# Machine Learning in Physics & Geometry

Yang-Hui<EMAIL_ADDRESS>Elli
<EMAIL_ADDRESS>Edward<EMAIL_ADDRESS>
(1London Institute for Mathematical Sciences, Royal Institution,
London W1S 4BS, UK
2Department of Mathematics, City, University of London,
EC1V 0HB, UK
3Merton College, University of Oxford, OX1 4JD, UK
4School of Physics, NanKai University, Tianjin, 300071, P.R. China )

###### Abstract

We survey some recent applications of machine learning to problems in geometry
and theoretical physics. Pure mathematical data has been compiled over the
last few decades by the community and experiments in supervised, semi-
supervised and unsupervised machine learning have found surprising success. We
thus advocate the programme of machine learning mathematical structures, and
formulating conjectures via pattern recognition, in other words using
artificial intelligence to help one do mathematics.

This is an invited chapter contribution to Elsevier’s Handbook of Statistics,
Volume 49: Artificial Intelligence edited by S. G. Krantz, A. S. R. Srinivasa
Rao, and C. R. Rao.

Keywords: String Theory, Machine Learning, Algebraic Geometry, Quantum Field
Theory, Elementary Particles

###### Contents

  1. 1 Introduction & Summary
    1. 1.1 Mathematical Data as Pure Data
    2. 1.2 The Inevitability of AI in Geometry & Physics
  2. 2 Background Physics and Mathematics
    1. 2.1 Polytopes
    2. 2.2 Amoebae
    3. 2.3 Quivers
    4. 2.4 Brane Webs
  3. 3 Supervised Learning
    1. 3.1 Neural Networks
    2. 3.2 Support Vector Machines
  4. 4 Unsupervised Learning
    1. 4.1 Principal Component Analysis
    2. 4.2 t-SNE
    3. 4.3 K-Means Clustering
    4. 4.4 Topological Data Analysis
  5. 5 Conclusion
  6. Acknowledgement

##  1 Introduction & Summary

The fields of mathematics and physics are intimately intertwined and have been
so since the dawn of philosophy. The ancient Greeks studied the conic
sections, curves obtained by slicing the cone by a plane. By Descartes, these
were realized to be the geometry of quadratic polynomials in two variables.
Finally, Newton combined his Second Law with the inverse square Law of
Universal Gravitation to show that the solution to the motion of planets were
precisely these conics. Gauss, Bolyai and Lobachevsky mused with an intrinsic,
non-Euclidean geometry, and Riemann was able to formulate it into the
foundations of differential geometry. Then, Einstein was able to marry this to
his General Relativity and demonstrated that gravity was the curvature of
spacetime.

Such examples abounded throughout the history of fundamental physics, where
there is a perpetual dance between the nature of reality and the structures of
mathematics. One could say that modern physics, from relativity to quantum
field theory, has been following a “Geometrization Programme”. Any unified
theory must follow this path. String theory, still the best contender for this
“theory of everything”, is favoured despite the lack of experimental evidence
thus far, precisely because it is a brain-child of this geometrization
programme.

Traditionally, computations in mathematics and physics were done solely by
hand. When computers first came about mathematicians and physicists were eager
to use them to speed up calculations. Think back to the days when the likes of
Euler had to find the digits of $\pi$ by hand! The development of computers
has been tremendous and one can now carry out calculations on a personal
laptop that would take years to complete by hand in just a matter of seconds.
This clearly was a huge advantage for contemporary mathematicians and
physicists. Imagine how much more Euler would have achieved if he had a
symbolic package like Mathematica [1] to help with the mundane tasks of
obtaining integrals and series.

And this brings us to the heart of the matter. We often forget, when reading
the final and polished journal papers, that theoretical research - even in
pure mathematics - begins with experimentation. Gauss would plot the number
$N(x)$ of primes less than a given real number $x$ and notice that it is
approximated by $x/\log(x)$, long before complex analysis was invented to
offer a proof. Einstein would add a $\Lambda$ constant to his field equations
just to toy with a cosmological model long before the theories of inflation or
evolution. This is what mathematicians do; we experiment and then we
formalize. In the words of the great Vladimir Arnol’d, “mathematics is a
branch of physics where experiments are cheap.” Henceforth, we will use the
term “mathematician” to include pure and applied mathematicians, as well as
theoretical physicists.

###  1.1 Mathematical Data as Pure Data

Here, then, lies the paradigmatic change. Whilst since at least the mid-
twentieth century computers have aided mathematicians, only the last decade or
so - considered the beginnings of the “Age of Data” - has brought in a
plethora of “mathematical data”. All of a sudden, any mathematician has free
and instant access to Gigabytes of data from across the disciplines, and,
importantly, the ability to experiment with this data on their laptops. There
is, by definition, something quite unique about mathematical data: distinct
from data coming from the “real world”, it is largely noiseless. Simulations
of numerical solutions aside, pure mathematical data such as the parity of
number of prime factors of the first one million integers, or the list of
inequivalent graphs with a given property up to 100 vertices, are exact.
Suddenly, we have an abundance of this sort of data at our fingertips and
finding patterns in them, like what Gauss did with $N(x)$, leads to
conjectures and mathematical discovery.

Treating large volumes of data inevitably brings us to the topic of artificial
intelligence. In the past few decades, the field of machine learning (ML) (a
subfield of artificial intelligence (AI)) has suddenly taken off and has
become an integral part of scientific research. In contrast to traditional
programming, where one tells the computer the exact steps to solve the
problem, machine learning algorithms are able to solve problems without this
instruction. Of course, AI and ML techniques have long been an indispensable
tool in the experimental sciences; in fundamental physics, for instance, the
Higgs particle could not have been found by CERN without deep-learning of the
detector data. It might seem counter-intuitive that a methodology grounded
upon the statistical inference of data should be of any use to the rigorous
world of proofs in mathematics and derivations in theoretical physics.

Nevertheless, from the introduction in 2017 of ML into the mathematics of
algebraic geometry and the physics of superstring theory [2, 3, 4, 5] (see
also the usage of genetic algorithms in searching for the correct vacuum [6]),
to the recent advances of Google’s DeepMind collaboration with mathematicians
in studying knot invariants in 2022 [7], as well as work to uncover physical
laws directly [8], there has been an explosion of activity in the past few
years.

###  1.2 The Inevitability of AI in Geometry & Physics

In this chapter we shall focus mainly on string theory as an example where ML
is helping push the boundaries of physics and geometry and shall present some
specific examples. String theory is currently the most promising candidate for
a theory of everything (a theory that combines quantum mechanics and general
relativity); it is a brain-child of the geometrization programme which modern
physics has followed.

The main premise is that everything in the universe is built from a
fundamental string object, whose motion gives rise to different particle
properties. A requirement for string theory to be self-consistent is that
there exists an extra six dimensions of space, on top of the three dimensions
of space and one dimension of time that we are familiar with. The geometry of
the extra dimensions together with the associated metric determines the
physics of the resulting universe, such as the masses and charges of
particles, the coupling between forces, etc. Therefore, the mathematical tools
from geometry used to construct these spaces are crucial for string theorists.

The biggest theoretical challenge to string theory is that there is a large
number of possible geometries that could describe the extra six dimensions and
we have no method for choosing one that resembles our universe. This is called
the vacuum degeneracy problem. We refer to the collection of possible
geometries as the “string landscape”. The size of the landscape is certainly
too big to study each by hand and even too big to exhaustively search using a
computer. To give an idea, a small corner of the string landscape, consisting
of so-called flux compatifications for Calabi-Yau manifolds, there are some
$10^{500}$ possibilities [9]. This is where ML can help. The above-mentioned
references brought machine-learning and AI techniques to the search of the
string landscape in order to identify regions that give rise to sensible
physics.

As well as choosing a geometry from the landscape, we must also find the
metric associated to the geometry, which is an equation which defines the
notion of distance on the geometry. There is currently no method for the
construction of an explicit metric on a Calabi-Yau. This is another area where
ML has proved useful. There have been attempts to approximate an appropriate
(Ricci flat) metric on a Calabi-Yau starting with direct numerical methods
through Donaldson’s algorithm [10, 11, 12], as well as with reinforcement
learning techniques [13, 14, 15], where suitable packages are now available
[16, 17].

In this work we review a selection of common ML techniques from both
supervised and unsupervised learning, and present some examples of their
applications in physics and geometry. The examples focus on four areas of
string theory and geometry, namely: Calabi-Yau manifolds from polytopes,
amoebae, quivers, and brane webs. We begin in Section 2 by introducing the
relevant physical and mathematical background. Section 3 focuses on supervised
learning, in particular neural networks (NN) §3.1 and support vector machines
(SVM) §3.2. In Section 4 we look at unsupervised learning, in particular
principal component analysis (PCA) §4.1, t-distributed stochastic
neighbourhood embedding (t-SNE) §4.2, $k$-means clustering §4.3, and
topological data analysis (TDA) §4.4. Finally leaving some concluding remarks
in Section 5.

##  2 Background Physics and Mathematics

As technology developed throughout the twentieth century, the equipment used
for experimentation became unimaginably more advanced. The scale of
experiments hence greatly expanded, in both directions, as mankind could now
probe both the deepest parts of space and the smallest scales of particles.
However, comfortable with a theory of atoms, these experiments at shorter
distance scales (and hence higher energy densities) instead of confirming the
atomic theory saw an explosion of new observed particles, as well as new
seemingly illogical effects.

It was clear that classical theory was no longer appropriate for describing
these unusual effects occurring at small distance scales, and so from the
ashes of atomic theory was born a new style of theory: quantum physics. At
these small distance scales the continuous nature of the particle properties
was lost, as energy levels became discrete and hence ‘quantised’; and over
time as the theory of quantum physics was developed more of this unusual
behaviour could be described. Despite this success, there was still desire to
incorporate the ideas which so succinctly explained the physics of long
distances into quantum theory, in particular Einstein’s relativity.

Relativity, as a theory, was the result of an exceptionally insightful
paradigm shift in how space and time were viewed. Instead of a three-
dimensional space ticking along through time, the playground on which our
physical theory lives was re-imagined as a manifestly four-dimensional
spacetime. Depending on our relative speeds, we saw the universe from a
different angle (a different ‘inertial frame’) and this could lead to
discrepancies in experiment – later confirmed through observation.
Relativity’s frames were hence based on speeds, with a maximum speed of light
which could not be exceeded in any frame. As special relativity was developed
into general relativity to incorporate accelerating objects and massive
bodies, the foundations of gravity in geometry were cemented. Inertial mass
dictating acceleration was now seen as equivalent to gravitational mass, and
introduction of mass warped the spacetime from a flat Minkowski manifold to a
curved Lorentzian manifold.

All physical theories rely heavily on the concept of symmetry, described
through the mathematical language of groups. Knowing a theory’s symmetry can
greatly simplify calculations, and provides the means for a sensible
development of the correct intuition. The symmetry of special relativity is
based on the Poincaré group of spacetime translation and rotation. The
rotation part is the Lorentz group, and knowing how the theory’s fundamental
objects transform under this group is essential for computation in the theory.
For our four-dimensional spacetime this group is $SO(1,3)$, but it is turns
out to be more natural to consider the double cover $Spin(1,3)$, where the
theory’s particles transform as representations of this group. These Lorentz
representations end up being classified according to a particle property known
as spin, which can take integer or half-integer values, categorising the
particle as a boson or fermion respectively. The most common representations
have lower spin values and hence their own special names: spin $0$ particles
are scalars, spin $\frac{1}{2}$ particles are spinors, and spin $1$ particles
are vectors.

To incorporate the central ideas of special relativity into quantum theory the
fundamental particle objects are changed to fields in particular
representations of the Lorentz group, so birthing quantum field theory. Beyond
the Poincaré symmetry somewhat associated to gravity, there are other symmetry
groups related to the other three fundamental forces. The current best theory
for our universe is the ‘standard model’ in flat spacetime with symmetry group
$SU(3)\times SU(2)\times U(1)$ to describe the strong nuclear force, the weak
nuclear force, and electromagnetism respectively. The standard model is truly
a gauge theory, and in the gauging process the symmetries described by the
given groups are generalised from global to local, and require the
introduction of gauge vector bosons which moderate the respective interactions
between the theory’s matter fields. Ultimately, physicists seek a theory of
everything, which unalike the standard model, works at all distance scales
describing all forces. The elusive piece for the standard model is gravity, as
although the techniques of quantum field theory work well with special
relativity, they fail with general relativity as gravity cannot be quantised.

The most famous candidate for a theory of everything is string theory. String
theory was developed out of another paradigm shifting idea, answering the
question: why do the fundamental objects of our universe have to be
0-dimensional particles? Relaxing this assumption that the building blocks of
the universe are particles (these 0-dimensional dots), a range of other
fundamental objects can be included, such as 1-dimensional strings,
2-dimensional membranes, etc. As one may guess from the name, the most
important objects in these theories at the low-energies of our reality turn
out to be strings. String theory is importantly self-consistent to all
energies, however as mentioned previously does require the introduction of
extra spatial dimensions into spacetime. In the limit of lower energies
different string theories with different spacetime geometries reduce to
different quantum field theories, and the task of quantising gravity is
reaffirmed concretely within the realm of geometry.

With the aim of application of ML techniques to theoretical physics the first
challenge is the identification and construction of relevant datasets. Each
dataset consists of a set of mathematical objects of a particular type, which
are represented in an appropriate way for computational processing. The type
of mathematical object used (as well as the representation choice) may change
the perspective with which the physical theory is viewed. Often is the case
that these dual viewpoints offer non-trivial insights unavailable from the
original representation. This duality concept leads to a large multitude of
objects which provide unique viewpoints on the same theory, a selection of
which are chosen for data generation, analysis, and machine learning for the
research work reviewed in this chapter. Further reviews of relevant work at
the intersection of machine learning, physics and geometry can be found at
[18, 19, 20, 21, 22].

###  2.1 Polytopes

The first mathematical dataset which we shall consider is that of polytopes. A
polytope is a geometrical object that is bounded by a finite set of faces.
Polytopes can exist in any number of dimensions, polytopes in 2 dimensions are
also called polygons and polytopes in 3 dimensions polytopes are also called
polyhedra. Simple examples of polytopes include the filled in triangle in 2
dimensions and the filled in cube in 3 dimensions.

Polytopes are of particular interest to string theorists because in 1993 it
was shown [23] that one can obtain a Calabi-Yau (CY) manifold from a
particular type of polytope, called a reflexive polytope, and CY manifolds
play an important role in string theory as candidate geometries for the
compactification space [24]. Roughly speaking, from a polytope $\Delta$ one
can construct a geometric space (toric variety), and it happens that if
$\Delta$ is reflexive then certain subspaces (hypersurfaces) of this space
describe CY manifolds.

Let’s now define polytopes and the notion of reflexivity in more detail. Let
$M$ and $N$ be a dual pair of $n$-dimensional integer lattices, meaning that
lattice points of $M$ are described by a set of $n$ integers
$X=(x_{1},...,x_{n})$, and lattice points of $N$ describe a linear map
$f:M\rightarrow\mathbb{Z}$. There exists a natural pairing between the two
lattices $\langle\cdot,\cdot\rangle:N\times M\rightarrow\mathbb{Z}$, where
$\langle f,X\rangle=f(X)$. We define a polytope $\Delta$ in $M$ as the convex
hull of a finite number of points. Such a polytope is called a lattice
polytope if all its vertices lie on lattice points. The dual polytope
$\Delta^{*}$ is defined as the set of all points $f$ in $N$ such that $\langle
f,X\rangle=f(X)\geq-1$ for all points $X$ in $M$. A polytope is said to
satisfy the interior point (IP) property when it contains only one interior
point at the origin. Let $\Delta$ be a lattice polytope satisfying the IP
property, then $\Delta$ is called reflexive if its dual $\Delta^{*}$ is also a
lattice polytope satisfying the IP property. An example of a reflexive
polytope in 2 dimensions is given in Figure 2.1.1.

Figure 2.1.1: An example 2-dimensional reflexive polytope, with vertices
$\\{(1,0),(0,1),(-1,-1)\\}$.

Motivated by the discovery that CY manifolds can be built from reflexive
polytopes, in 1997 Kreuzer and Skarke devised an algorithm to classify
reflexive polytopes of any dimension [25]. This algorithm consists of two
steps: Firstly construct a set $S$ of “maximal” polytopes, such that any
reflexive polytope is a subpolytope of a polytope in $S$. These maximal
polytopes are defined by a weight system or a combination of weight systems.
Secondly, given $S$, construct all subpolyhedra of all polyhedra in $S$ and
check for reflexivity. The complete classification of 4319 3-dimensional
reflexive polytopes by this algorithm was presented in [26]. The 184,026
weight systems giving rise to 4-dimensional reflexive polytopes was presented
in [27] and the classification of 473,800,776 4-dimensional reflexive
polytopes from these weight systems was given in [28].444Data of refleixve
polytopes obtained by Kreuzer and Skarke’s algorithm can be found on the
website [29]..

Included in the dataset of all 4-dimensional reflexive polytopes is the subset
of polytopes which give rise to the weighted projective space
$W\mathbb{P}^{4}$. There are 7555 such polytopes characterised by a set of
weights $w_{i=0,...,4}$, first computed in [30]. This database was analysed
with ML techniques in [31] and it was discovered that the topological data of
the resulting CYs depend almost linearly on the weights. Furthermore, in [32]
it was shown that supervised ML techniques are able to predict with high
accuracy standard properties of more general polytopes such as the volume,
dual volume, reflexivity, etc. Later on we shall give explicit example from
these two papers.

###  2.2 Amoebae

Amoebae, introduced by Gelfand, Kapranov and Zelevinsky in 1994 [33], are
regions in $\mathbb{R}^{n}$ with several holes and straight narrowing
tentacles reaching to infinity, constructed from polynomials in $n$ complex
variables.

Considering some $n$-dimensional lattice polytope in $\mathbb{Z}^{n}$, one can
construct a complex polynomial in $\mathbb{C}^{n}$, called the Newton
polynomial. Specifically, each polytope point is associated to a monomial term
in the polynomial equation, where each of the $n$ $\mathbb{C}^{n}$ coordinates
in the respective monomial is raised to the power of the respective entry in
the considered point’s lattice coordinates. The amoeba is then generated
through a log projection of the polynomial, using the polar representation of
the complex space coordinates, $z_{j}=e^{(s_{j}+i\theta_{j})}$, such that
$Log:\ z_{j}\longmapsto s_{j}=log|z_{j}|$, overall mapping the polynomial from
$\mathbb{C}^{n}$ to $\mathbb{R}^{n}$. Conversely, the algae map may be
considered onto the $\theta_{j}$ coordinates [34].

As an example, let’s consider the 2-dimensional lattice polytope with
coordinates $\\{(1,0),(0,1),(-1,-1)\\}$, as shown in Figure 2.1.1. To
construct the Newton polynomial, each of these three coordinates contributes a
monomial term defined using the equivalent $\mathbb{C}^{2}$ space
parameterised by $(z_{1},z_{2})$. This gives polynomial equation:
$0=c_{\alpha}z_{1}^{1}z_{2}^{0}+c_{\beta}z_{1}^{0}z_{2}^{1}+c_{\gamma}z_{1}^{-1}z_{2}^{-1}=c_{\alpha}z_{1}+c_{\beta}z_{2}+\frac{c_{\gamma}}{z_{1}z_{2}}$,
where each $c_{i}$ is some complex coefficient. After log projection we get
the resulting amoeba, drawn in Figure 2.2.1.

Figure 2.2.1: An example amoeba constructed as the amoeba projection of the
Newton polynomial for the lattice polytope with vertices
$\\{(1,0),(0,1),(-1,-1)\\}$; varying the polynomial coefficients can change to
hole’s size/existence.

This amoeba projection provides a visualisation tool for examining the modulus
properties of the complex polynomial. Importantly, how the map projects the
polynomial’s topology will change dependent on the choice of coefficients for
the polynomial. The complex polynomial will have a particular genus, and each
of the surface’s holes may or may not be projected under the amoeba map to the
respective amoeba. This leads to certain bounds in the coefficient space,
where different coefficient values will lead to an amoeba with different genus
values, but determining these bounds is computationally demanding. It is well
understood that ML techniques such as CNNs are very good image processing.
This therefore raises the question, could ML determine the bounds in
coefficient space from images of amoeba? This is precisely the topic of [35,
36]. We will discuss the results of this paper later on in this review.

###  2.3 Quivers

In 1972, Gabriel introduced the concept of a quiver [37], which is simply a
multi-digraph consisting of a set of nodes connected by directed edges. This
sparked a plethora of research in many branches of mathematics. Physicists
also discovered that these objects can be used to study gauge theories [38,
39]. They found that the gauge fields and matter content of some quantum field
theories can be neatly encoded in a quiver, where each node is a respective
$U(n_{i})$ symmetry group of the full theory with an implicit vector multiplet
arising from the gauging of the group, whilst each edge is a chiral multiplet
transforming under the symmetries of the two nodes it is connected to. An
example quiver is shown in Figure 2.3.1.

Figure 2.3.1: The trinian quiver, representing an $\mathcal{N}=1$
supersymmetric gauge theory with three $U(N_{i})$ gauge groups, each with an
implicit vector multiplet from gauging, and three chiral multiplets connecting
each pair of gauge groups. For a specific superpotential, and where all of the
$N_{i}=1$, the quiver theory’s moduli space is the toric variety constructed
from the lattice polytope with vertices $\\{(1,0),(0,1),(-1,-1)\\}$.

Infinitely many quivers can be drawn, each equivalent to some supersymmetric
gauge theory. However which of these can describe the universe we live in? To
probe this question we must examine the low-energy behaviour of the theory, at
scales reachable by our experimental technology. During the process of
reducing the energy of a theory there is a limiting point, called an infra-red
fixed point, below which the physics is fixed. One can categorise the infinite
space of potential quivers, into groups based on having the same infra-red
fixed point. Given one quiver gauge theory it is easy to generate another
infra-red equivalent theory through a process known as quiver mutation.

Quiver mutation was first discovered by Seiberg and hence also goes by the
name of Seiberg duality [40]. Given a quiver the process to generate an
equivalent quiver follows a simple series of steps:

  1. 1.

Select a node of the quiver.

  2. 2.

Reverse the direction of all edges incident to that node.

  3. 3.

Where there exists a 2-path passing through the selected node, introduce an
edge to extend this into a 3-cycle.

  4. 4.

Where introduction of edges leads to any 2-cycles, remove the 2-cycles.

  5. 5.

Update the selected node’s gauge group rank by subtracting the original rank
from the sum of the ranks of the gauge groups at the tail ends of any edges
the selected node was at the head end of prior to mutation.

The process of quiver mutation is shown explicitly for an example quiver in
Figure 2.3.2.

Figure 2.3.2: Mutation of an $\mathcal{N}=1$ quiver at the central node to
produce an infra-red equivalent theory, under the quiver-interpreted rules of
Seiberg duality. The node’s rank updates $N_{2}\longmapsto
N_{2}^{\prime}=N_{1}-N_{2}$.

Performing quiver mutation on any node of the quiver will produce an infra-red
equivalent quiver. Hence, all quivers produced from mutation of a given quiver
at each of its nodes are infra-red equivalent, and by extension any mutation
of those quivers are equivalent also. Iteration of the quiver mutation process
then leads to an entire class of equivalent quiver gauge theories, grouped
into a duality tree [41, 42]. For quiver mutation there are importantly two
mutation types: finite-mutation and infinite. For finite-mutation type quivers
exhaustive mutation of all nodes iteratively will only ever generated a finite
number for quivers, whilst for infinite type there will be some sequences of
mutation which will continue to generate new quivers.

While generating new quivers that are equivalent is relatively easy via quiver
mutation, determining whether or not any two given quivers are equivalent can
be very difficult. This leads one to wonder whether modern ML methods would be
able to detect dualities between quivers. This was the focus of the work in
[43], the results of which we will discuss later555The mathematical process of
mutation has roots in another field of mathematics, known as cluster algebras
[44]. The application of ML techniques and network science analysis to cluster
algebra mutation was initiated, in the papers [45, 46]..

###  2.4 Brane Webs

As we mentioned earlier, string theory relaxes the assumption that the
fundamental building blocks of the universe are 0-dimensional particles, and
introduces 1-dimensional strings. In fact, there is no need to stop at
1-dimension, we can consider building blocks of 2, 3 or even higher
dimensions. This is because in string theory we are dealing with 9 spatial
dimensions rather than 3. We refer to such $p$-dimensional objects as
‘$p$-branes’, and so in this terminology a string is also called a 1-brane. In
Type IIB (one of the five versions of string theory), an NS5-brane is a
5-brane that is magnetically charged under the Kalb-Ramond field and the
D5-brane is a 5-brane that is magnetically charged under the Ramond-Ramond
field. A $(p,q)$ 5-brane is then a bound state of $p\in\mathbb{Z}$ D5-branes
and $q\in\mathbb{Z}$ NS5-branes. Several $(p,q)$ 5-branes are allowed to meet
at a point so long as magnetic charge is conserved, i.e.,
$\sum_{i}p_{i}=\sum_{i}q_{i}=0$. We call the system of three or more such
5-branes meeting at a point, a 5-brane web. These brane webs can be
represented by a simple diagram where the $(p,q)$ 5-branes are represented by
line segments, as shown in Figure 2.4.1. These brane webs are interesting to
study because they describe certain quantum field theories at the junction.

(a)

(b)

Figure 2.4.1: Three-leg $(p,q)$ 5-brane webs, for (a) the general case, and
(b) the equivalent web for the trinian quiver theory with moduli space as the
lattice polygon with vertices $\\{(1,0),(0,1),(-1,-1)\\}$.

We can introduce $[p,q]$ 7-branes on which the $(p,q)$ 5-branes end, these are
shown as dots in Figure 2.4.1. Furthermore, there can be more than one $(p,q)$
5-brane ‘hanging’ from a $[p,q]$ 7-brane and therefore we must introduce
further variables $n_{i}$ that represent the number of 5-branes in each leg.
All the information of a 5-brane web with $N$ legs can then be encoded in a
$2\times N$ web matrix

$W=\begin{pmatrix}n_{1}p_{1}&n_{2}p_{2}&\cdots&n_{N}p_{N}\\\
n_{1}q_{1}&n_{2}q_{2}&\cdots&n_{N}q_{N}\end{pmatrix}$ (2.4.1)

An important feature of 7-branes is that they have an $SL(2,\mathbb{Z})$
monodromy associated to them, which is accounted for by the branch cut (dashed
line) in the brane web diagram. This means that when one 7-brane crosses the
branch cut of another 7-brane, the $[p,q]$ charges of the 7-brane crossing the
cut are changed by the monodromy of the other 7-brane. The monodromy matrix of
a $[p,q]$ 7-brane is given as

$M_{[p,q]}=\begin{pmatrix}1+pq&-p^{2}\\\ q^{2}&1-pq\end{pmatrix}.$ (2.4.2)

The position of the branch cut is irrelevant and therefore one can position
the cut of all 7-branes so that they go away from the brane web, such that no
5-branes cross them. The introduction of 7-branes also introduces further
constraints on brane webs. Namely, the self-intersection number $\mathcal{I}$
of the junction must satisfy the constraint $\mathcal{I}\geq-2$, where
$\mathcal{I}$ is defined as

$\mathcal{I}=\left|\sum_{1\leq i<j\leq
L}\text{det}\begin{pmatrix}p_{i}&p_{j}\\\
q_{i}&q_{j}\end{pmatrix}\right|-\sum_{i=1}^{L}[\text{gcd}(p_{i},q_{i})]^{2}$
(2.4.3)

The physics described by a brane web is not unique, in other words, there
exists equivalence relations between webs such that two webs may appear
different but in fact describe the physics. In [47] it was conjectured that
equivalent sets of 7-branes, which are equivalent to brane webs in the three-
leg case, are completely determined by three invariants: $\mathcal{I}$,
$M_{tot}$ and $\ell$. Here $\mathcal{I}$ is the self-intersection number
defined in (2.4.3), $M_{tot}$ is the total monodromy

$M_{tot}=M_{(p_{1},q_{1})}M_{(p_{2},q_{2})}M_{(p_{3},q_{3})},$ (2.4.4)

where $M_{(p_{i},q_{i})}$ is the monodromy of the $i$-th leg defined by
(2.4.2), and $\ell$ is the asymptotic charge

$\ell=\text{gcd}\biggl{\\{}\text{det}\begin{pmatrix}p_{i}&p_{j}\\\
q_{i}&q_{j}\end{pmatrix}\forall i,j\biggr{\\}}.$ (2.4.5)

It turns out that these three invariants are not enough to completely
determine equivalence between brane webs, but still provide a necessary
condition. The authors in [48] trained a neural network to determine whether
or not two three-leg 5-brane webs are “weakly equivalent”, meaning that they
share the same $\mathcal{I}$, $M_{tot}$ and $\ell$. We will present the
results of this paper in the following sections.

##  3 Supervised Learning

###  3.1 Neural Networks

To explain how neural networks work, we begin by introducing the building
block of any neural network; a neuron. A neuron takes in a set of input data
$\\{x_{i}\\}$ and produces a single numerical output $\hat{y}$. We can divide
the function of the neuron into three parts:

  1. 1.

Firstly, each input $x_{i}$ is multiplied by a weight $w_{i}$: $w_{i}x_{i}$.

  2. 2.

Next, all the weighted inputs are summed and a bias $b$ is added:
$\sum_{i}w_{i}x_{i}+b$.

  3. 3.

Finally, the sum is passed through an activation function which produces an
output: $\hat{y}=f(\sum_{i}w_{i}x_{i}+b)$.

A neural network is simply a collection of neurons connected together in a
series of layers. In between the input and output layers are one or more
‘hidden’ layers.

##### Training

The process of training a neural network involves repeatedly calculating the
‘error’ of the model outputs so that the weights and biases can be updated in
order to reduce the error. Computing the error requires a choice of loss
function. Typical loss functions for regression problems are mean absolute
error (MAE)

$MAE=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\hat{y}_{i}|,$ (3.1.1)

and mean square error (MSE)

$MSE=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2},$ (3.1.2)

where $y_{i}$ and $\hat{y}_{i}$ are the true and predicted values
respectively. For classification problems the default loss function is cross
entropy (CE)

$CE=-\sum_{i=1}^{n}y_{i}\log{(\hat{y}_{i})},$ (3.1.3)

for multiclass classification, and binary cross entropy (BCE)

$BCE=-\sum_{i=1}^{n}(y_{i}\log{(\hat{y}_{i})}+(1-y_{i}),\log{(1-\hat{y}_{i})})$
(3.1.4)

for binary classification.

The method by which we change the weights and biases to minimise the loss is
called the optimisation algorithm. The simplest optimisation algorithm is
called gradient descent, which determines which direction the model parameters
should be altered so that the loss function can reach the minima. In the
gradient decent optimisation algorithm, the model parameters $\theta$ are
adjusted as follows:

$\theta_{t+1}=\theta_{t}-\alpha\frac{\partial L}{\partial\theta}$ (3.1.5)

where $\alpha$ is a parameter called the learning rate that controls the speed
and accuracy of training. In mini-batch gradient descent, the training dataset
is divided into smaller batches, and the model parameters are updated after
every batch.

Gradient descent methods of optimisation present some challenges. Firstly,
choosing a value for the learning rate can be difficult. A value that is too
small can lead to painfully slow convergence. On the other hand, a value too
large can cause the loss function to overshoot the minimum and in some cases
diverge. Another problem that arises in gradient descent methods is getting
trapped at local minima. There exist more advanced optimisation methods, such
as Adam [49], that tackle these challenges but we wont explain these here.

##### Evaluation

For regression tasks, typical performance metrics include MAE (3.1.1), MSE
(3.1.2) and the $R^{2}$ score, which is defined as the proportion of the
variance in the dependent variable that is predictable from the independent
variable(s). Therefore, an $R^{2}$ score close to 1 means the regression model
is a good fit, whereas a score close to 0 means the model is a poor fit. The
equation for computing $R^{2}$ is as follows

$R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}},$
(3.1.6)

where

$\bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}.$ (3.1.7)

For classification tasks, the default performance metric is accuracy

$acc=\frac{TP+TN}{TP+FP+TN+FN},$ (3.1.8)

where $TP,FP,TN,FN$ denote the true positive, false positive, true negative,
and false negative counts respectively. Accuracy, however, can sometimes be
misleading. For example, consider the binary classification situation where
the first class occurs 99% of the time and the second class occurs only 1% of
the time. An algorithm that just predicts the first class every time would
give an accuracy score of 99% which looks great but this isn’t actually a good
model. A better metric in this case is Mathew’s correlation coefficient (MCC)

$MCC=\frac{TP\times TN-FP\times
FN}{\sqrt{(TP+FP)\times(TP+FN)\times(TN+FP)\times(TN+FN)}},$ (3.1.9)

which is similar to the Pearson correlation coefficient, where a score of 1
indicates complete agreement between predictions and truth, 0 indicates that
the predictions are no better than random guessing, and -1 indicates complete
disagreement between the predicted and truth. MCC is a better choice of metric
when dealing with classes of different sizes.

In order to obtain an unbiased evaluation of a neural network, one shuffles
and then splits the data into two sets: a training set and a test set. The
usual train:test split is 80:20. The training dataset is used the fit the
model and then the trained model is used to make predictions on the unseen
test set. Cross-validation is another method commonly used to get an unbiased
evaluation, whereby the data is shuffled and then split into $k$ groups, then
each group acts as the test group once and the remaining groups are combined
to create the training set. Each time the model is trained on the training
set, evaluated on the test set and the evaluation scores are recorded. The
mean and and standard deviation of the evaluation scores are then calculated
and used to measure the model performance.

##### Example: Polytope Properties

As introduced in §2.1, lattice polytopes can be used to construct CY
manifolds. The geometric properties of the resulting CY can be extracted
directly from the polytope and its dual. How these properties can be learnt
directly from the input polytopes is probed through the lens of ML in the work
of [32]. Here focus was put on 2 and 3-dimensional polytopes666ML analysis of
the 4-dimensional reflexive polytopes of the Kreuzer-Skarke dataset [28, 50]
is available at [51]., using the database [52]. A selection of simple
properties were investigated, with the aim that future work would develop the
application to more subtle properties777Extension may also consider the non-
compact toric variety construction as performed in [4] using lattice polytope
toric diagrams..

The properties focused on included: volume, dual volume, reflexivity,
Gorenstein index, and codimension, each having a simple interpretation in
terms of the input polytope. The volume is simply the volume of the polytope
in the lattice with an appropriate dimensional normalisation to ensure integer
outputs, whilst the dual volume is the same volume but for the dual polytope
of the input. Gorenstein index is a measure associated to reflexivity, and is
the minimum integer scaling of the dual polytope required such that it becomes
lattice. Finally the codimension dictates the difference in dimension between
the Fano variety and the projective variety it is embedded within. Each of
these properties relies on non-trivial formulas for computation, particularly
those relying on the computation of the respective dual. For example, the
codimension requires computing the Hilbert basis cardinality of cone of the
dual polytope, which is associated to the number of lattice points in and on
the dual polytope. We also note that due to the sparsity of reflexivity of
2-dimensional lattice polygons (only 16 unique reflexive polygons up to
transformation), reflexivity was only considered for 3-dimensional polyhedra.

The first key result of [32] arose from the subtleties in polytope
representation. The standard way to represent a polytope is the vertex list,
however as hinted to previously there are many vertex combinations that give
the same polytope and ultimately the same toric geometry. An example of this
is rotating a polytope through $\pi$ radians about one of the lattice
coordinate axes, the polytope is ultimately the same despite now having
different vertices. To circumvent this representation redundancy the Plücker
coordinates of the polytope were used as input. These coordinates are
$GL(n,\mathbb{Z})$-invariant, and when only considering polytopes whose
vertices generate the integer lattice this representation becomes unique. The
Plücker coordinates are the maximal minors of the vertex matrix kernel, and
although they do remove the $GL(2,\mathbb{Z})$ redundancy they still express
the permutation redundancy in the coordinate ordering – which was used for
data augmentation.

Use of the Plücker coordinates substantially improved learning results, in
some cases reducing the MAE measure by over an order of magnitude. To
exemplify the Plücker coordinate computation we return to the
$\\{(1,0),(0,1),(-1,-1)\\}$ polygon example. This vertex matrix has kernel:
$(1,1,1)$ from adding the three vertices’ coordinates, and since the kernel
matrix has dimension $1\times 3$ the maximal minors are simply the $1\times 1$
entries: $(1,1,1)$.

Training the dense NNs used to learn each of the considered properties from
Plücker coordinate input lead to a mix of results. For both the 2 and
3-dimensional datasets learning the volume and dual volumes lead to MAE scores
of order $\sim 1$, which was surprisingly strong considering the 2-dimensional
volume range was $\sim 500$. These result imply the existence of simple
formulas for the volume and dual volumes from the Plücker coordinates left to
be found in future work. This existence may be intuitively expected due to the
common determinant-nature of volume and minor computations. Reflexivity binary
classification for 3-dimensional polyhedra performed at accuracies $\sim 0.8$;
whilst for the 2-dimensional polygons learning the Gorenstein index had MAE
scores $\sim 5$ and for codimension $\sim 2$ relative to ranges of 29 and 40
respectively.

This disparate scores illustrate the relative functional complexity for
computing each of these properties. Gorenstein index relies on integer
truncation which the NNs struggle with, while the many steps of the
codimension computation seem to be learnt with some non-trivial success. How
these learnt NN functions can be interpreted for mathematical insight is an
prudent endeavour for future work.

##### Example: Amoebae Coefficients

The amoeba projection of complex polynomials, as discussed in §2.2, does not
necessarily preserve the polynomial’s topology. Where the geometry defined by
a polynomial in $\mathbb{C}^{n}$ has genus $g$, the respective amoeba will
have genus $\in\\{0,1,2,...,g\\}$ dependent on the coefficient choices in the
defining polynomial.

The aim of the work in [35, 36] involves the use of a NN architecture to learn
the genus of the respective amoeba given the vector of coefficients for the
polynomial. The classification over the range of genus values up to $g$ was
learnt with accuracies up to $0.99$ and $0.94$ for the simpler
$(\mathbb{P}^{1})^{\times n}$ examples respectively in $n=2,3$ dimensions.
Furthermore, where the set the coefficients were sampled from was less
restrictive, the learning performance lowered. For more complicated examples,
where the $g$ value was higher, the accuracies dropped to around $\sim 0.9$,
still showing strong performance.

These strong results then motivated the use of interpretable ML techniques to
extract the bounds in the coefficient space where the amoeba genus changed,
with satisfactory successes.

####  3.1.1 CNN

Convolutional neural networks (CNNs) are a specific type of NN, where the
connections between layers are restricted to connect neurons to those adjacent
to the equivalent neuron in the next layer. This way the architecture is
explicitly forced to focus on local structure, which in turn has lead to
significant advances in the field of image recognition.

As noted, the architecture is well suited for handling higher-dimensional data
structures, such as images. Hence, where the dimensionality of the objects
representation is important the use of CNNs may be well motivated. This
motivation also applies for graph-theoretic data which is simply represented
with the respective adjacency matrices, although this representation is in
some sense distinctly non-local due to graph isomorphism. The 2-dimensional
nature of complete intersection configuration matrices has also made this
architecture style more desirable when analysing this type of CY manifolds [2,
53, 54].

##### Example: Quiver Mutation

With the application to adjacency matrices in mind, we return to the question
of determining whether two given quivers are equivalent. In standard practise
one would have to exhaustively compute the mutations of one of the quivers
until the other was produced to confirm their equivalence. However as some
duality trees are infinite one could never exhaustively prove theories were
inequivalent. To attack this seemingly infeasible problem the work of [43]
initiates the implementation of tools from ML, more specifically CNNs888The
work in [43] dictates learning results with both the Naive Bayes and CNN
architectures, both performed comparatively so are quoted as CNN results
here., to learn how to identify equivalent theories.

In investigating the effectiveness of the ML architectures in learning whether
a given two quivers are equivalent, the learning performance was also compared
against the mutation type of quivers given. In performing the investigations a
sample of quivers from each type were taken, each quiver with a small number
of nodes such that mutation to a high depth was feasible. Importantly the
finite-mutation type quivers included those constructed from adding
orientations to the edges of simply-laced Dynkin diagrams, which are naturally
all of finite-mutation type.

Three primary investigations were carried out, each set up as a classification
task for the respective ML architectures. In each case the architecture
received the quivers as adjacency matrix data.

The first investigation sampled pairs of quivers from a database of quivers
formed from the combination of two duality trees. Each duality tree was
generated until an approximate number of 500 quivers were produced, then
approximately 20,000 pairs were sampled from all possible choices – ensuring a
balanced dataset of pairs being equivalent and not equivalent. The ML
architecture then performed binary classification on the pairs to determine
whether the two quivers were equivalent. For all combinations of types for the
trees: both finite-mutation, one finite-mutation and one infinite, and both
infinite; the classification was perfect with accuracy and MCC scores of 1.
Validating clearly that the architecture can identify invariants under the
mutation and learn how to identify quivers describing the same infra-red
theory.

The second investigation developed the previous line of inquiry to test the ML
architecture for binary classification on pairs now sampled from a database of
quivers from $>2$ duality trees. Where three trees were used the performance
measures dropped to accuracy $\sim 0.9$ and MCC $\sim 0.8$ for all tested
combinations of types, and where more trees were used the performance further
dropped down to a tested performance of 0.75 accuracy on a database of 6
trees. This implies that whilst the architectures can learn the equivalence
under mutation this becomes more difficult with more varied and sparser data,
as perhaps expected from standard data science intuition.

The third and final investigation performed the multiclassification directly.
In the direct multiclassification the CNN would receive a adjacency matrix of
a single quiver and have to assign it a label attributed to the duality tree
it came from. The performance reached accuracies and MCCs $\sim 0.9$ when
classifying between three finite-mutation type trees, and between a set of
three trees where two were finite-mutation and the other infinite. However
performance was not strong between three infinite type trees, dropping to
$\sim 0.4$ performance measures.

Further investigation extensions included enhancing the data with the gauge
group rank information of the quiver gauge theory which marginally improved
performances; extrapolating prediction to quivers many mutations away from the
low-mutation depth set trained on which performed well with accuracies $\sim
0.75$; and identifying quivers from a set including random antisymmetric
matrices with accuracies up to 1.

##### Example: Amoebae Images

Practically, the analytic computation of an amoeba is computationally
expensive. To bypass this they are often visualised through a Monte Carlo
sampling, whereby points on the complex polynomial are sampled and projected
onto the real space. However, due to the non-linearity of the projection,
creating a uniformly sampled amoeba image is exceptionally difficult. This
easily leads to erroneous identification of the amoeba genus, where certain
parts of the amoeba body are poorly sampled; and with a desire to identify the
genus for a large dataset of coefficients, computational techniques and ML
become a sensible route for streamlining the process.

In the work of [35], CNNs are used to learn the genus from the Monte Carlo
generated images directly, avoiding the need for expensive analytic
computation. Focusing on the prototypical $\mathbb{P}^{1}\times\mathbb{P}^{1}$
surface, leading to amoebae with genus values at most 1, the CNN binary
classifiers could reach accuracies up to $0.99$ and MCC scores up to $0.97$.
In performing the classification a coarse-graining of the filter was first
applied to the images, in the spirit of the pooling kernel layers typical for
CNN architectures, and the resolution reduction interestingly had an optimum
value where local structure was sufficiently averaged to smooth out the non-
linear sampling.

####  3.1.2 SNN

Siamese Neural Networks (SNNs), first introduced in [55] in the context of
signature verification, are neural network architectures consisting of two or
more identical sub-networks that determine the similarity of inputs. The sub-
networks $f_{w}$ map elements of a dataset $\mathcal{D}$ to points in
$\mathbb{R}^{d}$, where $w$ denotes the weights and biases of the network. The
goal is to train the network such that after training similar elements of
$\mathcal{D}$ are mapped close together in $\mathbb{R}^{d}$, and dissimilar
elements are mapped far apart. The weights and biases are determined by
extremising a loss function that depends on the squared Euclidean distance
between the embeddings $||f_{w}(W_{1})-f_{w}(W_{2})||^{2}$.

SNNs are commonly trained using the triplet loss, which takes an anchor (A),
positive (P) and negative (N) input triple, where the anchor-positive inputs
are similar and the anchor-negative inputs are dissimilar. If the network is
performing well, the distance between the anchor and positive embeddings will
be smaller than the distance between the anchor and negative embeddings:

$||f_{w}(A)-f_{w}(P)||^{2}\leq||f_{w}(A)-f_{w}(N)||^{2}.$ (3.1.10)

Equivalently, we could write

$||f_{w}(A)-f_{w}(P)||^{2}-||f_{w}(A)-f_{w}(N)||^{2}\leq 0.$ (3.1.11)

This inequality is satisfied if the $f_{w}$ always outputs a constant value,
say 0. To prevent the network from learning this, we modify the objective by
introducing a hyperparameter $\alpha\in\mathbb{R}_{>0}$:

$||f_{w}(A)-f_{w}(P)||^{2}-||f_{w}(A)-f_{w}(N)||^{2}+\alpha\leq 0$ (3.1.12)

The triplet loss function is then defined as:

$\mathcal{L}=\max{(||f_{w}(A)-f_{w}(P)||^{2}-||f_{w}(A)-f_{w}(N)||^{2}+\alpha,0)}$
(3.1.13)

This is the loss of a single triplet, the overall loss function of the network
is then the average of these individual losses over a batch of $\mu$ triplets:

$J=\frac{\sum_{i=1}^{\mu}\mathcal{L}(f_{w}(A^{(i)}),f_{w}(P^{(i)}),f_{w}(N^{(i)}))}{\mu}$
(3.1.14)

Using the triplet loss function, a network is made up of three identical sub-
networks that produce three embeddings for the anchor, positive and negative
inputs. The outputs of these sub-networks are fed to an external output layer
that computes the triplet loss. Figure 3.1.1 show the architecture of a
typical SNN trained with triplet loss.

Figure 3.1.1: Siamese neural network diagram using triplet loss.

##### Example: Brane Webs

It is difficult to determine by hand whether two 5-brane webs are equivalent
or not. It is therefore an interesting question to ask whether a neural
network could determine whether two brane webs are equivalent or not. This is
exactly what the authors in [48] tested for three leg brane webs using an SNN.

An SNN was trained, using triplet loss, to determine whether two webs are
equivalent, where the input data was in the format of $2\times 3$ web matrices
(2.4.1). After training on triplets taken from the training set, the trained
subnetwork was then used to produce an embedding for every web in the
remaining test set. Two webs taken from the test set were predicted to be
equivalent if the squared Euclidean distance between their embeddings was less
than some threshold value and inequivalent otherwise. The result of these
pairwise equivalence predictions was that the SNN was able to detect
equivalence between webs taken from two equivalence classes with 100%
accuracy.

###  3.2 Support Vector Machines

Support vector machines (SVMs) are simple architectures designed specifically
for binary classification of high-dimensional data in $\mathbb{R}^{n}$. The
SVM seeks to optimise the position of an $(n-1)$-dimensional linear hyperplane
in $\mathbb{R}^{n}$ which best separates the two classes. The linearity
requirement for the hyperplane can be dropped through the use of kernel
methods, which introduce non-linear terms into the hyperplane equation by
embedding the data in a $\tilde{n}$-dimensional space (for $\tilde{n}>n)$
where the extra dimensions correspond to nonlinear terms, then performing the
linear hyperplane optimisation there and projecting back down to the original
$n$-dimensions.

Since an explicit hyperplane is generated through the SVM algorithm this
hyperplane has the potential for direct interpretation in the physical context
of the mathematical data; as well as simple interpretable extrapolation of the
classification to objects beyond the sampling range.

##### Example: Calabi-Yau Threefolds

The Calabi-Yau threefold database constructed from hypersurfaces in
$\mathbb{P}^{4}$ is represented by a set of 7555 5-vectors of weights [30]. In
the work of [31], to probe the importance of each necessary condition on these
5-vectors, conjugate datasets which did not satisfy each condition were
generated.

A selection of supervised ML architectures were then used to binary classify
whether a given 5-vector of weights (and hence a given 4-dimensional weighted
projective space) admitted a CY threefold hypersurface. Despite NNs performing
the best, simpler architectures were concentrated on due to their
interpretability. Notably including the use of SVMs.

SVMs could differentiate 5-vectors which admitted a CY hypersurface with
accuracies up to $0.75$. The consistency across the binary classification of
the CY dataset against each of datasets with progressively more necessary
conditions included indicated that the CY property occurred more subtly in the
5-vectors than for the properties considered.

Further investigation of the correlation between the SVM misclassifications
and the CY Hodge numbers, indicated which of the non-CY datasets used for
training made the classifier better over different ranges of these topological
Hodge numbers. In particular, if the transverse property999For the 5-vector’s
weights $w_{i}$ the transversity property implies that $\forall w_{i}\ \exists
w_{j}\ s.t.\ \frac{\sum_{k}(w_{k})-w_{j}}{w_{i}}\in\mathbb{Z}^{+}$. was not
included in the non-CY dataset the classifiers performed perfectly across the
high $h^{2,1}$ range, while if it was included they instead only performed
perfectly across the high $h^{1,1}$ range. These unexpected correlations
indicate the intimate link between transversity of a weighted projective space
and the Hodge number of the subsequent hypersurfaces. A relationship that
would be interesting to solidify in future work.

##  4 Unsupervised Learning

The program of mathematical research often follows the paradigm of conjecture
formulation followed by proof. The first step of conjecture formulation
involves testing example cases of the examined mathematical objects to
identify phenomena which they appear to obey. However to be confident in a
conjecture a large number of examples must be confirmed for sensible
extrapolation of the conjectured relation. This step lends itself naturally to
techniques from big data, not only for the processing of large sets of
examples, but also for the identification of the patterns which lead to the
conjectures.

Unsupervised learning includes a large class of techniques, perhaps most
importantly including methods for processing datasets to extract patterns and
fundamental degrees of freedom. The unsupervised subfield of machine learning
can hence be further subdivided into techniques for feature extraction which
identify the most significant and relevant parts of the data, and techniques
for clustering which appropriately groups the data and assesses similarities
and symmetries.

Each of these classes of methods have spirit manifestly derived from the
scientific method in physics. To truly understand the universe we live in we
require efficient prediction, which relies on knowledge of the relative
importance’s of each component in a calculation – naturally associating to
feature extraction. Whilst to most appropriately develop intuition for
extension of theorised models an understanding of the theory’s symmetries
helps reduce the amount of work required – naturally extractable from
clustering methods.

In this section, we discuss how some of the fundamental techniques within
unsupervised learning are applied to selected problems in high-energy theory.
The feature extraction methods include principal component analysis (PCA);
t-distributed stochastic neighbour embedding (t-SNE); and topological data
analysis (TDA); all useful for analysis of the datasets of mathematical
objects and their visualisation. Additionally, the application of the
prototypical clustering method, $k$-Means clustering, is discussed to aid with
classification of the respective objects.

###  4.1 Principal Component Analysis

Approximation is an integral part of computation in physics. Crucial to
accurate approximation is the identification of the importance of each
contributing term to a calculation. Principal component analysis, or PCA,
provides a method to quantify the relative importance of data features.
Specifically, for a vector representation it identifies the linear
combinations of the vectors’ entries which best describe the data’s variation.

Also important for data representation is the independence of the components
used, a property that is implicitly enforced in the PCA method of covariance
matrix diagonalisation. Independence of components ensures that the components
used are truly fundamental, and that the physical representation is maximally
efficient for the simplicity of representation. This effect is ubiquitous in
physical theory, perhaps most commonly introduced through the normal mode
representation for string vibration.

PCA therefore provides a method of linear feature extraction of objects
represented by vectors, and the linearity restriction can be further
generalised to non-linear combinations with kernel methods akin to those used
for SVMs; as discussed in §3.2.

##### Example: Calabi-Yau Threefolds

The weighted $\mathbb{P}^{4}$ database of 5-vectors admitting a CY
hypersurface of [30], analysed in [31], is too high-dimensional for direct
visualisation. Since the 5-vector’s weights have no implicit logical order,
they are naturally listed in ascending order. This leads to the most dominant
components for the variation naturally being the later entries. This behaviour
was reflected in the PCA decomposition of the datasets considered.

For the 4 datasets considered: (a) random 5-vectors, (b) coprime 5-vectors,
(c) transverse 5-vectors, (d) CY dataset; the most dominant principal
component in each case had normalised eigenvalue entries of: 0.76, 0.75, 0.91,
0.98 respectively. These indicate that a 1-dimensional projection would be
sufficient for preserving the majority of the data, however to consistently
account for $>0.9$ of the data information a 2-dimensional projection was used
in each case. Despite the latter entries dominating these principal components
as expected, the lack of a block diagonal form for the covariance matrices
indicated a slightly more subtle structure using all the vector information.
These projections are shown in Figure 4.1.1, where the cone-like structure is
an artefact of the vector sorting, however the separation into linear clusters
in the CY dataset case is unexplained and surprising.

This linear clustering of the CY weights was a novel fundamental structure,
and its lack of occurrence in the conjugate non-CY datasets highlights its
uniqueness to the CY property. Further analysis in this work showed consistent
linear correlation to the cluster separation formed by plotting the weights
(especially the largest weight), against the hypersurface’s $h^{1,1}$ value,
as shown in Figure 4.1.2. These PCA results hence nicely corroborate the
existence of a more direct connection to the hypersurface’s $h^{1,1}$ value
within these linear clusters, and petitions for investigation into the
fundamental structure behind this apparently simple linear clustering
behaviour.

(a) Random 5-vectors

(b) Coprime 5-vectors

(c) Transverse 5-vectors

(d) CY 5-vectors

Figure 4.1.1: 2-dimensional PCA plots for the 4 considered datasets. As more
of the conditions are added, more structure appears, in particular there is
some form of distinct class separation for the CY weights. Figure 4.1.2: The
equivalent linear clustering behaviour of the Calabi-Yau weights with the
$h^{1,1}$ Hodge number corroborated by the PCA results.

###  4.2 t-SNE

t-distributed stochastic neighbour embedding, or t-SNE [56], is a
dimensionalilty reduction method that reduces high-dimensional data by
embedding it in a low-dimensional space. It calculates similarities between
data points in the high dimensional space and in the low dimensional space and
tries to minimise the divergence between the two. The process can be broken
down into three steps:

  1. 1.

Given a set of $N$ points, $\textbf{x}_{1},...,\textbf{x}_{N}$ in high
dimensional space, compute the similarities between pairs of points. The
similarity of data point $\textbf{x}_{i}$ to data point $\textbf{x}_{j}$ is
defined as:

$p_{ij}=\frac{p_{j|i}+p_{i|j}}{2N}\,,$ (4.2.1)

where $p_{j|i}$ is the conditional probability that $\textbf{x}_{i}$ would
pick $\textbf{x}_{j}$ as its neighbour. This probability is proportional to
the probability density under a Gaussian centred at $\textbf{x}_{i}$:

$\displaystyle
p_{j|i}=\begin{cases}\frac{\text{exp}(-||\textbf{x}_{i}-\textbf{x}_{j}||^{2}/2\sigma_{i}^{2})}{\sum_{k\neq
i}{\text{exp}(-||\textbf{x}_{i}-\textbf{x}_{k}||^{2}/2\sigma_{i}^{2})}}&\text{if
}i\neq j\\\ 0&\text{if }i=j\end{cases}\;,$ (4.2.2)

where $\sigma_{i}$ are the standard deviations.

  2. 2.

In the low dimensional space compute the similarity measures between data
points, but instead of using a Gaussian distribution use a Student’s
t-distribution with one degree of freedom. In this case, the similarity of
data point $\textbf{y}_{i}$ to data point $\textbf{y}_{j}$ is defined as:

$\displaystyle
q_{ij}=\begin{cases}\frac{(1+||\textbf{y}_{i}-\textbf{y}_{j}||^{2})^{-1}}{\sum_{k\neq
l}{(1+||\textbf{y}_{k}-\textbf{y}_{l}||^{2})^{-1}}}&\text{if }i\neq j\\\
0&\text{if }i=j\end{cases}\;.$ (4.2.3)

  3. 3.

Compute the Kullback-Liebler divergence between the two probability
distributions $p_{ij}$ and $q_{ij}$:

$\displaystyle D_{KL}(P||Q)=\sum_{i\neq
j}{p_{ij}\log{\frac{p_{ij}}{q_{ij}}}}\;.$ (4.2.4)

This divergence is minimised with respect to the data points $\textbf{y}_{i}$
using gradient descent.

##### Example: Brane Webs

In §3.1.2 we discussed how an SNN was used in [48] to predict whether two
brane webs are equivalent. Recall, that two webs are equivalent if they share
the same self-intersection number (2.4.3), total monodromy (2.4.4) and
asymptotic charge invariant (2.4.5). It would nice to visualise the embeddings
of the webs in the test set to see whether equivalent webs appear to cluster
together as we would like them to. The dimension of the embedding space used
in the SNN was $d=10$ and so t-SNE is applied to the resulting embeddings in
order to reduce the embeddings down to 2 dimensions. The result are shown in
Figure 4.2.1. We can see clearly that the embeddings coming from the two
equivalence classes $\textbf{X}_{1}\cup\textbf{X}_{2}$ cluster together into
two distinct sets. This aligns with the pairwise equivalence prediction result
we saw earlier in §3.1.2.

Figure 4.2.1: t-SNE plot of the 10-dimensional web embeddings generated by the
SNN, reduced to 2 dimensions.

###  4.3 K-Means Clustering

$k$-means clustering is a clustering algorithm that groups data points into
$k$ clusters, where each data point belongs to the cluster with the nearest
centroid. The objective is to minimise the sum of squared distances between
the data points and the nearest centroid, known as the inertia $\mathcal{I}$.
In other words, given a set of data points
$(\textbf{x}_{1},...,\textbf{x}_{n})$, the $k$-means clustering algorithm
partitions the data points into $k$ sets $\textbf{S}=(S_{1},...,S_{k})$ so as
to minimise the sum:

$\displaystyle\mathcal{I}=\sum_{i=1}^{k}{\sum_{\textbf{x}\in
S_{i}}{||\textbf{x}-\mu_{i}||^{2}}}\,,$ (4.3.1)

where $\mu_{i}$ denotes the centroid of the set $S_{i}$.

The first step is to choose $k$, i.e. the number of clusters. The next step is
to randomly select $k$ points from the data and label them as centroids. One
then assigns each data point to the cluster with the closest centroid. Once
data points are sorted into clusters, the new centroid of each cluster is
computed and again the data points are assigned to the nearest cluster. This
two step process of computing the new centroid and then assigning data points
to clusters is repeated until the centroids do not change their position.
Where this input $k$ number of clusters is unknown, an elbow method can be
implemented whereby the clustering is run until completion for a range of
integer values and the final inertia is added to a penalty factor for the
number of clusters used; leading to a convex optimisation to easily extract
the discrete minimum $k$ value for the optimum number of clusters.

One common metric used to evaluate clustering performance is Rand index (RI).
RI measures the similarity between two partitions by considering all pairs of
elements and counting pairs where both elements are assigned to the same
subset, or different subsets. The two partitions compared are the predicted
partition and the true partition. Let $X=\\{x_{1},...,x_{n}\\}$ be a set of
$n$ elements and $A=\\{A_{1},...,A_{r}\\}$, $B=\\{B_{1},...,B_{s}\\}$ be two
partitions of $X$ into $r$ and $s$ subsets respectively. Then the Rand index
is given by

$\displaystyle RI=\frac{a+b}{a+b+c+d}\;,$ (4.3.2)

where

  * •

$a$ is the number of pairs of elements in $X$ that are in the same subset in
$A$ and the same subset in $B$.

  * •

$b$ is the number of pairs of elements in $X$ that are in different subsets in
$A$ and different subsets in $B$.

  * •

$c$ is the number of pairs of elements in $X$ that are in the same subset in
$A$ and different subsets in $B$.

  * •

$d$ is the number of pairs of elements in $X$ that are in different subsets in
$A$ and the same subset in $B$.

Therefore RI exists in the range [0,1], where 1 corresponds to a perfect
agreement between the partitions and 0 corresponds to complete disagreement.

##### Example: Brane Webs

Since the idea of an SNN is to map elements in a dataset to an embedding
space, such that similar elements are mapped close together in the embedding
space and dissimilar elements are mapped far apart, it makes sense to apply
$k$-means cluster on the embeddings in order to group the data into
equivalence classes. This is precisely what the authors in [48] did to the
embeddings of 5-brane webs produced by the SNN, see §3.1.2 for more details.
Setting $k=2$ the Rand index score of the results of $k$-means clustering was
$1.0000\pm 0.0000$ which means that the $k$-means clustering matched perfectly
with the true clustering. These results also support the conclusions made from
the t-SNE plot in Figure 4.2.1.

##### Example: Calabi-Yau Threefolds

The linear clustering behaviour observed in the PCA and Hodge plots, Figures
4.1.1 and 4.1.2 in §4.1, advocates for the application of unsupervised
clustering methods. The authors of [31] chose to use $k$-means to analyse this
clustering behaviour.

Since each cluster in Figure 4.1.2 appears to lie on a line through the
origin, the data can be projected down onto the ratio $h^{1,1}/w_{5}$ to
create a 1-dimensional clustering problem for the K-Means algorithm. Running
the $k$-means elbow method dictates that there is an optimum number of 10
clusters, matching the observed number from the plots. In addition, the
normalised inertia value with respect to the number of vectors considered and
the range of ratios across this dataset took a value of $0.00084$; indicating
that each ratio was $<0.1\%$ of the ratio-range away from its nearest cluster,
corroborating the existence of a linear clustering in these CY 5-vectors.

###  4.4 Topological Data Analysis

Data visualisation is useful for the development of one’s intuition when
working with a dataset. However, our ability to picture data is limited to at
most three dimensions. This motivates the use of techniques that project data
to a lower dimension, such as PCA (§4.1) and t-SNE (§4.2). Any data projection
is inevitably going to lose some information and hence some perspective for
data interpretation.

Topological data analysis (TDA) is the name given to the field of techniques
that analyse the topological properties of a higher-dimensional datasets
without projection. This direct analysis is in a sense more physical, and
naturally evades the errors that occur with more standard visitation. The most
prominent technique within TDA uses persistent homology. Here, the
$n$-dimensional data in $\mathbb{R}^{n}$ space is used to build a filtration
of simplicial complexes on which persistent homology can be applied. The
computation process initially generates an equivalent simplicial complex
analogous to the data distribution with a 0-simplex (point) for each
datapoint. Then it centres an $n$-dimensional ball on each data point of
radius $\delta$ and smoothly increases the value of $\delta$ from 0 to
$\infty$. At each new $\delta$ where there is a new intersection of $k\leq n$
balls an equivalent $k$-simplex is drawn in the simplicial complex between the
respective 0-simplices; creating a discrete filtration of complexes over the
range of $\delta$ values.

The algorithm keeps track of the persistent features of each dimension
throughout the filtration, whereby a $H_{k}$ feature is a set of $k$-simplices
which do not form the boundary of a set of $(k+1)$-simplices. The dominant
persistent features which exist for a large range of $\delta$ values indicate
intrinsic structure to the data. These features can be represented on a
persistence diagram, which is a 2-dimensional plot with coordinates (birth,
death), where ‘birth’ is the $\delta$ value the feature was created in the
filtration, and ‘death’ the $\delta$ value that it disappeared / was filled.

##### Example: Amoebae Images

Considering amoebae in dimension $n$, the most relevant topological properties
are the existence of $H_{n-1}$ features counted by the relevant Betti number
$b_{n-1}$. In the 2-dimensional examples considered in [35] these are $H_{1}$
features which are 2-dimensional holes, for the 3-dimensional examples
considered in [36] these are $H_{2}$ features which are 3-dimensional
cavities. Their existence is identified through TDA’s persistent homology by
the presence of persistent features far from the diagonal of the amoeba’s
persistence diagram.

In [35] persistent homology was used to reanalyse a set of misclassified
amoeba, and the procedure managed to classify the genus more accurately than
the CNN architectures on these more difficult examples. In [36] the persistent
homology process was tested more thoroughly as the automated computation
measure for determining the label ahead of the CNN learning. Although the
computation of the persistent features is expensive, the procedure is simply
automatable, and thus more practical for implementation than the heuristic
identification of the suitable bounds using lopsidedness. An example Monte
Carlo sampled 3-dimensional amoeba image is given in Figure 4.4.1, along with
its respective persistence diagram – demonstrating the correct identification
of a $H_{2}$ feature where $b_{2}=1$.

(a) Monte Carlo sampled amoeba

(b) Amoeba persistence diagram

Figure 4.4.1: For the $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
geometry, the respective 3-dimensional amoeba is shown in (a) where the points
are sampled from the complex polynomial and projected to the amoeba. In (b)
the equivalent persistence diagram is shown for the same sampling, where the
green dot far from the diagonal ($\sim(10.6,14.6)$) indicates a $H_{2}$
feature.

##  5 Conclusion

Through a selection of research works, this chapter has exemplified the
successes of machine learning methods on problems within physics and
geometry.As computational power and proficiency continues to develop, the
abundance of mathematical data will further the demands for statistical
methods of analysis, particularly the range of techniques available under the
umbrella of machine learning.

The techniques of focus here have been those of simpler structure, and
motivate the use of more advanced (and appropriately specialised) methods for
development of the analysis, and extraction of concrete mathematical insights.
Whilst this work highlights the initial successes of ML in the field of
physics, it is also worth noting that there are now multiple programs of work
to apply ideas from quantum field theory and high-energy physics to understand
the ubiquitous successes of NNs [57, 58, 59].

As mathematical physics and machine learning continue to intertwine, their
breadth of cross-application will no doubt lead to exciting insightful mutual
development.

## Acknowledgement

YHH would like to thank STFC for grant ST/J00037X/2. E. Heyes would like to
thank SMCSE at City, University of London for the PhD studentship, as well as
the Jersey Government for a postgraduate grant. E. Hirst would like to thank
STFC for a PhD studentship.

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