# Comparative visualization of epidemiological data during various stages of a
pandemic

Thomas Kreuz<EMAIL_ADDRESS>Institute for Complex Systems, CNR, Sesto
Fiorentino, Italy

###### Abstract

After COVID-19 was first reported in China at the end of 2019, it took only a
few months for this local crisis to turn into a global pandemic with
unprecedented disruptions of everyday life. However, at any moment in time the
situation in different parts of the world is far from uniform and each country
follows its own epidemiological trajectory. In order to keep track of the
course of the pandemic in many different places at the same time, it is vital
to develop comparative visualizations that facilitate the recognition of
common trends and divergent behaviors. Similarly, it is important to always
focus on the information that is most relevant at any given point in time. In
this study we look at exactly one year of daily numbers of new cases and
deaths and present data visualizations that compare many different countries
and are adapted to the overall stage of the pandemic. During the early stage
when cases and deaths still rise we focus on the time lag relative to the
current epicenter of the pandemic and the doubling times. Later we monitor the
rise and fall of the daily numbers via wave detection plots. The transition
between these two stages takes place when the daily numbers stop rising for
the first time.

##  I  Introduction

Human cases of COVID-19, the disease caused by the novel coronavirus Severe
Acute Respiratory Syndrome Corona-Virus 2 (SARS-CoV-2), were first reported in
Wuhan City, China, in December 2019 (Huang et al., 2020). After initial
transmissions were restricted to Central China’s Hubei province, already by
January 2020 first cases had been reported not only in other Asian countries
(starting with Taiwan, South Korea and Japan) but also in Australia, the US
and several European countries (World Health Organization: Coronavirus disease
2019 Situation Report 11, January 31, 2020(2020), COVID-19). Similarly to what
had happened before in China (Zhu et al., 2020; Guan et al., 2020), within
February 2020 first cases turned into first deaths (Zhou et al., 2020) and
South Korea, Iran and increasingly Italy emerged as early hotspots outside of
China where the epidemic now seemed to be under control (World Health
Organization: Coronavirus disease 2019 Situation Report 40, February 29,
2020(2020), COVID-19). On March 11, 2020, the Director General of the World
Health Organisation (WHO) declared the novel coronavirus outbreak a worldwide
pandemic (World Health Organization: Coronavirus disease 2019 Situation Report
51, March 11, 2020(2020), COVID-19). A few days later Italy became the clear
global epicenter as the first country to surpass China in number of deaths
(World Health Organization: Coronavirus disease 2019 Situation Report 60,
March 20, 2020(2020), COVID-19). Since then the coronavirus has spread across
the globe with an unprecedented impact on healthcare (Shamasunder et al.,
2020), economy (McKibbin and Fernando, 2020), finances (Zhang et al., 2020),
science (Palayew et al., 2020; Myers et al., 2020), education (Marinoni et
al., 2020), travel (Škare et al., 2020), sports (Garcia-Garcia et al., 2020),
mental health (Dubey et al., 2020) and basically all other sectors of society.

Already on January 22, 2020, the Center for Systems Science and Engineering
(CSSE) at John Hopkins University (Baltimore, MD, USA) started publishing a
freely available COVID-19 Data Repository that was updated daily (Dong et al.,
2020). Once global datasets like this one became publicly available, the
scientific community sprang into action and within a short time a host of
studies appeared that focused to a large extent on modeling the data and using
these models to predict the future course of the pandemic (e.g., (Fanelli and
Piazza, 2020; Flaxman et al., 2020; Dehning et al., 2020; de Sousa et al.,
2020)).

While modelling the Covid-19 pandemic has already yielded some important
results and policy recommendations (Sy et al., 2020; Moon, 2020; Jia and Lu,
2020) and much work still remains to be done (Vespignani et al., 2020; Prakash
et al., 2020), extrapolation based on often incomplete, inaccurate or
unreliable data does also have its limitations and pitfalls (Sridhar and
Majumder, 2020). In this article we refrain from making inferences about the
future but rather restrict ourselves to pure visualization of past and present
data (Callaghan, 2020). The aim, in a nutshell, is to develop a simple and
consistent comparative data visualization framework that is general and
adapted to the various stages of a pandemic, thereby summarizing in one sweep
the dynamic and heterogeneous situation worldwide.

First, we focus on visualizations that allow a meaningful comparison of the
course of the pandemic for many different countries (or on smaller spatial
scales: states, regions etc.) at the same time. This is in contrast to the
commonly used histograms in which the temporal profile of the data is plotted
for one country at a time. Second, we argue that the most relevant information
to be found in the data changes between the early and the later stages of a
pandemic and accordingly we present two different kind of data visualizations.

In the early stage (as long as cases and deaths continue to rise) it is most
important to monitor both the time lag compared to the current epicenter
(typically the country where the epidemic first took hold but later this can
shift) and the severity of the spread of the disease (usually expressed by
means of the doubling time). Both of these quantities provide very useful
information about the urgency of the situation (The Lancet, 2020) and can help
with general decision making in order to find the right moment for the
implementation of preventive measures (Phua et al., 2020; Kennedy et al.,
2020).

On the other hand, the later stages are more about monitoring the course of
the pandemic in each country in terms of peaks, valleys and plateaus. This is
when there is often a back and forth between imposing, tightening and relaxing
of contact restrictions depending on the trajectory of the epidemiological
dynamics in the population at that point in time (Goldsztejn et al., 2020;
Charpentier et al., 2020). The transition between these two stages takes place
when the daily numbers of cases and deaths stop rising for the first time
(Leung et al., 2020).

The remainder of this article is organized as follows: First in Section II we
describe the dataset and the preprocessing performed. The two Method Sections
III.1 and III.2 illustrate the quantities, graphs and sorting criteria we will
use to visualize the data in the early and the later stages of the pandemic,
respectively. In Sections IV.1 and IV.2 we show the data plots for $42$
selected countries from all over the world as well as two more local examples,
the US states and the regions of Italy. Finally, in Section V we summarize and
conclude.

##  II  Data

Figure 1: The dataset: Absolute numbers of cumulative deaths (a) and cases (b)
versus time for 42 selected countries and the one year interval from January
22, 2020 to January 21, 2021. This plot like all subsequent plots that show
cumulative numbers, uses a log scale to facilitate the comparison also between
countries at different stages of their pandemic course. The legend lists the
countries and their cumulative number of cases and deaths after one year,
respectively. Source: John Hopkins COVID-19 Data Repository.

Dataset: We illustrate our data visualizations using the freely available
dataset from the COVID-19 Data Repository by the Center for Systems Science
and Engineering (CSSE) at Johns Hopkins University (Github Webpage:
https://github.com/CSSEGISandData/COVID-19) (Dong et al., 2020). We here use
the daily cumulative data for both cases and deaths right from its first
publication on January 22, 2020, until January 21, 2021, thus covering exactly
one year of data. We selected $42$ representative countries focusing on Europe
and larger countries in other parts of the world. The data for the US states
were available within the same repository. The data for the Italian regions
can be found on the webpage https://github.com/pcm-dpc/COVID-19. The data from
Liguria were not available and this region was thus not included. The
population sizes used in the normalization were taken from the webpage
https://www.worldometers.info/coronavirus/ (data as of January 21st, 2021).

Preprocessing: Datasets of both the overall number of cases and deaths up to a
certain date can be expected to increase monotonically with time, but not
strictly monotonously since there might be days without any new cases or
deaths. However, occasionally some of the datasets do contain negative jumps
from one day to the next, typically due to elimination of double counting or
other kinds of retrospective reevaluations such as fundamental changes in the
way cases and deaths were defined (see the COVID-19 Data Repository (Dong et
al., 2020) for details). To clean the data and eliminate these negative jumps
we follow the reasonable assumption that later data are more correct than
earlier data (after all that is what corrections are for) and each time
decrease all the spuriously high early data points to the corrected later
value. As a positive side effect this smoothing also eliminates most of the
plateaus that were created due to the aforementioned corrections, and this
helps in the later calculations of the doubling times. Finally, we apply a
moving average of order 7 days in order to smooth out weekday variations such
as the tow-day dips that often occur due to reporting delays on weekends
(Bergman et al., 2020).

The cumulative absolute (not normalized by population) number of deaths and
cases for the selected $42$ countries and the year from January 22, 2020, to
January 21, 2021, obtained in this way are shown in Fig. 1a and Fig. 1b,
respectively.

##  III  Methods

Figure 2: Definition of time lags: The monotonously increasing thick curves
depict normalized number of deaths (per one million inhabitants) versus time
for three selected countries during the time interval from February 23 to
March 30, 2020. Horizontal lines illustrate the changing time lag of Spain
(red) and the US (blue) with respect to the pandemic course of Italy (thick
black curve), first for March 11, 2020 (label 1 and thin lines on the left)
and then for the end of March (label 2 and thick lines on the right). During
these twenty days the time lag between Spain and Italy decreased from $-9$ to
$-3$ days, while the time lag between the US and Italy increased from $-17$
days to $-19$ days. The time lags on March 31, together with the number of
deaths up to that day, are reported in the legend.

In the Method Section we illustrate the various data visualization plots using
the number of deaths as an example. Deaths tend to be more reliable
(Subbaraman, 2020) since they are not affected by the number of tests
performed which itself depends on a variety of factors, not only the number of
either symptomatic or essential people (compare, e.g., Johansson et al.
(2021); Pullano et al. (2021)) but also healthcare system capacities and
political decisions (Callaghan, 2020). However, for completeness, in the
Results Section IV we also show two plots based on the number of cases.

###  III.1  Early stages of a pandemic

In the early stages when cases and deaths are still in their first rising
phase, it is important to monitor the initial spread of the pandemic (Phua et
al., 2020; Kennedy et al., 2020): How far behind are different countries
compared to the country where the epidemic started and how fast is the current
spread in each country? The most relevant quantities are the time lag with
respect to the current epicenter of the pandemic and the doubling time.

####  III.1.1  Time Lag

In Fig. 2 we show the course of the number of deaths for three selected
countries (Italy, Spain and the US) during the initial period of the pandemic,
from the first reported death in any of these three countries to the end of
March. For better comparability the numbers were normalized to the overall
population of the respective country.

Italy was the country in the Western hemisphere with the earliest onset of an
epidemic (Nacoti et al., 2020; Indolfi and Spaccarotella, 2020) and also the
first Western country with an officially recorded COVID-19 fatality which, as
the graph shows, occurred on February 23, 2020, eight days earlier than in the
US and eleven days earlier than Spain. In order to compare the course of the
epidemic in different countries it thus makes sense to use Italy as the
reference country.

The first important information to know is by how many days your country lags
behind Italy’s curve or, at a later stage, after a potential reversal of
fortune, by how many days it is ahead in its epidemic course. To this aim, we
define the time lag of a country with respect to Italy as follows:

  1. 1.

Start with the last data point of the country and then check when Italy’s
curve crossed this value.

  2. 2.

Do vice versa in case Italy is behind.

Figure 3: Doubling times of number of deaths for the same three countries and
the same time interval used in Fig. 2. All three curves show quite large
fluctuations early but are much smoother after that. At the end of March the
doubling time was longest for Italy followed by Spain and the US. However,
they were all trending towards larger values corresponding to a slowing down
of the exponential growth in numbers of deaths.

In Fig. 2 this is illustrated for three different countries, using the end of
March (marked as thick dashed vertical line 2) as an example. By construction
the time lag of Italy to itself is always $0$. Spain reached $129$ deaths per
one million inhabitants three days later than Italy. Accordingly, the time lag
is $-3$. The US reached $9$ deaths per one million inhabitants $19$ days later
than Italy. So here the time lag is $-19$.

Note that the time lag defined this way is a time-dependent variable. Instead
of extracting via some kind of double fitting one value for the pair of two
entire curves it is estimated such that it can change day by day. In our
example from Fig. 2 in the particular interval from March 11, 2020 (marked as
thin vertical line 1) to March 31, 2020 (again line 2) the time lag between
Spain and Italy decreased from $-9$ days to $-3$ days (Spain was catching up),
while the time lag between the US and Italy increased from $-17$ days to $-19$
days (at that time the US was falling further behind).

####  III.1.2  Doubling time

The second variable of importance is the doubling time, the characteristic
unit for exponential growth. It is the time it takes for the number of deaths
(or cases) to double. The lower its value the faster the spread of the
epidemic. The first step in calculating this number is to determine the
percentage growth rate $p(t)$ from one day to the next:

$p(t)=\frac{d(t)-d(t-1)}{d(t-1)}.$ (1)

Here d(t) refers to the cumulative number of deaths until day t. From the
percentage growth the doubling time $T_{d}$ is calculated as:

$T_{d}(t)=\frac{ln(2)}{ln[1+p(t)]}.$ (2)

Like the time lag the doubling time is also a time-local quantity that changes
day by day. In Fig. 3 we depict the course of the doubling times for the same
three countries and during the same time interval that was already used in
Fig. 2. In the beginning of the pandemic in each country the doubling time
exhibits quite large fluctuations due to the low absolute numbers, but after a
few weeks when the numbers rise this tends to stabilize into a more smooth
course.

Articles that have applied the doubling time in the context of the
Covid19-pandemic include Nunes-Vaz (2020); Lurie et al. (2020).

Figure 4: Doubling time versus time lag with respect to Italy for $10$
different countries. Large markers indicate the position on March 31, 2020,
the tail shows the development over the previous seven days. The four corners
are marked by letters A to D and the background is shaded to indicate more or
less preferable regions (from bright to dark). The legend states overall
number of deaths until the end of March 2020 as well as time lag, doubling
time and trend over the last seven days.

####  III.1.3  Overview plots

In the next step we combine these two time-local quantities, time lag and
doubling time, in one large overview graph that allows for an easy comparison
of the current state of the epidemic in many different countries. In Fig. 4 we
plot the doubling time for deaths versus the time lag with respect to Italy
for ten countries (including the ones from Figs. 2 and 3).

The countries that are compared with Italy were selected as follows: Four of
them (Russia, South Korea, China and Spain) were at that moment in time
closest to one of the four corners in the plot. The remaining five countries
were on different stages of a rather typical curve of a country on this graph
(World Health Organization: Coronavirus disease 2019 Situation Report 72,
April 1, 2020(2020), COVID-19).

In this plot we use the brightness of the background to indicate the preferred
order of the corners from Bright (B) to Dark (D): B, A, C, D. But our
description begins with the typical starting point A:

Figure 5: Number of reported new deaths per day for the ten countries from
Fig. 4 and for the year from January 22, 2020 to January 21, 2021. Each row
represents one country and colours are normalised from 0 (white) to 1 (black)
by the maximum number of deaths of that country over the whole interval. The
last row depicts the sum over all these countries together. Local maxima are
marked by white bullet points and the ordinal number of the wave peak. On the
left we indicate the absolute number of deaths on the last day and on the
right the trend over the last week (‘+’,‘-’,‘o’ for upwards, downwards,
constant, respectively). Finally, the histogram plot on the right shows for
each country the number of deaths per one million inhabitants over the whole
year. In order to provide information on the relative overall impact of the
epidemic in different countries we here use this number as the criterion for
the sorting of rows.

A – Large relative time lag, low doubling time.

Basically all countries start on the lower left of the graph. There it is
still early days, presumably the virus has started to spread within the
country not that long ago so the time lag to the epicenter is typically quite
big (depending on how long it took for the virus to reach the country, the
later a country enters the plot the bigger the time lag). The initial doubling
times are rather low so the spreading advances quickly but the numbers are
still such that it might appear as if there is not yet that much to worry
about. However, this is actually the time where measures should be taken as
soon as possible in order to make a big difference later. At the end of March
2020 the closest country to this corner A was Russia which had just entered
the plot with its first registered deaths.

B – Large relative time lag, high doubling time.

If a country is close to corner B it means that it has basically contained the
virus in its earliest days and the doubling times are so high that one can
hardly speak of an epidemic. On this day no country had really gotten there
yet, but among the countries selected here South Korea was the one that was
slowly getting closer.

C – Relative time lag close to zero, high doubling time.

This means that for the moment the worst is over but also that it was very
bad. Eventually all countries tend to go up towards larger doubling times but
of course it is much better to do it earlier rather than later. On the 31st of
March 2020 China was the country closest to this corner but since less and
less new deaths were reported it was actually moving towards corner B.

D – Relative time lag close to zero, low doubling time.

This is the situation to avoid at all costs (literally). Here countries are
already right in the middle of an epidemic but the doubling times are still
very low. This can be very bad because of the characteristics of unabated
epidemic spread. At A it might take a few days to double the number of cases
from $100$ to $200$ but at D it would take exactly the same time to double
from $10.000$ to $20.000$ or even from $100.000$ to $200.000$ (depending on
the overall stage of the epidemic). At the end of March 2020 the country
closest to this situation was Spain (apart from the reference country Italy)
and in fact it was right on its way of catching up with Italy.

The remaining countries were at that point in time positioned somewhere
between A and C. Like Spain, the US and the UK were moving closer to Italy.
France was basically time-locked with Italy which means that its death curve
was following Italy’s with a constant time lag. By contrast, Germany and
Canada were moving further away from Italy.

In Fig. 4 we show the position of all the countries in this two-dimensional
plot at the end of March 2020 but to each country we also append a tail that
depicts the development over the previous seven days. The direction of the
movement over that week is captured in the trend which can be found as the
very last entry in the legend:

A \- towards lower doubling times but larger time lags with respect to Italy

$\leftarrow$ \- no change in doubling time but towards larger time lags

B \- towards higher doubling times and larger time lags (best possible course)

$\uparrow$ \- towards higher doubling times, no change in time lag

C \- towards higher doubling times but shorter time lags

$\rightarrow$ \- no change in doubling time but towards shorter time lags

D \- towards lower doubling times and shorter time lags (worst possible
course)

$\downarrow$ \- towards lower doubling times but no change in time lag

$\bullet$ \- no change in either direction

Over this week, apart from Russia and Canada, the doubling time of most of
these countries had increased. Regarding the time lag, there were three
groups: for the countries on the right (Spain, the US and the UK) it had
increased, for the countries on the left (Germany, South Korea, Canada and
Russia) it had decreased and for the two countries in the middle (UK and
France) it had remained constant. Accordingly, overall the trends were
dominated by B, $\uparrow$ and C with only two countries on a downward trend
towards A.

###  III.2  Later stages of a pandemic

While the plots for the earlier stages are designed to provide a comparative
overview of the initial rise, in the later stages the focus shifts to
monitoring the course of the pandemic in each country in terms of peaks,
valleys and plateaus in order to be able to react accordingly (Charpentier et
al., 2020). The transition between these two stages takes place around the
time the daily numbers of cases and deaths stop rising for the first time
(Leung et al., 2020).

The left side of Fig. 5 shows a 2D color plot of the number of new deaths per
day for the same ten countries already depicted in Fig. 4 and for one year
starting on January 22, 2020. Each row is normalized individually in order to
provide an overview of the course of the epidemics for each country
separately. This means that for each country the color scale ranges from zero
daily deaths (white) to the maximum daily number of deaths over the whole
interval (black). As a consequence the course of the pandemic even in
countries with numbers of different orders of magnitude can be compared in the
same plot. It also becomes immediately apparent whether a country is already
over its peak and whether there are new waves; the brighter the colors on the
last day, the further away a given country is from its peak value.

We also use the color plot as a wave detector by identifying for each country
all the local maxima that fulfill the following two criteria:

\- Minimum prominence $P_{min}$

The prominence $P$ is defined as the smaller of the largest decrease in value
on both side of the local maximum before encountering the next local maximum.
For any given sequence of daily increases $D$ the largest possible prominence
is $P_{max}=max(D)-min(D)$.

\- Minimum separation $S_{min}$ between consecutive local maxima

The separation $S$ is defined in units of sample points (here days). For a
given $S>0$, we select the largest local maximum and ignore all other local
maximum within $S$ units of it. This process is repeated until no more local
maxima are detected.

There is no unique and unambiguous definition of a wave peak, so varying these
two parameter values will lead to different detections. Here we set the
minimum prominence to $P_{min}=0.5$ and the minimum separation to $S_{min}=10$
days. This selection eliminates all minor bumps and maintains only the large-
scale peaks that appear to be significant.

We also added trend indicators right next to the value of the current day that
show whether the numbers from the last day are more than $5\%$ higher than
they were the week before (‘+’, upward trend), whether they are within $5\%$
of that value (‘o’, plateau) or whether they are more than $5\%$ lower (‘-’,
downward trend).

The individual normalization used in the color plot facilitates tracking the
course of the pandemic within each country and allows to infer relative time
lags of peaks and valleys between different countries. However, it does not
provide any information about the overall severity of the situation in each
country. To rectify this we add a histogram (right) with the overall numbers
for every country normalized by population size.

Figure 6: Doubling time versus time lag with respect to Italy as in Fig. 4 but
now for $42$ different countries and one month later, on April 30, 2020. For
clarity, only the seven day tails of the eight countries that are most
advanced in the pandemic (in terms of time lag) are shown, however, the seven
day trends for all countries can be found in the legend. In deaths (a), at
this point in time, the US had surpassed Italy as the worldwide leader of the
pandemic, with the UK, Spain and France slightly behind, while for cases (b)
not only the US but also Spain had surpassed Italy, and

Finally, depending on which information should be stressed, countries can be
sorted in various ways:

\- The value of the histogram on the right hand side sorts countries by the
overall severity of the situation (e.g., deaths per one million inhabitants).
This sorting was used in Fig. 5 and will also be used in Figs. 7a and 7b for
both worldwide deaths and worldwide cases.

\- The value of normalized new number of deaths/cases on the last day allows a
comparison of the current state of the pandemic compared to the peak value of
each country. Which countries are currently at their absolute peak and for
which countries the worst is behind? This sorting will be used in Fig. 9 to
compare the situation for all the US states.

\- The occurrence of the first death/case or the position of the peak of the
first wave provides information about the gradual or sudden spatio-temporal
propagation of the pandemic. Where did the pandemic start and where did it
arrive last? In Fig. 9 we will use the sorting based on first cases to trace
the initial spread of the virus in the Italian regions.

\- The similarity of the daily new deaths/cases profiles. We use a
straightforward combination of correlation coefficient analysis and single
linkage algorithm to cluster countries according to the similarity of their
temporal profiles. From the resulting hierarchical dendrogram we obtain an
order that starts with the countries that are most similar to each other and
ends with those that are least similar to any of the other countries. This
sorting will be used in Fig. 10 for the worldwide data.

##  IV Results

###  IV.1  Early stages of a pandemic

In the early stages cases and deaths are still on their first rise and
typically there is an early epicenter which becomes a very useful reference to
which to compare the state of the pandemic in any given country. The most
important indicator of this state is the doubling time. Thus, the two relevant
quantities are the time lag with respect to that epicenter (here, Italy) and
the doubling time and in Fig. 6 we plot one against the other.

First, in Fig. 6a we look at deaths numbers for all the $42$ countries on
April 30, 2020 which was around the time the first worldwide peak in deaths
had just passed (World Health Organization: Coronavirus disease 2019 Situation
Report 102, May 1, 2020(2020), COVID-19). Fig. 6a is accompanied by
Supplementary Movie 1 which contains the development from the day Italy
reported its second death up to the end of April (such that the final frame of
the movie corresponds to this Fig. 6a).

At this moment in time countries could basically be divided into three
different groups. The most severe situation was found in the US which actually
had already surpassed Italy as the front runner of the pandemic and even at
that advanced stage had a rather low doubling time of $20$ days and was thus
continuing its course towards higher positive time lags. The other members of
that group were the European countries Italy, the UK, Spain and France which
were all more or less phase-locked with Italy, i.e. the time lags remained
quite constant.

The second and by far largest group of countries was showing a trend towards
increasing negative time lags and higher doubling times, thus getting closer
to corner B (compare last entries in the legend). Typically this meant that
for those countries the initial rise was already flattening considerably. This
group included Iran, Germany and Belgium, which were all closest to Italy but
still moving away.

The third and last group consisted of Taiwan, Iceland, and China which had all
basically stopped reporting any new deaths. At least for that specific moment
in time these five countries had brought the pandemic under control.

Fig. 6b depicts the same kind of plot as Fig. 6a, but now for cases instead of
deaths. As before, Supplementary Movie 2 shows the whole history of this plot
from the beginning of February (first reported cases in Italy) to the end of
April. Overall, both the groupings and trends for cases are very similar to
the ones seen for deaths. One notable difference is that for cases the
separation between the three groups is much less pronounced.

###  IV.2  Later stages of a pandemic

Figure 7: Number of reported new deaths (a) and cases (b) per day for the year
from January 22, 2020 to January 21, 2021 for all $42$ selected countries.
Layout as in Fig. 5. Countries are again sorted by the number of deaths per
one million inhabitants over the whole year (histogram on the right). At this
moment in time more than half of the countries were peaking or close to
peaking and typically it was either the second or the third peak.

Figure 8: Number of reported new deaths per day for all the US states during
the year from January 22, 2020 to January 21, 2021. Layout as in Fig. 5 but in
contrast to the previous wave detection plots (Figs. 5 and 7) this one is
sorted according to the (normalized) value obtained for the very last day of
the plot. This provides information about the relative severity of the
situation in each state at this point in time, while the histogram on the
right still shows the overall severity up to that day.

Figure 9: Number of reported new deaths per day for the Italian regions during
the same year as before. Layout as in Fig. 5 with the only difference that in
this wave detection plot the regions are sorted according to the time of the
first reported death (from bottom to top). This allows tracing the course of
the pandemic in Italy from early epicenters such as Lombardy and Campania to
regions like Basilicata that were reached only much later.

Figure 10: Similarity of countries in terms of daily new deaths. (a) Pairwise
Correlation coefficient matrix calculated from the temporal profiles as
depicted in Fig. 7a. (b) Hierarchical cluster tree (dendrogram) obtained from
(a) via the single linkage clustering algorithm. Countries that are most
similar to each other (lowest distances between them) are on the left,
countries that are more unique (higher distances) are on the right. (c) Same
data as in the wave plot of Fig. 7a but with the countries sorted according to
the similarity criterion. For clarity the same sorting was already used in
(a). Note that the three colors in (b) and the corresponding separating lines
in (a) and (c) are just visual aids, they are not based on any strict
criteria.

Once the initial rising phase of a pandemic has been passed, it becomes more
important to monitor the rise and fall of deaths and cases via the
characteristic peaks, valleys and plateaus in the profile of each country.
Accordingly, we continue with Fig. 7a which follows Fig. 5 in showing the
number of reported daily new deaths from the beginning of the dataset up to a
year later, but it does so for all $42$ countries. Similar to before, Fig. 7a
is the last frame of Supplementary Movie 3, which tracks the development of
the number of deaths over the whole year.

After this one year, the countries hit the hardest were predominantly in
Europe, but also countries like the US, Peru and Mexico had suffered very high
numbers of deaths. On the other hand, many countries in Asia and Oceania had
fared quite will. Among these countries were Taiwan, South Korea, Japan as
well as New Zealand and Australia.

At this specific point in time there was almost equal division among upwards,
constant and downwards trends, but there were just a few more countries
trending downwards rather than upwards. So most countries were either
plateauing on an unprecedented high level or had just started to slightly
decrease (for confirmation just refer to the data for all countries together
in the last row). The relatively highest peaks on that day were obtained for
the US and Mexico, whereas a few countries (Taiwan, New Zealand, Singapore,
Australia and Iceland) were reporting no deaths at all.

Fig. 7b shows the reported number of daily cases for the same countries and
the same time interval. The overall situation was quite similar to the one
reported for deaths (Fig. 7a). Also here half the countries were peaking or
close to peaking and again only a handful of countries (Taiwan, China, New
Zealand, Australia, and Iceland) reported almost no new cases at all. Fig. 7b
corresponds to the last frame of the final Supplementary Movie 4.

When comparing the relative positions of the countries in the two graphs for
deaths (Fig. 7a) and cases (Fig. 7b) we find a rather high correlation
coefficient of $0.84$, which means that typically countries that rank high in
deaths also rank high in cases. The two most notable exceptions are Israel
(which ranked much higher in cases than in deaths) and Mexico (the other way
around, it ranked much higher in deaths than in cases).

So far we looked at global plots containing a comparison of different
countries from all over the world. The last two plots depict the same kind of
data but now for smaller regions within a country - first the US states and
then the regions of Italy - and each time using a new kind of sorting
criterion.

First, in Fig. 9 we visualize the data of the US states (plus the country as a
whole), again over the same first year of data availability. In this graph we
sort the states according to the number of deaths on the very last day (in
this case January 21, 2021) normalized to the maximum value obtained for each
state so far. This gives us a good idea of the relative severity of the
situation on that day since it sorts states according to how close countries
are to their absolute peak. The states that are currently peaking can be found
at the top whereas the states that have passed their peak(s) appear at the
bottom.

On this very day in the US mostly southern states like South Carolina,
Oklahoma, Kentucky, Texas, and Georgia were still peaking in deaths, while the
state furthest away from its own past peak was New York (followed by Colorado
and Nebraska). On the other hand, as the histogram on the right shows, the
highest overall numbers of deaths had been obtained in north-eastern states
such as New Jersey, New York, Massachusetts and Rhode Island.

Fig. 9 depicts the regions of Italy sorted by the time of their first reported
death. It started with Lombardy in the North which became the early epicenter
and then gradually reached the whole country with Basilicata being the last
region to report a death.

This plot also shows quite nicely how the severity of the situation in each
region can vary for different waves. Marche and Lombardia, two of the regions
that were hit early and hard during the first wave (in fact were among the
first places in Europe to report Corona-related deaths), were relatively
speaking spared during the the second wave of the next winter, 2020/2021. On
the other hand, regions such as Basilicata, Molise and Toscana had much worse
trajectories the second time around.

Finally, in Fig. 10 we return to the worldwide data and combine a correlation
coefficient analysis (Fig. 10a) with the single linkage algorithm to arrange
the countries in a hierarchical cluster tree (dendrogram, Fig. 10b) according
to the similarity of their daily new deaths profiles (from Fig. 7a). Note that
during this mapping from the pairwise distance matrix to a one-dimensional
order important information gets lost. Vicinity in the ordered list does not
correspond to a direct measure of distance. However, what hold is that
countries that are most similar to each other appear on the left and countries
with the most unique profiles can be found on the right (as reflected my the
monotonous increase of the distances from left to right).

Using the resulting order from the dendrogram we get a new wave plot (Fig.
10c). Now we have on top the countries closest to each other which seem to be
those with a rather weak first wave (spring 2020) but a very strong second
wave (winter 2020/21). The next group contains the countries with two rather
strong waves. Both of these groups are predominantly European and North
American. They are followed by countries with a wave in summer 2020. This
might be due to a later arrival time of the virus but there are also many
countries from the Southern hemisphere which also points to an explanation in
terms of anti-phase seasonal variations between the two hemispheres (Merow and
Urban, 2020). The last group include those that apart from a few sporadic
eruptions had brought the pandemic mostly under control and ends which China
with its very early outbreak followed by a consistent flattening of the curve.

##  V Discussion

We presented visualizations of epidemiological data (such as the number of
deaths and cases) that take into account many countries (states, regions,
etc.) at the same time and are tailored to the specific stage of the epidemic.
During the initial rising phase we focus on time lag with respect to the
current epicenter and doubling time since this combination can inform about
the timeliness and the urgency of interventions needed to curb the spread. In
contrast, during the later stages we monitor the state of the pandemic by
following the wavelike profile of daily new death or case numbers with regard
to peaks, valleys and plateaus. This, together with other epidemiological
quantities such as attack rate (Liu et al., 2020) or basic reproduction number
(Park et al., 2020; Viceconte and Petrosillo, 2020; Tao, 2020), can help to
decide about appropriate counter measures, i.e. whether to impose, tighten or
relax contact restrictions in the population (Anderson et al., 2020; Valba et
al., 2020).

The visualizations used here are universal and can easily be applied to other
kinds of epidemiological data. The spatial scale is flexible as well. While we
focused on countries, states and regions, it would of course also work with
smaller areas. Moreover, in the two-dimensional plots designed for the initial
stages of an epidemic (Figs. 4 and 6) we use a country of reference, namely
Italy, the early European epicenter of the pandemic (Nacoti et al., 2020;
Indolfi and Spaccarotella, 2020). However, this is certainly a matter of
choice. Different countries could be chosen, e.g., in order to test different
hypotheses. Or you could select your own country and then the plot can
basically be seen from your countries’ point of view and provide more detailed
information about its relative position in the pandemic. Finally, our
methodology based on time lags and doubling times was only applied to the
rising phase of the first wave (and indeed there it is most useful), but of
course in cases where the individual waves are separate enough (e.g., due to
seasonal variations (Merow and Urban, 2020)) it would also be possible to look
at second, third or even later waves.

The color-coded wave detection plots used during the later stages can easily
be modified to be sensitive to several other traits in the data. Here we used
four different sorting criterions (Figs. 7, 9, 9 and 10c) but also the
normalization of the color scale could be altered to stress certain other
aspects of the data. Similarly, one could adapt the wave detection parameters
in order to focus on specific time scales and resolutions.

Note that the current study itself is not concerned with drawing specific
conclusions from the data, rather we focus on efficient and informative ways
of presenting the data to then be in a better position to actually draw
specific conclusions. We restrict ourselves to examinations of the past up to
the present day, but there is no extrapolation into the future based on any
kind of model, assumption or parameter selections. This way we avoid any
pitfalls caused by potential deficiencies in either completeness or accuracy
of the data (Sridhar and Majumder, 2020). However, given reliable data these
visualizations could be used to, e.g., correlate the data with different
containment strategies (Lurie et al., 2020), or to perform other more extended
analyses (Callaghan, 2020; Edwards and Lessler, 2021).

We would like to close with the following appeal: Please always be aware of
the tragic real-life consequences behind these numbers and do whatever you can
to keep them low or bring them down again.

##  VI  Source codes and outreach

Matlab source codes will be available at this webpage:

http://wwwold.fi.isc.cnr.it/users/thomas.kreuz/Source-Code/Corona.html

Since the worsening of the pandemic in Italy (March 2020) regular updates
based on the data analysis methods described in this article have been posted
on this Facebook page:

https://www.facebook.com/tk.corona.updates/

This will continue as long as the pandemic causes significant damage all over
the world. Regular updates show data for the same $42$ selected countries used
here. On demand the data from the regions/provinces of some individual
countries with publicly available data (specifically the US, Italy, Germany,
Spain, the UK, and Canada) are displayed as well. For now this is mostly the
US, the only major country with continuously elevated numbers and strong
regional diversity.

##  VII  Supplementary Movies

Caption Supplementary Movie 1

(https://youtu.be/2BMZw5PHmK0):

The first supplementary movie depicts the doubling time for deaths versus the
time lag with respect to Italy for all $42$ countries from February 21, 2020
(the day Italy reported its second death) to April 30, 2020. The layout is
identical to the one used in Fig. 6a, in fact, it ends with Fig. 6, the state
of the pandemic at the end of April 2020. Over the course of time most
countries tend to slowly move towards corner B (larger time lags with respect
to Italy, higher doubling times). But there are a few exceptions, notably
Spain, France, the UK, and in particular the US which during April can be seen
to slowly overtake Italy.

Caption Supplementary Movie 2

(https://youtu.be/vAsSQfWvJwQ):

The second supplementary movie is similar to the first but depicts cases
instead of deaths. It runs from February 2, 2020 (the day Italy reported its
second case) to April 30, 2020. Its last frame is identical to Fig. 6b.

Caption Supplementary Movie 3

(https://youtu.be/UC13sP7gHeU):

The third supplementary movie shows the course of the pandemic for all $42$
countries over one year, from January 22, 2020 to January 21, 2021, in terms
of number of reported new deaths per day. The layout is identical to the one
used in Fig. 7a. Countries are again sorted by the relative number of deaths
over the whole year (histogram on the right) which makes it easier to follow
the overall impact of the epidemic for different regions. During the initial
stages China, Italy, South Korea, Iran and Spain were the epicenters of the
pandemic, at the later stages Belgium, Czechia, Peru, the UK and the US were
among those countries that exhibited the highest relative numbers of deaths.

Caption Supplementary Movie 4

(https://youtu.be/WCyhIEQJc7Q):

The fourth supplementary movie is similar to the third but instead of deaths
it depicts cases for all $42$ countries over one year. Its last frame
corresponds to Fig. 7b. This movie shows even more clearly how the pandemic
spread from China and its neighboring countries all over the world.

Further movies (various combinations of deaths/cases with/without
normalization and different kinds of sortings for the $42$ selected countries,
the US states, and the Italian regions can be found on the tk.corona.updates
Youtube channel:

https://www.youtube.com/channel/UCASTaaV9CKZEpsFRhnYF-EQ

The first four movies on the channel correspond to the Supplementary Movies
1-4 of this article.

###### Acknowledgements.

T.K. would like to thank Alban Levy for useful discussions, his thorough
reading of the manuscript as well as his very early efforts in rising
awareness about the upcoming pandemic. T.K. would also like to thank Don
MacLeod, Kaare Bjarke Mikkelsen, and Sabine Raphael for feedback on the data
analysis and Benedetta Moschitta for many inspiring discussions.

## References

  * Huang et al. (2020) C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, L. Zhang, G. Fan, J. Xu, X. Gu, et al., The Lancet 395, 497 (2020).
  * World Health Organization: Coronavirus disease 2019 Situation Report 11, January 31, 2020(2020) (COVID-19) World Health Organization: Coronavirus disease 2019 (COVID-19) Situation Report 11, January 31, 2020 (2020).
  * Zhu et al. (2020) N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, X. Zhao, B. Huang, W. Shi, R. Lu, et al., New England journal of medicine 382, 727 (2020).
  * Guan et al. (2020) W.-J. Guan, Z.-y. Ni, Y. Hu, W.-H. Liang, C.-q. Ou, J.-X. He, L. Liu, H. Shan, C.-L. Lei, D. S. Hui, et al., New England Journal of Medicine 382, 1708 (2020).
  * Zhou et al. (2020) F. Zhou, T. Yu, R. Du, G. Fan, Y. Liu, Z. Liu, J. Xiang, Y. Wang, B. Song, X. Gu, et al., The Lancet 395, 1054 (2020).
  * World Health Organization: Coronavirus disease 2019 Situation Report 40, February 29, 2020(2020) (COVID-19) World Health Organization: Coronavirus disease 2019 (COVID-19) Situation Report 40, February 29, 2020 (2020).
  * World Health Organization: Coronavirus disease 2019 Situation Report 51, March 11, 2020(2020) (COVID-19) World Health Organization: Coronavirus disease 2019 (COVID-19) Situation Report 51, March 11, 2020 (2020).
  * World Health Organization: Coronavirus disease 2019 Situation Report 60, March 20, 2020(2020) (COVID-19) World Health Organization: Coronavirus disease 2019 (COVID-19) Situation Report 60, March 20, 2020 (2020).
  * Shamasunder et al. (2020) S. Shamasunder, S. M. Holmes, T. Goronga, H. Carrasco, E. Katz, R. Frankfurter, and S. Keshavjee, Global Public Health 15, 1083 (2020).
  * McKibbin and Fernando (2020) W. McKibbin and R. Fernando, Economics in the Time of COVID-19 45 (2020).
  * Zhang et al. (2020) D. Zhang, M. Hu, and Q. Ji, Finance Research Letters 36, 101528 (2020).
  * Palayew et al. (2020) A. Palayew, O. Norgaard, K. Safreed-Harmon, T. H. Andersen, L. N. Rasmussen, and J. V. Lazarus, Nature Human Behaviour 4, 666 (2020).
  * Myers et al. (2020) K. R. Myers, W. Y. Tham, Y. Yin, N. Cohodes, J. G. Thursby, M. C. Thursby, P. Schiffer, J. T. Walsh, K. R. Lakhani, and D. Wang, Nature Human Behaviour 4, 880 (2020).
  * Marinoni et al. (2020) G. Marinoni, H. Van’t Land, and T. Jensen, IAU Global Survey Report (2020).
  * Škare et al. (2020) M. Škare, D. R. Soriano, and M. Porada-Rochoń, Technological Forecasting and Social Change p. 120469 (2020).
  * Garcia-Garcia et al. (2020) B. Garcia-Garcia, M. James, D. Koller, J. Lindholm, D. Mavromati, R. Parrish, and R. Rodenberg, Int Sports Law J 20, 115 (2020).
  * Dubey et al. (2020) S. Dubey, P. Biswas, R. Ghosh, S. Chatterjee, M. J. Dubey, S. Chatterjee, D. Lahiri, and C. J. Lavie, Diabetes & Metabolic Syndrome: Clinical Research & Reviews 14, 779 (2020).
  * Dong et al. (2020) E. Dong, H. Du, and L. Gardner, The Lancet Infectious Diseases 20, 533 (2020).
  * Fanelli and Piazza (2020) D. Fanelli and F. Piazza, Chaos, Solitons & Fractals 134, 109761 (2020).
  * Flaxman et al. (2020) S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland, C. Whittaker, H. Zhu, T. Berah, J. W. Eaton, et al., Nature 584, 257 (2020).
  * Dehning et al. (2020) J. Dehning, J. Zierenberg, F. P. Spitzner, M. Wibral, J. P. Neto, M. Wilczek, and V. Priesemann, Science 369, eabb9789 (2020).
  * de Sousa et al. (2020) L. E. de Sousa, P. H. de Oliveira Neto, and D. A. da Silva Filho, Physical Review E 102, 032133 (2020).
  * Sy et al. (2020) C. Sy, E. Bernardo, A. Miguel, J. L. San Juan, A. P. Mayol, P. M. Ching, A. Culaba, A. Ubando, and J. E. Mutuc, Process Integration and Optimization for Sustainability 4, 497 (2020).
  * Moon (2020) M. J. Moon, Public Administration Review 80, 651 (2020).
  * Jia and Lu (2020) Z. Jia and Z. Lu, The Lancet. Infectious Diseases 20, 757 (2020).
  * Vespignani et al. (2020) A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, S. V. Scarpino, B. Gutierrez, M. U. Kraemer, J. Wu, et al., Nature Reviews Physics 2, 279 (2020).
  * Prakash et al. (2020) M. K. Prakash, S. Kaushal, S. Bhattacharya, A. Chandran, A. Kumar, and S. Ansumali, Physical Review E 102, 021301 (2020).
  * Sridhar and Majumder (2020) D. Sridhar and M. S. Majumder, BMJ 369 (2020).
  * Callaghan (2020) S. Callaghan, Patterns 1 (2020).
  * The Lancet (2020) The Lancet, Lancet (London, England) 395, 1011 (2020).
  * Phua et al. (2020) J. Phua, L. Weng, L. Ling, M. Egi, C.-M. Lim, J. V. Divatia, B. R. Shrestha, Y. M. Arabi, J. Ng, C. D. Gomersall, et al., The Lancet Respiratory Medicine 8, 506 (2020).
  * Kennedy et al. (2020) D. M. Kennedy, G. J. Zambrano, Y. Wang, and O. P. Neto, Journal of Clinical Virology 128, 104440 (2020).
  * Goldsztejn et al. (2020) U. Goldsztejn, D. Schwartzman, and A. Nehorai, PLOS ONE 15, e0244174 (2020).
  * Charpentier et al. (2020) A. Charpentier, R. Elie, M. Laurière, and V. C. Tran, Mathematical Modelling of Natural Phenomena 15, 57 (2020).
  * Leung et al. (2020) K. Leung, J. T. Wu, D. Liu, and G. M. Leung, The Lancet 395, 1382 (2020).
  * Bergman et al. (2020) A. Bergman, Y. Sella, P. Agre, and A. Casadevall, mSystems 5 (2020).
  * Subbaraman (2020) N. Subbaraman, Nature, DOI: 10.1038/d41586-020-01008-1 (2020).
  * Johansson et al. (2021) M. A. Johansson, T. M. Quandelacy, S. Kada, P. V. Prasad, M. Steele, J. T. Brooks, R. B. Slayton, M. Biggerstaff, and J. C. Butler, JAMA network open 4, e2035057 (2021).
  * Pullano et al. (2021) G. Pullano, L. Di Domenico, C. E. Sabbatini, E. Valdano, C. Turbelin, M. Debin, C. Guerrisi, C. Kengne-Kuetche, C. Souty, T. Hanslik, et al., Nature 590, 134 (2021).
  * Nacoti et al. (2020) M. Nacoti, A. Ciocca, A. Giupponi, P. Brambillasca, F. Lussana, M. Pisano, G. Goisis, D. Bonacina, F. Fazzi, R. Naspro, et al., NEJM Catalyst Innovations in Care Delivery 1 (2020).
  * Indolfi and Spaccarotella (2020) C. Indolfi and C. Spaccarotella, Journal of the American College of Cardiology: Case Reports 2, 1414 (2020).
  * Nunes-Vaz (2020) R. Nunes-Vaz, Global Biosecurity 1 (2020).
  * Lurie et al. (2020) M. N. Lurie, J. Silva, R. R. Yorlets, J. Tao, and P. A. Chan, The Journal of Infectious Diseases 222, 1601 (2020).
  * World Health Organization: Coronavirus disease 2019 Situation Report 72, April 1, 2020(2020) (COVID-19) World Health Organization: Coronavirus disease 2019 (COVID-19) Situation Report 72, April 1, 2020 (2020).
  * World Health Organization: Coronavirus disease 2019 Situation Report 102, May 1, 2020(2020) (COVID-19) World Health Organization: Coronavirus disease 2019 (COVID-19) Situation Report 102, May 1, 2020 (2020).
  * Merow and Urban (2020) C. Merow and M. C. Urban, Proceedings of the National Academy of Sciences 117, 27456 (2020).
  * Liu et al. (2020) Y. Liu, R. M. Eggo, and A. J. Kucharski, The Lancet 395, e47 (2020).
  * Park et al. (2020) S. W. Park, B. M. Bolker, D. Champredon, D. J. Earn, M. Li, J. S. Weitz, B. T. Grenfell, and J. Dushoff, Journal of the Royal Society Interface 17, 20200144 (2020).
  * Viceconte and Petrosillo (2020) G. Viceconte and N. Petrosillo, Infectious Disease Reports 12 (2020).
  * Tao (2020) Y. Tao, Physical Review E 102, 032136 (2020).
  * Anderson et al. (2020) R. M. Anderson, H. Heesterbeek, D. Klinkenberg, and T. D. Hollingsworth, The Lancet 395, 931 (2020).
  * Valba et al. (2020) O. Valba, V. Avetisov, A. Gorsky, and S. Nechaev, Physical Review E 102, 010401 (2020).
  * Edwards and Lessler (2021) J. K. Edwards and J. Lessler, American Journal of Epidemiology 190, 17 (2021).


# Machine Learning Symmetry

Shailesh Lal<EMAIL_ADDRESS>
###### Abstract

We review recent work in machine learning aspects of conformal field theory
and Lie algebra representation theory using neural networks.

Faculdade de Ciencias, Universidade do Porto

687 Rua do Campo Alegre, Porto, Portugal

##  1 Introduction

One of the most striking examples of the success of machine learning and
neural networks in analyzing physical systems has been in the context of the
string landscape problem, initiated in [1, 2, 3, 4], see [5, 6] for reviews.
Neural networks have been used with great accuracy to predict various
properties of compactification manifolds such as topological numbers [7, 8, 9,
10], bundle cohomology [3, 11, 12], metrics [13, 14], and symmetry properties
[15]. Machine Learning has also proved useful to find patterns in particle
masses [16] and more generally to statistical predictions and model-building
in string theory [17, 18, 19, 20]. Interestingly, neural networks have been
also used to even quantify the degree of likelihood that two string vacua will
give rise to similar phenomenology [21]. At the heart of these successes is
the robust pattern-finding ability of neural networks which allow us to
directly infer the output given an input by learning heuristic association
rules from previously available data.

If we, somewhat reductively, pose the landscape problem as the problem of
scanning across a large parameter space to search for points where specific
properties are realized, coupled with the computational complexity of probing
even a single point, then it is apparent that the landscape problem is not
unique to string theory but is quite generic to mathematical physics. For
example, one might ask: ‘among which in the set of representations of
$\mathfrak{su}(5)$ may one embed the matter content of the standard model’? Or
equally well, ‘where in the set of conformal field theories does a given OPE
coefficient take a particular value’? For an individual $\mathfrak{su}(5)$
representation or an individual conformal field theory, the answer may be
obtained through a direct, albeit possibly involved, computation. We however,
would like to address these questions across an in principle infinite set of
possibilities. Doing so might unveil new structures and organizing principles
for the given parameter space, and eventually to new insight. In the
following, we will report on some initial progress in using neural networks
for these purposes [22, 23].

The works [22, 23] are in a sense part of a more general program of applying
methods from data science and machine learning to study problems in
mathematical physics, see [24, 25] for a pedagogical treatment. By now,
machine learning and data science have been used to learn various aspects of
number theory [26, 27, 28], quiver gauge theories and cluster algebras [29],
knot theory [30, 31, 32] as well as graph theory [33] and commutative algebra
[34]. Importantly, and in hindsight somewhat as a precursor of [23], support
vector machines and neural networks were used to learn the algebraic
structures of discrete groups and rings [35]. It was explicitly shown there
that these machine learning frameworks are able to solve difficult problems in
the representation theory of discrete groups without having to appeal to
computationally expensive methods like Sylow theorems. In addition, neural
network regressors have also been trained to identify Lie algebra symmetries
in datasets in [36].

##  2 Machine Learning Conformal Field Theory

Consider $d$-dimensional Euclidean space with the inner product

$\vec{x}^{2}=x_{1}^{2}+x_{2}^{2}+\ldots+x_{d}^{2}$ (1)

and the symmetry transformations

$\vec{x}\mapsto\vec{x}+\vec{c},\,\vec{x}\mapsto
R\cdot\vec{x},\,\vec{x}\mapsto\lambda\vec{x},\,\vec{x}\mapsto\frac{\vec{x}}{|\vec{x}|^{2}}\,,$
(2)

A conformal field theory, or CFT, is a quantum field theory which is invariant
under these symmetries. These theories are in ubiquitous physics, being the
formalism to describe systems as diverse as phase transitions in a table-top
experiment to quantum gravity. The past few years have seen a rapid spurt of
new numerical and analytical results for CFT by means of the conformal
bootstrap. In particular, one starts with a few minimal assumptions about a
putative CFT, such as the spectrum of light operators. Then crossing symmetry
of four-point functions, along with unitarity, strongly constrain the possible
dynamics of such theories, see [37] for a review. These developments have
opened up the possibility of exploring the landscape of these theories by
robust numerical methods.

It further turns out that various defining properties of Conformal Field
Theory are machine learnable to a near hundred percent accuracy [22]. We will
focus here on the conformal block expansion of four point correlation
functions of scalar conformal primaries. These are operators that transform
under conformal transformations as

$\mathcal{O}_{\Delta}\left(\lambda\,x\right)=\lambda^{-\Delta}\,\mathcal{O}_{\Delta}\left(\lambda\,x\right)\,.$
(3)

Here $\Delta$ is known as the conformal dimension of the operator
$\mathcal{O}_{\Delta}$. In the following, we will set $d=1$. The 4-point
function of the operator $\mathcal{O}_{\Delta}$ may then be expressed as [38]

$\left\langle\mathcal{O}_{\Delta}\left(x_{1}\right)\mathcal{O}_{\Delta}\left(x_{2}\right)\mathcal{O}_{\Delta}\left(x_{3}\right)\mathcal{O}_{\Delta}\left(x_{4}\right)\right\rangle=\frac{\sum_{\\{h\\}}c_{h}\,^{2}\,z^{h}\,_{2}F_{1}(h,h,2h;z)}{|x_{12}|^{2\Delta}|x_{34}|^{2\Delta}}\,.$
(4)

Here $z$ is the cross ratio

$z=\frac{|x_{12}||x_{34}|}{|x_{13}||x_{24}|}\,,$ (5)

and the sum runs over all operators $\mathcal{O}_{h}$ that appear in the OPE
$\mathcal{O}_{\Delta}\times\mathcal{O}_{\Delta}$, and $c_{h}$ is the
corresponding OPE coefficient. Here the conformal dimention of
$\mathcal{O}_{h}$ is $h$ and the functions $z^{h}\,_{2}F_{1}(h,h,2h;z)$ are
the conformal blocks.

In the following, we will demonstrate how neural networks distinguish between
two classes of four-point functions, namely ones where a particular
coefficient $c_{h}$ are zero and non-zero respectively. To do so, it is useful
to define an auxiliary function

$f\left(z\right)=\sum_{\\{h\\}}c_{h}\,^{2}z^{h}\,_{2}F_{1}(h,h,2h;z)\,.$ (6)

The spectrum of conformal dimensions $\\{h\\}$ and the OPE coefficients
$c_{h}$ are not arbitrary and are constrained so that the overall four-point
function is crossing symmetric. We will however consider a toy setting where
the $h$ are fixed to be positive integers and the $c_{h}$ are generated
randomly. We then define two classes of functions, viz.

$f_{1}\left(z\right)=\sum_{n=0}^{10}c_{n}\,^{2}z^{n}\,_{2}F_{1}(n,n,2n;z)\,,\quad
f_{2}\left(z\right)=\sum_{n=0,\,n\neq
2}^{10}c_{n}\,^{2}z^{n}\,_{2}F_{1}(n,n,2n;z)\,,$ (7)

Some examples of these two classes of functions are visualized in Figures 1(a)
and 1(b).

(a) Functions with $c_{n}\neq 0$. (b) Functions with $c_{2}=0$.

Figure 1: Toy four-point functions for CFT1.

To prepare this problem for machine learning, we discretize $z$ to lie on a
grid of 100 equidistant points in $|z|\leq 0.8$ and generate 100000 instances
each of functions $f_{1}$ and $f_{2}$ by randomly sampling the $c_{h}$ in the
interval $(0,1)$. A relatively simple $\mathtt{relu}$ activated MLP
architecture trained on this data achieves near perfect performance [22]. In
particular, when evaluating on 2000 test functions, we obtain the confusion
matrix

$\begin{pmatrix}1000&0\\\ 67&933\end{pmatrix}\,.$ (8)

In sum, simple MLP architectures are able to distinguish between two classes
of functions which look quite similar at first glance. In fact, it is possible
to understand this difference quite explicitly. Principal component analysis
applied to the randomly generated point clouds of Equation 7 clearly
visualizes the two classes of functions as being separated from each other.

Figure 2: The point clouds corresponding to $f_{1}$, shown as blue, and
$f_{2}$, shown as yellow, represented in two dimensions by principal component
analysis. Each point corresponds to a particular function $f$.

##  3 Machine Learning Lie Algebras

One of the most fundamental problems in physics is the description of
symmetries of a system, i.e. the set of operations that leaves the system
invariant. This set has a natural group structure that may be imposed on it.
The composition of two symmetry operations is quite clearly a symmetry
operation, and any symmetry operation can also be inverted to restore the
system to its original state. We will focus on the groups which have elements
continuously connected to the identity, i.e. these elements may be
parametrized as

$U\left(\theta\right)=\exp\left(i\theta^{a}\,T_{a}\right)\,,\theta^{a}\in\mathbb{R}\,.$
(9)

These are the Lie groups, whose generators $T^{a}$ form a Lie algebra, i.e.
they are endowed with a composition rule $[\cdot,\cdot]$ known as the Lie
bracket that obeys, among other properties,

$\left[T_{a},T_{b}\right]=i\,f_{ab}\,^{c}\,T_{c}\,.$ (10)

Physicists usually focus on unitary representations of the group $U$ over a
vector space $\mathcal{V}$. The $T_{a}$ then are represented by hermitian
matrices on $\mathcal{V}$, and the Lie bracket is the usual matrix commutator.

A representation for which all the $T_{a}$ cannot be simultaneously brought to
block diagonal form is called irreducible. Representations which are not
irreducible are called reducible, and may be expressed as a direct sum of
irreducible representations. If a subset of the Lie algebra closes within
itself under the same Lie bracket, it is known as a subalgebra. Further,
irreducible representations of a Lie algebra decompose in general into a
direct sum of irreducible representations of a given subalgebra. Carrying out
such decompositions is in general a complicated analytical task [39, 40]. We
refer the reader to [41] for more details regarding Lie algebras and their
representation theory.

Given that particles carry unitary irreducible representations of the symmetry
group, see e.g. [42], such computations are of key relevance to particle
physics model building. As an example, consider the Lie algebras
$\mathfrak{su}(N)$. These are the algebras corresponding to the group of
unitary $N\times N$ matrices of determinant 1. The algebra $\mathfrak{su}(5)$
contains as a subalgebra
$\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)$, the symmetry
algebra of the standard model. Classifying which representations of
$\mathfrak{su}(5)$ contain the matter content of the standard model is an
important problem for particle physics model building. Such computations are
seen to run on exponential time scales [23] when carried out on LieART, a
Mathematica software for computations in Lie algebra representation theory
[40].

To machine-learn this decomposition, we note that unitary irreducible
representations of $\mathfrak{su}(5)$ are labeled by an ordered $4$-tuple of
integers, see e.g. [41]. We restrict ourselves to those $4$-tuples for which
each individual entry ranges from 0 to 4. The dimensions of these
representations range from 1, for $(0,0,0,0)$ to 9765625 for $(4,4,4,4)$.

(a) Accuracy

(b) Loss

Figure 3: Training curves for $\mathfrak{su}(5)$ decomposition.

This data is again fed to a relu activated MLP [23]. The training curves for
this classification are shown in Figure 3. The neural network trains to 90%
test accuracy with a Matthews’ phi-coefficient of 0.81. The confusion matrix
obtained is

$\begin{pmatrix}43&0\\\ 8&29\end{pmatrix}\,.$ (11)

##  4 Outlook

In summary we see that neural networks are able to learn important facts about
mathematical physics landscapes, both for conformal field theory and for the
representation theory of Lie algebras. We hope that these promising results
pave the way for a more exhaustive exploration of these landscapes using
methods from data science and machine learning. We note especially the recent
works on two dimensional CFTs using classifiers as well as autoencoders [43],
along with the application of reinforcement learning to solve the equations of
the conformal bootstrap itself [44]. It would also be interesting to examine
if the methods of [21] could now be extended to these cases, and to glean
interesting organizing principles by doing so. The pictorial representation of
putative CFTs obtained by PCA, shown in Figure 2 is already suggestive in this
direction.

Finally, we note that symmetry in datasets was also characterized using neural
networks in [36, 45, 46, 47, 48]. It would be interesting to study how those
methods apply to the examples studied here. More generally, we also note that
ideas and methods from mathematical physics, in particular quantum field
theory, have been used to explicate the dynamics of neural networks (see e.g.
[49, 50, 51, 52]). Indeed, the very structures of quantum field theory and
holography appear to be closely connected to certain neural networks [53]. We
therefore expect that the interplay between mathematics, physics and data
science will yield many new insights of mutual interest and benefit over the
long term as well.

### Acknowledgements:

Figures 1 and 2 are reproduced from the originals in [22]. SL’s research was
supported by the Simons Collaboration on the Non-perturbative Bootstrap and
the Faculty of Sciences, University of Porto while [22, 23] were carried out.

## References

  * [1] Y. H. He, [arXiv:1706.02714 [hep-th]].
  * [2] D. Krefl and R. K. Seong, Phys. Rev. D 96 (2017) no.6, 066014 [arXiv:1706.03346 [hep-th]].
  * [3] F. Ruehle, JHEP 08 (2017), 038 [arXiv:1706.07024 [hep-th]].
  * [4] J. Carifio, J. Halverson, D. Krioukov and B. D. Nelson, JHEP 09 (2017), 157 [arXiv:1707.00655 [hep-th]].
  * [5] Y. H. He, Int. J. Mod. Phys. A 36 (2021) no.29, 2130017 [arXiv:2011.14442 [hep-th]].
  * [6] F. Ruehle, Phys. Rept. 839 (2020), 1-117
  * [7] Y. H. He, Phys. Lett. B 774 (2017), 564-568
  * [8] H. Erbin and R. Finotello, Mach. Learn. Sci. Tech. 2 (2021) no.2, 02LT03 [arXiv:2007.13379 [hep-th]].
  * [9] H. Erbin and R. Finotello, Phys. Rev. D 103 (2021) no.12, 126014 [arXiv:2007.15706 [hep-th]].
  * [10] H. Erbin, R. Finotello, R. Schneider and M. Tamaazousti, Mach. Learn. Sci. Tech. 3 (2022) no.1, 015006 [arXiv:2108.02221 [hep-th]].
  * [11] D. Klaewer and L. Schlechter, Phys. Lett. B 789 (2019), 438-443 [arXiv:1809.02547 [hep-th]].
  * [12] C. R. Brodie, A. Constantin, R. Deen and A. Lukas, Fortsch. Phys. 68 (2020) no.1, 1900087 [arXiv:1906.08730 [hep-th]].
  * [13] A. Ashmore, Y. H. He and B. A. Ovrut, Fortsch. Phys. 68 (2020) no.9, 2000068 [arXiv:1910.08605 [hep-th]].
  * [14] M. Larfors, A. Lukas, F. Ruehle and R. Schneider, [arXiv:2111.01436 [hep-th]].
  * [15] X. Gao and H. Zou, [arXiv:2112.04950 [hep-th]].
  * [16] Y. Gal, V. Jejjala, D. K. Mayorga Pena and C. Mishra, [arXiv:2003.10445 [hep-ph]].
  * [17] R. Deen, Y. H. He, S. J. Lee and A. Lukas, [arXiv:2003.13339 [hep-th]].
  * [18] J. Halverson, B. Nelson and F. Ruehle, JHEP 06 (2019), 003 [arXiv:1903.11616 [hep-th]].
  * [19] J. Halverson and C. Long, Fortsch. Phys. 68 (2020) no.5, 2000005 [arXiv:2001.00555 [hep-th]].
  * [20] M. Larfors and R. Schneider, Fortsch. Phys. 68 (2020) no.5, 2000034 [arXiv:2003.04817 [hep-th]].
  * [21] Y. H. He, S. Lal and M. Z. Zaz, [arXiv:2111.04761 [hep-th]].
  * [22] H. Y. Chen, Y. H. He, S. Lal and M. Z. Zaz, [arXiv:2006.16114 [hep-th]].
  * [23] H. Y. Chen, Y. H. He, S. Lal and S. Majumder, Phys. Lett. B 817 (2021), 136297 [arXiv:2011.00871 [hep-th]].
  * [24] Y. H. He, [arXiv:1812.02893 [hep-th]].
  * [25] Y. H. He, [arXiv:2101.06317 [cs.LG]].
  * [26] L. Alessandretti, A. Baronchelli and Y. H. He, [arXiv:1911.02008 [math.NT]].
  * [27] Y. H. He, E. Hirst and T. Peterken, J. Phys. A 54 (2021) no.7, 075401 [arXiv:2004.05218 [hep-th]].
  * [28] Y. H. He, K. H. Lee and T. Oliver, [arXiv:2010.01213 [math.NT]].
  * [29] J. Bao, S. Franco, Y. H. He, E. Hirst, G. Musiker and Y. Xiao, Phys. Rev. D 102 (2020) no.8, 086013 [arXiv:2006.10783 [hep-th]].
  * [30] V. Jejjala, A. Kar and O. Parrikar, Phys. Lett. B 799 (2019), 135033 [arXiv:1902.05547 [hep-th]].
  * [31] S. Gukov, J. Halverson, F. Ruehle and P. Sułkowski, Mach. Learn. Sci. Tech. 2 (2021) no.2, 025035 [arXiv:2010.16263 [math.GT]].
  * [32] A. Davies, P. Veličković, L. Buesing, et. al. Nature 600, 70–74 (2021).
  * [33] Y. H. He and S. T. Yau, [arXiv:2006.16619 [math.CO]].
  * [34] L. Amorós, O. Gasanova and L. Jakobsson, [arXiv:2109.11417 [math.AC]].
  * [35] Y. H. He and M. Kim, [arXiv:1905.02263 [cs.LG]].
  * [36] S. Craven, D. Croon, D. Cutting and R. Houtz, [arXiv:2112.07673 [hep-ph]].
  * [37] D. Poland, S. Rychkov and A. Vichi, Rev. Mod. Phys. 91 (2019), 015002 [arXiv:1805.04405 [hep-th]].
  * [38] M. Hogervorst and B. C. van Rees, JHEP 11 (2017), 193 [arXiv:1702.08471 [hep-th]].
  * [39] R. Slansky, Phys. Rept. 79 (1981), 1-128
  * [40] R. Feger and T. W. Kephart, Comput. Phys. Commun. 192 (2015), 166-195 [arXiv:1206.6379 [math-ph]].
  * [41] J. Fuchs and C. Schweigert, “Symmetries, Lie algebras and representations: A graduate course for physicists,” Cambridge University Press.
  * [42] S. Weinberg, “The Quantum Theory of Fields, Volume 1: Foundations,” Cambridge University Press.
  * [43] E. J. Kuo, A. Seif, R. Lundgren, S. Whitsitt and M. Hafezi, [arXiv:2106.13485 [cond-mat.str-el]].
  * [44] G. Kántor, V. Niarchos and C. Papageorgakis, [arXiv:2108.09330 [hep-th]].
  * [45] S. Krippendorf and M. Syvaeri, [arXiv:2003.13679 [physics.comp-ph]].
  * [46] B. M. Dillon, G. Kasieczka, H. Olischlager, T. Plehn, P. Sorrenson and L. Vogel, [arXiv:2108.04253 [hep-ph]].
  * [47] G. Barenboim, J. Hirn and V. Sanz, SciPost Phys. 11 (2021), 014 [arXiv:2103.06115 [cs.LG]].
  * [48] K. Desai, B. Nachman and J. Thaler, [arXiv:2112.05722 [hep-ph]].
  * [49] E. De Mello Koch, R. De Mello Koch and L Cheng, IEEE Access, vol. 8, pp. 106487-106505, 2020, [arXiv:1906.05212 [cs.LG]].
  * [50] N. Halmagyi and S. Lal, [arXiv:2007.10181 [math-ph]].
  * [51] J. Halverson, A. Maiti and K. Stoner, Mach. Learn. Sci. Tech. 2 (2021) no.3, 035002 [arXiv:2008.08601 [cs.LG]].
  * [52] H. Erbin, V. Lahoche and D. O. Samary, [arXiv:2108.01403 [hep-th]].
  * [53] K. Hashimoto, Phys. Rev. D 99 (2019) no.10, 106017 [arXiv:1903.04951 [hep-th]].