-signature theorem, with 0.3 conjectured as an upper bound — “Across the datasets we generated, we can place a lower bound of c ≥ 0.23392, and it would be reasonable to conjecture that c is at most 0.3, which gives a tight relationship in the regions in which we have calculated.” | content_tags.quantities[2] |
| quantity | 24,322 — number of non-isomorphic Bruhat interval graphs retained for S9 after deduplication by KL polynomial — “We further reduce the dataset by including a single interval for each distinct KL polynomial for all graphs with the same number of nodes, resulting in 24,322 non-isomorphic graphs for S 9 .” | content_tags.quantities[3] |
| quantity | 2.8% to 7.8% — rate at which the injectivity radius computation failed, rising with knot crossing number — “The computation of the injectivity radius fails between 2.8% of the time on smaller knots up to 7.8% of the time on datasets of knots with a higher number of crossings.” | content_tags.quantities[4] |
| quantity | 80%/20% — train/test partition ratio for the Bruhat interval dataset — “We split the intervals randomly into train/test partitions at 80%/20%.” | content_tags.quantities[5] |
| quantity | several hours on a single GPU — training cost for the models behind both mathematical results — “In each of these cases, the necessary models can be trained within several hours on a machine with a single graphics processing unit.” | content_tags.quantities[6] |
| quantity | two results, in topology and representation theory — the demonstrations the generality claim rests on — “We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory.” | content_tags.quantities[7] |
| variable | hyperbolic invariants (independent) — “The specific hypothesis we investigated was that the geometry is predictive of the signature.” | content_tags.variables[0] |
| variable | accuracy of the learned function (dependent) — “We consider a relationship between X ( z ) and Y ( z ) to be found if the accuracy of the learned function f ˆ is statistically above chance on further samples from P Z on which the model was not trained.” | content_tags.variables[1] |
| variable | model architecture (control) — “The model architecture used for the experiments was a fully connected, feed-forward neural network, with hidden unit sizes” | content_tags.variables[2] |
| variable | threshold C k (control) — “The threshold C k for salient nodes was chosen a priori as the 99th percentile of attribution values across the dataset” | content_tags.variables[3] |
| definition | conjecture — “One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjectures: statements that are suspected to be true but have not been proven to hold in all cases.” | content_tags.definitions[0] |
| definition | invariant — “One of the principal ways that this is carried out is through invariants, which are algebraic, geometric or numerical quantities that are the same for any two equivalent knots.” | content_tags.definitions[1] |
| definition | natural slope — “We introduce the ‘natural slope’, defined to be slope( K )” | content_tags.definitions[2] |
| definition | test bed for intuition — “Conceptually, this framework provides a ‘test bed for intuition’—quickly verifying whether an intuition about the relationship between two quantities may be worth pursuing and, if so, guidance as to how they may be related.” | content_tags.definitions[3] |
| definition | combinatorial invariance conjecture — “It states that the KL polynomial of two elements in a symmetric group S N can be calculated from their unlabelled Bruhat interval 30 , a directed graph.” | content_tags.definitions[4] |
| definition | conjecture — “One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjectures: statements that are suspected to be true but have not been proven to hold in all cases.” | content_tags.definitions[5] |
| definition | gradient saliency — “The attribution technique we use in our work, gradient saliency, does this by calculating the derivative of outputs of f ˆ , with respect to the inputs.” | content_tags.definitions[6] |
| definition | knots — “Knots, which are simple closed curves in R 3 , are one of the key objects that are studied” | content_tags.definitions[8] |
| key-terms | conjecture, intuition, supervised learning, attribution techniques, gradient saliency, knots, representation theory, machine learning, pattern recognition, counterexamples |
| concept | conjecture — not defined in this paper's own words; a statement suspected true but not proven in all cases - the output the framework is built to produce — “One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjectures: statements that are suspected to be true but have not been proven to hold in all cases.” | concepts[0] |
| concept | intuition — not defined in this paper's own words; the mathematician's sense of which relationships are worth pursuing, treated here as the thing to be guided rather than replaced — “A mathematician’s intuition plays an enormously important role in mathematical discovery—“It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems” 25 .” | concepts[1] |
| concept | supervised learning — not defined in this paper's own words; the stage that tests whether a hypothesised relationship exists at all, by learning it from generated pairs — “In the supervised learning stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). By” | concepts[2] |
| concept | attribution techniques — not defined in this paper's own words; methods that expose which parts of the input a learned function depends on, so a person can reason about what it found — “Attribution techniques can be used to understand which aspects of f ˆ are relevant for predictions of Y ( z ).” | concepts[3] |
| concept | gradient saliency — not defined in this paper's own words; the specific attribution method used: the derivative of the model's output with respect to its inputs — “The attribution technique we use in our work, gradient saliency, does this by calculating the derivative of outputs of f ˆ , with respect to the inputs.” | concepts[4] |
| concept | knots — not defined in this paper's own words; simple closed curves in three-space, the objects of the topology demonstration — “Knots, which are simple closed curves in R 3 , are one of the key objects that are studied” | concepts[5] |
| concept | machine learning — not defined in this paper's own words; the source of the nonlinear function classes that make the detection step possible where earlier data-driven methods failed — “The key contributions of machine learning in this regression process are the broad set of possible nonlinear functions that can be learned given a sufficient amount of data.” | concepts[6] |
| classification accuracy | as written: test classification accuracy as a metric of performance — “The task was framed as a multi-class classification problem, with the distinct values of the signature as classes, cross-entropy loss as an optimizable loss function and test classification accuracy as a metric of performance.” | content_tags.statistics[0] |
| t-test t | as written: two-sample two-sided t -test — “In this diagram, the two-sample two-sided t -test statistics are as follows—simple edges: t = 25.7, P = 4.0 × 10 −10 ; extremal edges: t = −13.8, P = 1.1 × 10 −7 ; other edges: t = −3.2, P = 0.01.” | content_tags.statistics[1] |
| confidence interval | as written: 95% confidence interval error bars — “The 95% confidence interval error bars are across 10 retrainings of the model.” | content_tags.statistics[2] |
| bootstrap | as written: 10 bootstrapped samples of the same size from the dataset — “The percentage of observed edges of each type in the salient subgraph for 10 retrainings of the model compared to 10 bootstrapped samples of the same size from the dataset.” | content_tags.statistics[3] |
| train/test split (held-out evaluation) | as written: train/test partitions at 80%/20% — “We split the intervals randomly into train/test partitions at 80%/20%.” | content_tags.statistics[4] |
| standardization (mean and variance) | as written: normalized by subtracting the mean and dividing by the variance — “All features are normalized by subtracting the mean and dividing by the variance.” | content_tags.statistics[5] |
| regression f̂ | — “The key contributions of machine learning in this regression process are the broad set of possible nonlinear functions that can be learned given a sufficient amount of data.” | content_tags.statistics[6] |
| accuracy better than chance | — “If f ˆ is more accurate than would be expected by chance, it indicates that there may be such a relationship to explore.” | content_tags.statistics[7] |
| phenomenon | machine-assisted formation of mathematical conjectures — a second reading DIFFERING from the recorded value; recorded: machine-learning-guided mathematical conjecture discovery — “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.” | alternatives.phenomenon |
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Advancing mathematics by guiding human intuitionterm·dom with AI Alex DaviespersName Alex DaviespersName 1 DeepMind, London, UK 1, ✉ , Petar VeličkovićpersName Petar VeličkovićpersName 1 DeepMind, London, UK 1 , Lars BuesingpersName Lars BuesingpersName 1 DeepMind, London, UK 1 , Sam BlackwellpersName Sam BlackwellpersName 1 DeepMind, London, UK 1 , Daniel ZhengpersName Daniel ZhengpersName 1 DeepMind, London, UK 1 , Nenad TomaševpersName Nenad TomaševpersName 1 DeepMind, London, UK 1 , Richard TanburnpersName Richard TanburnpersName 1 DeepMind, London, UK 1 , Peter BattagliapersName Peter BattagliapersName 1 DeepMind, London, UK 1 , Charles Blundell Charles Blundell 1 DeepMind, London, UK 1 , András Juhász András Juhász 2 University of Oxford, Oxford, UK 2 , Marc Lackenby Marc Lackenby 2 University of Oxford, Oxford, UK 2 , Geordie Williamson Geordie Williamson 3 University of Sydney, Sydney, New South Wales Australia 3 , Demis Hassabis Demis Hassabis 1 DeepMind, London, UK 1 , Pushmeet Kohli Pushmeet Kohli 1 DeepMind, London, UK 1, ✉ 1 DeepMind, London, UK 2 University of Oxford, Oxford, UK 3 University of Sydney, Sydney, New South Wales Australia ✉ Corresponding author.
Received 2021 Jul 10; Accepted 2021 Sep 30; Issue date 2021.
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PMCID: PMC8636249 PMID: 34853458 Abstract The practice of mathematics involves discovering patterns and using these to formulate and prove conjecturesterm·dom, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjecturesterm·dom 1 , most famously in the Birch and Swinnerton-Dyer conjectureterm·dom 2 , a Millennium Prize Problem 3 . Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learningterm·mth—demonstrating a method by which machine learningterm·mth can aid mathematicians in discovering new conjecturesterm·dom and theorems. We propose a process of using machine learningterm·mth to discover potential patterns and relations between mathematical objects, understanding them with attribution techniquesterm·mth and using these observations to guide intuitionterm·dom and propose conjecturesterm·dom. 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 knotsterm·dom, and a candidate algorithm predicted by the combinatorial invariance conjectureterm·def for symmetric groups 4 . 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 learningterm·mth.
Subject terms: Computer science, Pure mathematics, Statistics A framework through which machine learningterm·mth can guide mathematicians in discovering new conjecturesterm·dom and theorems is presented and shown to yield mathematical insight on important open problems in different areas of pure mathematics.
Main One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjecturesterm·dom: statements that are suspected to be true but have not been proven to hold in all cases. Mathematicians have always used data to help in this process—from the early hand-calculated prime tables used by Gauss and others that led to the prime number theorem 5 , to modern computer-generated data 1 , 5 in cases such as the Birch and Swinnerton-Dyer conjectureterm·dom 2 . The introduction of computers to generate data and test conjecturesterm·dom afforded mathematicians a new understanding of problems that were previously inaccessible 6 , but while computational techniques have become consistently useful in other parts of the mathematical process 7 , 8 , artificial intelligence (AI) systems have not yet established a similar place. Prior systems for generating conjecturesterm·dom have either contributed genuinely useful research conjecturesterm·dom 9 via methods that do not easily generalize to other mathematical areas 10 , or have demonstrated novel, general methods for finding conjecturesterm·dom 11 that have not yet yielded mathematically valuable results.
AI, in particular the field of machine learningterm·mth 12 – 14 , offers a collection of techniques that can effectively detect patterns in data and has increasingly demonstrated utility in scientific disciplines 15 . In mathematics, it has been shown that AI can be used as a valuable tool by finding counterexamplesterm·key to existing conjecturesterm·dom 16 , accelerating calculations 17 , generating symbolic solutions 18 and detecting the existence of structure in mathematical objects 19 . In this work, we demonstrate that AI can also be used to assist in the discovery of theorems and conjecturesterm·dom at the forefront of mathematical research. This extends work using supervised learningterm·mth to find patterns 20 – 24 by focusing on enabling mathematicians to understand the learned functions and derive useful mathematical insight. We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognitionterm·key and interpretation methods from machine learningterm·mth and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theoryterm·key. Our contribution shows how mature machine learningterm·mth methodologies can be adapted and integrated into existing mathematical workflows to achieve novel results.
Guiding mathematical intuitionterm·dom with AI A mathematician’s intuitionterm·dom plays an enormously important role in mathematical discovery—“It is only with a combination of both rigorous formalism and good intuitionterm·dom that one can tackle complex mathematical problems” 25 . The following framework, illustrated in Fig. 1 , describes a general method by which mathematicians can use tools from machine learningterm·mth to guide their intuitionsterm·dom concerning complex mathematical objects, verifying their hypotheses about the existence of relationships and helping them understand those relationships. We propose that this is a natural and empirically productive way that these well-understood techniques in statistics and machine learningterm·mth can be used as part of a mathematician’s work.
Fig. 1. Flowchart of the framework.
The process helps guide a mathematician’s intuitionterm·dom about a hypothesized function f , by training a machine learningterm·mth model to estimate that function over a particular distribution of data P Z . The insights from the accuracy of the learned functionterm·var f ˆ and attribution techniquesterm·mth applied to it can aid in the understanding of the problem and the construction of a closed-form f ′. The process is iterative and interactive, rather than a single series of steps.
Concretely, it helps guide a mathematician’s intuitionterm·dom about the relationship between two mathematical objects X ( z ) and Y ( z ) associated with z by identifying a function f ˆ such that f ˆ ( X ( z )) ≈ Y ( z ) and analysing it to allow the mathematician to understand properties of the relationship. As an illustrative example: let z be convex polyhedra, X ( z ) ∈ Z 2 × R 2 be the number of vertices and edges of z , as well as the volume and surface area, and Y ( z ) ∈ ℤ be the number of faces of z. Euler’s formula states that there is an exact relationship between X ( z ) and Y ( z ) in this case: X ( z ) · (−1, 1, 0, 0) + 2 = Y ( z ). In this simple example, among many other ways, the relationship could be rediscovered by the traditional methods of data-driven conjectureterm·dom generation 1 . However, for X ( z ) and Y ( z ) in higher-dimensional spaces, or of more complex types, such as graphs, and for more complicated, nonlinear f ˆ , this approach is either less useful or entirely infeasible.
The framework helps guide the intuitionterm·dom of mathematicians in two ways: by verifying the hypothesized existence of structure/patterns in mathematical objects through the use of supervised machine learningterm·mth; and by helping in the understanding of these patterns through the use of attribution techniquesterm·mth.
In the supervised learningterm·mth stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). By generating a dataset of X ( z ) and Y ( z ) pairs, we can use supervised learningterm·mth to train a function f ˆ that predicts Y ( z ), using only X ( z ) as input. The key contributions of machine learningterm·mth in this regressionterm·stat process are the broad set of possible nonlinear functions that can be learned given a sufficient amount of data. If f ˆ is more accurate than would be expected by chance, it indicates that there may be such a relationship to explore. If so, attribution techniquesterm·mth can help in the understanding of the learned function f ˆ sufficiently for the mathematician to conjectureterm·dom a candidate f ′. Attribution techniquesterm·mth can be used to understand which aspects of f ˆ are relevant for predictions of Y ( z ). For example, many attribution techniquesterm·mth aim to quantify which component of X ( z ) the function f ˆ is sensitive to. The attribution techniqueterm·mth we use in our work, gradient saliencyterm·mth, does this by calculating the derivative of outputs of f ˆ , with respect to the inputs. This allows a mathematician to identify and prioritize aspects of the problem that are most likely to be relevant for the relationship. This iterative process might need to be repeated several times before a viable conjectureterm·dom is settled on. In this process, the mathematician can guide the choice of conjecturesterm·dom to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.
Conceptually, this framework provides a ‘test bed for intuitionterm·def’—quickly verifying whether an intuitionterm·dom about the relationship between two quantities may be worth pursuing and, if so, guidance as to how they may be related. We have used the above framework to help mathematicians to obtain impactful mathematical results in two cases—discovering and proving one of the first relationships between algebraic and geometric invariantsterm·def in knotterm·dom theory and conjecturing a resolution to the combinatorial invariance conjectureterm·def for symmetric groups 4 , a well-known conjectureterm·dom in representation theoryterm·key. In each area, we demonstrate how the framework has successfully helped guide the mathematician to achieve the result. In each of these cases, the necessary models can be trained within several hours on a machine with a single graphics processing unit.
Topology Low-dimensional topology is an active and influential area of mathematics. Knotsterm·dom, which are simple closed curves in R 3 , are one of the key objects that are studied, and some of the subject’s main goals are to classify them, to understand their properties and to establish connections with other fields. One of the principal ways that this is carried out is through invariantsterm·def, which are algebraic, geometric or numerical quantities that are the same for any two equivalent knotsterm·dom. These invariantsterm·def are derived in many different ways, but we focus on two of the main categories: hyperbolic invariantsterm·var and algebraic invariantsterm·def. These two types of invariantsterm·def are derived from quite different mathematical disciplines, and so it is of considerable interest to establish connections between them. Some examples of these invariantsterm·def for small knotsterm·dom are shown in Fig. 2 . A notable example of a conjectured connection is the volume conjectureterm·dom 26 , which proposes that the hyperbolic volume of a knotterm·dom (a geometric invariantterm·def) should be encoded within the asymptotic behaviour of its coloured Jones polynomials (which are algebraic invariantsterm·def).
Fig. 2. Examples of invariantsterm·def for three hyperbolic knotsterm·dom.
We hypothesized that there was a previously undiscovered relationship between the geometric and algebraic invariantsterm·def.
Our hypothesis was that there exists an undiscovered relationship between the hyperbolic and algebraic invariantsterm·def of a knotterm·dom. A supervised learningterm·mth model was able to detect the existence of a pattern between a large set of geometric invariantsterm·def and the signature σ ( K ), which is known to encode important information about a knotterm·dom K , but was not previously known to be related to the hyperbolic geometry. The most relevant features identified by the attribution techniqueterm·mth, shown in Fig. 3a , were three invariantsterm·def of the cusp geometry, with the relationship visualized partly in Fig. 3b . Training a second model with X ( z ) consisting of only these measurements achieved a very similar accuracy, suggesting that they are a sufficient set of features to capture almost all of the effect of the geometry on the signature. These three invariantsterm·def were the real and imaginary parts of the meridional translation μ and the longitudinal translation λ . There is a nonlinear, multivariate relationship between these quantities and the signature. Having been guided to focus on these invariantsterm·def, we discovered that this relationship is best understood by means of a new quantity, which is linearly related to the signature. We introduce the ‘natural slopeterm·def’, defined to be slope( K ) = Re( λ/μ ), where Re denotes the real part. It has the following geometric interpretation. One can realize the meridian curve as a geodesic γ on the Euclidean torus. If one fires off a geodesic γ ⊥ from this orthogonally, it will eventually return and hit γ at some point. In doing so, it will have travelled along a longitude minus some multiple of the meridian. This multiple is the natural slopeterm·def. It need not be an integer, because the endpoint of γ ⊥ might not be the same as its starting point. Our initial conjectureterm·dom relating natural slopeterm·def and signature was as follows.
Fig. 3. Knotterm·dom theory attribution.
a , Attribution values for each of the input X ( z ). The features with high values are those that the learned function is most sensitive to and are probably relevant for further exploration. The 95% confidence intervalterm·stat error bars are across 10 retrainings of the model. b , Example visualization of relevant features—the real part of the meridional translation against signature, coloured by the longitudinal translation.
Conjectureterm·dom: There exist constants c 1 and c 2 such that, for every hyperbolic knotterm·dom K , | 2 σ ( K ) − slope ( K ) | < c 1 vol ( K ) + c 2 1 While this conjectureterm·dom was supported by an analysis of several large datasets sampled from different distributions, we were able to construct counterexamplesterm·key using braids of a specific form. Subsequently, we were able to establish a relationship between slope( K ), signature σ ( K ), volume vol( K ) and one of the next most salient geometric invariantsterm·def, the injectivity radius inj( K ) (ref. 27 ).
Theorem: There exists a constant c such that, for any hyperbolic knotterm·dom K , | 2 σ ( K ) − slope ( K ) | ≤ c vol ( K ) inj ( K ) − 3 2 It turns out that the injectivity radius tends not to get very small, even for knotsterm·dom of large volume. Hence, the term inj( K ) −3 tends not to get very large in practice. However, it would clearly be desirable to have a theorem that avoided the dependence on inj( K ) −3 , and we give such a result that instead relies on short geodesics, another of the most salient features, in the Supplementary Information. Further details and a full proof of the above theorem are available in ref. 27 . Across the datasets we generated, we can place a lower bound of c ≥ 0.23392, and it would be reasonable to conjectureterm·dom that c is at most 0.3, which gives a tight relationship in the regions in which we have calculated.
The above theorem is one of the first results that connect the algebraic and geometric invariantsterm·def of knotsterm·dom and has various interesting applications. It directly implies that the signature controls the non-hyperbolic Dehn surgeries on the knotterm·dom and that the natural slopeterm·def controls the genus of surfaces in R + 4 whose boundary is the knotterm·dom. We expect that this newly discovered relationship between natural slopeterm·def and signature will have many other applications in low-dimensional topology. It is surprising that a simple yet profound connection such as this has been overlooked in an area that has been extensively studied.
Representation theoryterm·key Representation theoryterm·key is the theory of linear symmetry. The building blocks of all representations are the irreducible ones, and understanding them is one of the most important goals of representation theoryterm·key. Irreducible representations generalize the fundamental frequencies of Fourier analysis 28 . In several important examples, the structure of irreducible representations is governed by Kazhdan–Lusztig (KL) polynomials, which have deep connections to combinatorics, algebraic geometry and singularity theory. KL polynomials are polynomials attached to pairs of elements in symmetric groups (or more generally, pairs of elements in Coxeter groups). The combinatorial invariance conjectureterm·def is a fascinating open conjectureterm·dom concerning KL polynomials that has stood for 40 years, with only partial progress 29 . It states that the KL polynomial of two elements in a symmetric group S N can be calculated from their unlabelled Bruhat interval 30 , a directed graph. One barrier to progress in understanding the relationship between these objects is that the Bruhat intervals for non-trivial KL polynomials (those that are not equal to 1) are very large graphs that are difficult to develop intuitionterm·dom about. Some examples of small Bruhat intervals and their KL polynomials are shown in Fig. 4 .
Fig. 4. Two example dataset elements, one from S 5 and one from S 6 .
The combinatorial invariance conjectureterm·def states that the KL polynomial of a pair of permutations should be computable from their unlabelled Bruhat interval, but no such function was previously known.
We took the conjectureterm·dom as our initial hypothesis, and found that a supervised learningterm·mth model was able to predict the KL polynomial from the Bruhat interval with reasonably high accuracy. By experimenting on the way in which we input the Bruhat interval to the network, it became apparent that some choices of graphs and features were particularly conducive to accurate predictions. In particular, we found that a subgraph inspired by prior work 31 may be sufficient to calculate the KL polynomial, and this was supported by a much more accurate estimated function.
Further structural evidence was found by calculating salient subgraphs that attribution techniquesterm·mth determined were most relevant and analysing the edge distribution in these graphs compared to the original graphs. In Fig. 5a , we aggregate the relative frequency of the edges in the salient subgraphs by the reflection that they represent. It shows that extremal reflections (those of the form (0 , i ) or ( i , N − 1) for S N ) appear more commonly in salient subgraphs than one would expect, at the expense of simple reflections (those of the form ( i , i + 1)), which is confirmed over many retrainings of the model in Fig. 5b . This is notable because the edge labels are not given to the network and are not recoverable from the unlabelled Bruhat interval. From the definition of KL polynomials, it is intuitive that the distinction between simple and non-simple reflections is relevant for calculating it; however, it was not initially obvious why extremal reflections would be overrepresented in salient subgraphs. Considering this observation led us to the discovery that there is a natural decomposition of an interval into two parts—a hypercube induced by one set of extremal edges and a graph isomorphic to an interval in S N −1 .
Fig. 5. Representation theoryterm·key attribution.
a , An example heatmap of the percentage increase in reflections present in the salient subgraphs compared with the average across intervals in the dataset when predicting q 4 . b , The percentage of observed edges of each type in the salient subgraph for 10 retrainings of the model compared to 10 bootstrapped samples of the same size from the dataset. The error bars are 95% confidence intervalsterm·stat, and the significance level shown was determined using a two-sided two-sample t -test. * p < 0.05; **** p < 0.0001. c , Illustration for the interval 021435–240513 ∈ S 6 of the interesting substructures that were discovered through the iterative process of hypothesis, supervised learningterm·mth and attribution. The subgraph inspired by previous work 31 is highlighted in red, the hypercube in green and the decomposition component isomporphic to an interval in S N − 1 in blue.
The importance of these two structures, illustrated in Fig. 5c , led to a proof that the KL polynomial can be computed directly from the hypercube and S N −1 components through a beautiful formula that is summarized in the Supplementary Information . A further detailed treatment of the mathematical results is given in ref. 32 .
Theorem: Every Bruhat interval admits a canonical hypercube decomposition along its extremal reflections, from which the KL polynomial is directly computable.
Remarkably, further tests suggested that all hypercube decompositions correctly determine the KL polynomial. This has been computationally verified for all of the ∼3 × 10 6 intervals in the symmetric groups up to S 7 and more than 1.3 × 10 5 non-isomorphic intervals sampled from the symmetric groups S 8 and S 9 .
Conjectureterm·dom: The KL polynomial of an unlabelled Bruhat interval can be calculated using the previous formula with any hypercube decomposition.
This conjectured solution, if proven true, would settle the combinatorial invariance conjectureterm·def for symmetric groups. This is a promising direction as not only is the conjectureterm·dom empirically verified up to quite large examples, but it also has a particularly nice form that suggests potential avenues for attacking the conjectureterm·dom. This case demonstrates how non-trivial insights about the behaviour of large mathematical objects can be obtained from trained models, such that new structure can be discovered.
Conclusion In this work we have demonstrated a framework for mathematicians to use machine learningterm·mth that has led to mathematical insight across two distinct disciplines: one of the first connections between the algebraic and geometric structure of knotsterm·dom and a proposed resolution to a long-standing open conjectureterm·dom in representation theoryterm·key. Rather than use machine learningterm·mth to directly generate conjecturesterm·dom, we focus on helping guide the highly tuned intuitionterm·dom of expert mathematicians, yielding results that are both interesting and deep. It is clear that intuitionterm·dom plays an important role in elite performance in many human pursuits. For example, it is critical for top Go players and the success of AlphaGo (ref. 33 ) came in part from its ability to use machine learningterm·mth to learn elements of play that humans perform intuitively. It is similarly seen as critical for top mathematicians—Ramanujan was dubbed the Prince of Intuitionterm·dom 34 and it has inspired reflections by famous mathematicians on its place in their field 35 , 36 . As mathematics is a very different, more cooperative endeavour than Go, the role of AI in assisting intuitionterm·dom is far more natural. Here we show that there is indeed fruitful space to assist mathematicians in this aspect of their work.
Our case studies demonstrate how a foundational connection in a well-studied and mathematically interesting area can go unnoticed, and how the framework allows mathematicians to better understand the behaviour of objects that are too large for them to otherwise observe patterns in. There are limitations to where this framework will be useful—it requires the ability to generate large datasets of the representations of objects and for the patterns to be detectable in examples that are calculable. Further, in some domains the functions of interest may be difficult to learn in this paradigm. However, we believe there are many areas that could benefit from our methodology. More broadly, it is our hope that this framework is an effective mechanism to allow for the introduction of machine learningterm·mth into mathematicians’ work, and encourage further collaboration between the two fields.
Methods Framework Supervised learningterm·mth In the supervised learningterm·mth stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). In this work we assume that there is no known function mapping from X ( z ) to Y ( z ), which in turn implies that X is not invertible (otherwise there would exist a known function Y ° X −1 ). While there may still be value to this process when the function is known, we leave this for future work. To test the hypothesis that X and Y are related, we generate a dataset of X ( z ), Y ( z ) pairs, where z is sampled from a distribution P Z . The results of the subsequent stages will hold true only for the distribution P Z , and not the whole space Z . Initially, sensible choices for P Z would be, for example, uniformly over the first N items for Z with a notion of ordering, or uniformly at random where possible. In subsequent iterations, P Z may be chosen to understand the behaviour on different parts of the space Z (for example, regions of Z that may be more likely to provide counterexamplesterm·key to a particular hypothesis). To first test whether a relation between X ( z ) and Y ( z ) can be found, we use supervised learningterm·mth to train a function f ˆ that approximately maps X ( z ) to Y ( z ). In this work we use neural networks as the supervised learningterm·mth method, in part because they can be easily adapted to many different types of X and Y and knowledge of any inherent geometry (in terms of invariances and symmetries) of the input domain X can be incorporated into the architecture of the network 37 . We consider a relationship between X ( z ) and Y ( z ) to be found if the accuracy of the learned functionterm·var f ˆ is statistically above chance on further samples from P Z on which the model was not trained. The converse is not true; namely, if the model cannot predict the relationship better than chance, it may mean that a pattern exists, but is sufficiently complicated that it cannot be captured by the given model and training procedure. If it does indeed exist, this can give a mathematician confidence to pursue a particular line of enquiry in a problem that may otherwise be only speculative.
Attribution techniquesterm·mth If a relationship is found, the attribution stage is to probe the learned function f ˆ with attribution techniquesterm·mth to further understand the nature of the relationship. These techniques attempt to explain what features or structures are relevant to the predictions made by f ˆ , which can be used to understand what parts of the problem are relevant to explore further. There are many attribution techniquesterm·mth in the body of literature on machine learningterm·mth and statistics, including stepwise forward feature selection 38 , feature occlusion and attention weights 39 . In this work we use gradient-based techniques 40 , broadly similar to sensitivity analysis in classical statistics and sometimes referred to as saliency maps. These techniques attribute importance to the elements of X ( z ), by calculating how much f ˆ changes in predictions of Y (z) given small changes in X ( z ). We believe these are a particularly useful class of attribution techniquesterm·mth as they are conceptually simple, flexible and easy to calculate with machine learningterm·mth libraries that support automatic differentiation 41 – 43 . Information extracted via attribution techniquesterm·mth can then be useful to guide the next steps of mathematical reasoning, such as conjecturing closed-form candidates f ′, altering the sampling distribution P Z or generating new hypotheses about the object of interest z , as shown in Fig. 1 . This can then lead to an improved or corrected version of the conjectured relationship between these quantities.
Topology Problem framing Not all knotsterm·dom admit a hyperbolic geometry; however, most do, and all knotsterm·dom can be constructed from hyperbolic and torus knotsterm·dom using satellite operations 44 . In this work we focus only on hyperbolic knotsterm·dom. We characterize the hyperbolic structure of the knotterm·dom complement by a number of easily computable invariantsterm·def. These invariantsterm·def do not fully define the hyperbolic structure, but they are representative of the most commonly interesting properties of the geometry. Our initial general hypothesis was that the hyperbolic invariantsterm·var would be predictive of algebraic invariantsterm·def. The specific hypothesis we investigated was that the geometry is predictive of the signature. The signature is an ideal candidate as it is a well-understood and common invariantterm·def, it is easy to calculate for large knotsterm·dom and it is an integer, which makes the prediction task particularly straightforward (compared to, for example, a polynomial).
Data generation We generated a number of datasets from different distributions P Z on the set of knotsterm·dom using the SnapPy software package 45 , as follows.
All knotsterm·dom up to 16 crossings (∼1.7 × 10 6 knotsterm·dom), taken from the Regina census 46 .
Random knotterm·dom diagrams of 80 crossings generated by SnapPy’s random_link function (∼10 6 knotsterm·dom). As random knotterm·dom generation can potentially lead to duplicates, we calculate a large number of invariantsterm·def for each knotterm·dom diagram and remove any samples that have identical invariantsterm·def to a previous sample, as they are likely to represent that same knotterm·dom with very high probability.
Knotsterm·dom obtained as the closures of certain braids. Unlike the previous two datasets, the knotsterm·dom that were produced here are not, in any sense, generic. Instead, they were specifically constructed to disprove Conjectureterm·dom 1. The braids that we used were 4-braids ( n = 11,756), 5-braids ( n = 13,217) and 6-braids ( n = 10,897). In terms of the standard generators σ i for these braid groups, the braids were chosen to be ( σ i 1 n 1 σ i 2 n 2 . . . σ i k n k ) N . The integers i j were chosen uniformly at random for the appropriate braid group. The powers n j were chosen uniformly at random in the ranges [−10, −3] and [3, 10]. The final power N was chosen uniformly between 1 and 10. The quantity ∑ | n i | was restricted to be at most 15 for 5-braids and 6-braids and 12 for 4-braids, and the total number of crossings N ∑ | n i | was restricted to lie in the range between 10 and 60. The rationale for these restrictions was to ensure a rich set of examples that were small enough to avoid an excessive number of failures in the invariantterm·def computations.
For the above datasets, we computed a number of algebraic and geometric knotterm·dom invariantsterm·def. Different datasets involved computing different subsets of these, depending on their role in forming and examining the main conjectureterm·dom. Each of the datasets contains a subset of the following list of invariantsterm·def: signature, slope, volume, meridional translation, longitudinal translation, injectivity radius, positivity, Chern–Simons invariantterm·def, symmetry group, hyperbolic torsion, hyperbolic adjoint torsion, invariantterm·def trace field, normal boundary slopes and length spectrum including the linking numbers of the short geodesics.
The computation of the canonical triangulation of randomly generated knotsterm·dom fails in SnapPy in our data generation process in between 0.6% and 1.7% of the cases, across datasets. The computation of the injectivity radius fails between 2.8% of the time on smaller knotsterm·dom up to 7.8% of the time on datasets of knotsterm·dom with a higher number of crossings. On knotsterm·dom up to 16 crossings from the Regina dataset, the injectivity radius computation failed in 5.2% of the cases. Occasional failures can occur in most of the invariantterm·def computations, in which case the computations continue for the knotterm·dom in question for the remaining invariantsterm·def in the requested set. Additionally, as the computational complexity of some invariantsterm·def is high, operations can time out if they take more than 5 min for an invariantterm·def. This is a flexible bound and ultimately a trade-off that we have used only for the invariantsterm·def that were not critical for our analysis, to avoid biasing the results.
Data encoding The following encoding scheme was used for converting the different types of features into real valued inputs for the network: reals directly encoded; complex numbers as two reals corresponding to the real and imaginary parts; categoricals as one-hot vectors.
All features are normalized by subtracting the mean and dividing by the varianceterm·stat. For simplicity, in Fig. 3a , the salience values of categoricals are aggregated by taking the maximum value of the saliencies of their encoded features.
Model and training procedure The model architectureterm·var used for the experiments was a fully connected, feed-forward neural network, with hidden unit sizes [300, 300, 300] and sigmoid activations. The task was framed as a multi-class classification problem, with the distinct values of the signature as classes, cross-entropy loss as an optimizable loss function and test classification accuracy as a metric of performanceterm·stat. It is trained for a fixed number of steps using a standard optimizer (Adam). All settings were chosen as a priori reasonable values and did not need to be optimized.
Process First, to assess whether there may be a relationship between the geometry and algebra of a knotterm·dom, we trained a feed-forward neural network to predict the signature from measurements of the geometry on a dataset of randomly sampled knotsterm·dom. The model was able to achieve an accuracy of 78% on a held-out test set, with no errors larger than ±2. This is substantially higher than chance (a baseline accuracy of 25%), which gave us strong confidence that a relationship may exist.
To understand how this prediction is being made by the network, we used gradient-based attribution to determine which measurements of the geometry are most relevant to the signature. We do this using a simple sensitivity measure r i that averages the gradient of the loss L with respect to a given input feature x i over all of the examples x in a dataset X : r i = 1 | X | ∑ x ∈ X | ∂ L ∂ x i | 3 This quantity for each input feature is shown in Fig. 3a , where we can determine that the relevant measurements of the geometry appear to be what is known as the cusp shape: the meridional translation, which we will denote μ , and the longitudinal translation, which we will denote λ . This was confirmed by training a new model to predict the signature from only these three measurements, which was able to achieve the same level of performance as the original model.
To confirm that the slope is a sufficient aspect of the geometry to focus on, we trained a model to predict the signature from the slope alone. Visual inspection of the slope and signature in Extended Data Fig. 1a, b shows a clear linear trend, and training a linear model on this data results in a test accuracy of 78%, which is equivalent to the predictive power of the original model. This implies that the slope linearly captures all of the information about the signature that the original model had extracted from the geometry.
Extended Data Fig. 1. Empirical relationship between slope and signature.
a Signature vs slope for random dataset. b Signature vs slope for Regina dataset.
Evaluation The confidence intervalsterm·stat on the feature saliencies were calculated by retraining the model 10 times with a different train/test split and a different random seed initializing both the network weights and training procedure.
Representation theoryterm·key Data generation For our main dataset we consider the symmetric groups up to S 9 . The first symmetric group that contains a non-trivial Bruhat interval whose KL polynomial is not simply 1 is S 5 , and the largest interval in S 9 contains 9! ≈ 3.6 × 10 5 nodes, which starts to pose computational issues when used as inputs to networks. The number of intervals in a symmetric group S N is O ( N ! 2 ), which results in many billions of intervals in S 9 . The distribution of coefficients of the KL polynomials uniformly across intervals is very imbalanced, as higher coefficients are especially rare and associated with unknown complex structure. To adjust for this and simplify the learning problem, we take advantage of equivalence classes of Bruhat intervals that eliminate many redundant small polynomials 47 . This has the added benefit of reducing the number of intervals per symmetric group (for example, to ~2.9 million intervals in S 9 ). We further reduce the dataset by including a single interval for each distinct KL polynomial for all graphs with the same number of nodes, resulting in 24,322 non-isomorphic graphs for S 9 . We split the intervals randomly into train/test partitions at 80%/20%.
Data encoding The Bruhat interval of a pair of permutations is a partially ordered set of the elements of the group, and it can be represented as a directed acyclic graph where each node is labelled by a permutation, and each edge is labelled by a reflection. We add two features at each node representing the in-degree and out-degree of that node.
Model and training procedure For modelling the Bruhat intervals, we used a particular GraphNet architecture called a message-passing neural network (MPNN) 48 . The design of the model architectureterm·var (in terms of activation functions and directionality) was motivated by the algorithms for computing KL polynomials from labelled Bruhat intervals. While labelled Bruhat intervals contain privileged information, these algorithms hinted at the kind of computation that may be useful for computing KL polynomial coefficients. Accordingly, we designed our MPNN to algorithmically align to this computation 49 . The model is bi-directional, with a hidden layer width of 128, four propagation steps and skip connections. We treat the prediction of each coefficient of the KL polynomial as a separate classification problem.
Process First, to gain confidence that the conjectureterm·dom is correct, we trained a model to predict coefficients of the KL polynomial from the unlabelled Bruhat interval. We were able to do so across the different coefficients with reasonable accuracy (Extended Data Table 1 ) giving some evidence that a general function may exist, as a four-step MPNN is a relatively simple function class. We trained a GraphNet model on the basis of a newly hypothesized representation and could achieve significantly better performance, lending evidence that it is a sufficient and helpful representation to understand the KL polynomial.
Extended Data Table 1.
Model accuracies at predicting KL coefficients from Bruhat intervals in S 9 To understand how the predictions were being made by the learned function f ˆ , we used gradient-based attribution to define a salient subgraph S G for each example interval G , induced by a subset of nodes in that interval, where L is the loss and x v is the feature for vertex v : S G = { v ∈ G | | ∂ L ∂ x v | > C k } 4 We then aggregated the edges by their edge type (each is a reflection) and compared the frequency of their occurrence to the overall dataset. The effect on extremal edges was present in the salient subgraphs for predictions of the higher-order terms ( q 3 , q 4 ), which are the more complicated and less well-understood terms.
Evaluation The threshold C kterm·var for salient nodes was chosen a priori as the 99th percentile of attribution values across the dataset, although the results are present for different values of C k in the range [95, 99.5]. In Fig. 5a , we visualize a measure of edge attribution for a particular snapshot of a trained model for expository purposes. This view will change across time and random seeds, but we can confirm that the pattern remains by looking at aggregate statistics over many runs of training the model, as in Fig. 5b . In this diagram, the two-sample two-sided t -testterm·stat statistics are as follows—simple edges: t = 25.7, P = 4.0 × 10 −10 ; extremal edges: t = −13.8, P = 1.1 × 10 −7 ; other edges: t = −3.2, P = 0.01. These significance results are robust to different settings of the hyper-parameters of the model.
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