Advancing mathematics by using artificial intelligence.
Alex Davies orcid.org/0000-0003-4917-5341.
Petar Velikovi1.
The person is Lars Buesing.
Sam Blackwell.
The person is Daniel Zheng.
Nenad Tomaev orcid.org/0000-0003-1624-02201.
Richard Tanburn.
Peter Battaglia.
Charles Blundell1.
Andrs Juhsz2.
There is a man named Lackenby2.
The person is Geordie Williamson.
...
The orcid.org/0000-0003-2812-99171 contains information about the person.
Pushmeet Kohli is at orcid.org.
This article is about Nature 600*, 70–74.
The practice of mathematics involves discovering patterns and using them to form and prove hypotheses. The Birch and Swinnerton-Dyer conjecture is one of the most well-known examples of the use of computers in mathematics. Here we show examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning. 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. In each case, we show how the machine-learning-guided framework led to meaningful mathematical contributions on important open problems, such as a new connection between the algebraic and geometric structure of knots. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence that can achieve surprising results by using the strengths of mathematicians and machine learning.
One of the main drivers of mathematical progress is the discovery of patterns and the creation of useful conjectures, statements that are suspected to be true but have not been proven to hold in all cases. Data has always been used to help in this process, from the early hand-calculated prime tables used by Gauss and others to the modern computer-generated data used in the Birch and Swinnerton-Dyer conjecture. Computational techniques have become more useful in other parts of the mathematical process, but artificial intelligence systems have not yet established a similar place. Prior systems for generating conjectures have contributed useful research by using methods that are not easy to generalize to other mathematical areas, or have demonstrated novel, general methods for finding conjectures that have not yet yielded useful results.
Artificial intelligence, in particular the field of machine learning, offers a collection of techniques that can effectively detect patterns in data and has become a utility in scientific disciplines. Artificial intelligence can be used in mathematics to find counterexamples to existing conjectures, accelerate calculations, generate symbolic solutions and detect the existence of structure in mathematical objects. In this work, we show that artificial intelligence can be used to assist in the discovery of mathematics. This extends work by using supervised learning to find patterns and enable mathematicians to understand the learned functions. 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 to two fundamental new discoveries. Our contribution shows how mature machine learning methodologies can be integrated into existing mathematical workflows to achieve novel results.
It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems. The framework shows how mathematicians can use machine learning to help them understand mathematical relationships and how they can use machine learning to guide their intuitions. This is a natural and productive way that these well-understood techniques in statistics and machine learning can be used in a mathematician's work.
The framework is depicted in fig. 1.
A machine learning model is trained to estimate a function over a particular distribution of data PZ, which helps guide a mathematician's intuition. The insights from the accuracy of the learned function can help in understanding the problem and the construction of a closed-form f. The process is iterative and interactive.
It helps guide a mathematician's intuition about the relationship between two mathematical objects X(z) and Y(z) associated with Z. The number of edges and vertices of Z can be used as an example. There is an exact relationship between X(z) and Y(z) in this case, according to the formula. The relationship could be rediscovered by using traditional methods of data-driven conjecture generation. This approach is less useful for X(z) and Y(z) in higher-dimensional spaces, or for more complex types, such as graphs.
The framework helps guide the intuition of mathematicians by helping to verify the existence of structure/patterns in mathematical objects through the use of supervised machine learning, and by helping to understand these patterns through the use of attribution techniques.
The mathematician proposes a hypothesis that there is a relationship between X and Y. We can use supervised learning to train a function that predicts Y(z) using only X(z) as input. The broad set of possible nonlinear functions that can be learned from a sufficient amount of data are the key contributions of machine learning in this regression process. If (hatf) is more accurate than would be expected, it means there may be a relationship to explore. Attribution techniques can help the mathematician understand the learned function (hatf) sufficiently for him to guess a candidate f Attribution techniques can be used to understand which aspects of (hatf) are relevant for predictions. Many techniques aim to quantify which component of the function is sensitive to. The method we use to calculate the derivative of outputs of (hatf)) with respect to the inputs is called the gradient saliency. This allows a mathematician to look at the problem in a different way. This process might need to be repeated several times. The mathematician can guide the choice of conjectures to those that are plausibly true and suggestive of a proof strategy.
This framework provides a test bed for intuition, which can be used to verify whether an intuition about the relationship between two quantities is worth pursuing and, if so, guidance as to how they may be related. We used the above framework to help mathematicians get results in two cases, one of which was the discovery and proving of a relationship between a pair of invariants in knot theory. The framework helped guide the mathematician to achieve the result. The models can be trained on a single graphics processing unit in each case.
The low-dimensional area of mathematics is influential. One of the key objects that are studied is the knot, which is a simple closed curve, and the subject's main goals are to classify them, to understand their properties and to establish connections with other fields. One of the ways that this is done is through invariants, which are the same for any two equivalent knots. There are many different ways in which these invariants are derived. These two types of invariants are derived from different mathematical disciplines, and so it is of great interest to establish connections between them. There are some examples of invariants for small knots shown in the second figure. A notable example of a conjectured connection is the volume conjecture26, which proposes that the hyperbolic volume of a knot should be hidden within the coloured Jones polynomials.
There are invariants for three hyperbolic knots.
There was a previously undiscovered relationship between the geometric and algebraic invariants.
Our hypothesis was that there is a relationship between the hyperbolic and algebraic invariants of a knot. A supervised learning model was able to detect the existence of a pattern between a large set of geometric invariants and the signature (K), which was not previously known to be related to the hyperbolic geometry. The relationship between the three invariants of the cusp geometry was visualized partly in fig. 3b. 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. The real and imaginary parts of the meridional and longitudinal translation were the three invariants. There is a relationship between the quantities and the signature. The relationship is best understood by means of a new quantity, which is related to the signature. The natural slope is defined as slope(K) + Re(/) where Re is the real part. It has a geometric interpretation. One can see the curve as a geodesic on the torus. It will return and hit at some point if one fires off a geodesic from this. It will have traveled along a longitude and a multiple of the meridian. The natural slope is the multiple. The endpoint of might not be the same as its starting point. Our initial hypothesis was about natural slope and signature.
The knot theory is attributed in fig. 3.
Attribution values for each input. The features with high values are those that the learned function is most sensitive to. There are 10 retrainings of the model and the 95% confidence interval error bars.
There are constants for every hyperbolic knot.
2sigma (K)-rmslope(K)
The edges were aggregated by their edge type and compared to the overall dataset. The effect on the edges was shown in the subgraphs for the higher-order terms.
Evaluation.
The threshold Ck was chosen as the 99th percentile of the values in the range 95, 99.5, although the results are present for different values of Ck in the range. The measure of edge attribution is visualized in the figure. We can confirm the pattern by looking at aggregate statistics over many runs of training the model, as shown in fig. 5b. The two-sided t-test statistics are as follows: simple edges: t is 25.7, P is 4.0, and the other is t is 13.8. The significance results are robust to different settings of the model.
There are notebooks that can be used to regenerate the results for knot theory and representation theory.
The generated datasets used in the experiments have been made available for download.
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References can be downloaded.
We appreciate the help and feedback from J. Ellenberg, S. Mohamed, O. Vinyals, A. Gaunt, A. Fawzi and D. Saxton. DeepMind funded this research.
The authors do not have competing interests.
A slope vs signature for a random dataset.
The table has accuracies for predicting KL coefficients from Bruhat intervals.
Supplementary text is in this file. hyperbolic knots and references.
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