# Feminist Perspective on Robot Learning Processes

Juana Valeria Hurtado∗1 and Valentina Mejía∗ * These authors have contributed
equally to this work1 Department of Computer Science, University of Freiburg,
Freiburg im Breisgau, Germany

###### Abstract

As different research works report and daily life experiences confirm,
learning models can result in biased outcomes. The biased learned models
usually replicate historical discrimination in society and typically
negatively affect the less represented identities. Robots are equipped with
these models that allow them to operate, performing tasks more complex every
day. The learning process consists of different stages depending on human
judgments. Moreover, the resulting learned models for robot decisions rely on
recorded labeled data or demonstrations. Therefore, the robot learning process
is susceptible to bias linked to human behavior in society. This imposes a
potential danger, especially when robots operate around humans and the
learning process can reflect the social unfairness present today. Different
feminist proposals study social inequality and provide essential perspectives
towards removing bias in various fields. What is more, feminism allowed and
still allows to reconfigure numerous social dynamics and stereotypes
advocating for equality across people through their diversity. Consequently,
we provide a feminist perspective on the robot learning process in this work.
We base our discussion on intersectional feminism, community feminism,
decolonial feminism, and pedagogy perspectives, and we frame our work in a
feminist robotics approach. In this paper, we present an initial discussion to
emphasize the relevance of feminist perspectives to explore, foresee, en
eventually correct the biased robot decisions.

##  I INTRODUCTION

In the history of robotics, an important goal has been to equip robots with
the ability to make complex decisions, and in some situations, with better
effectiveness and precision than humans. These skills have already been
demonstrated in various applications such as space exploration, medical
applications, scene understanding, transport of materials, robots in the
industry, and manufacturing and assembly, among many others For robots to
correctly operate and achieve a required task, they need to sense their
surroundings, identify and understand the elements and actions occurring
around them, and interact with the world to perform a specific task.
Additionally, in social environments, besides completing the task, we also
expect that robots behave within certain social parameters similar to the
decisions humans make, so the shared space is safe and comfortable for us.
Machine learning algorithms have played an essential role in developing
robots, making learning processes widely used in the field.

With the increased precision and effectiveness a robot can execute a task, at
some point, it is given higher autonomy to make decisions executing more
complex tasks nearer to humans. This allows robots to plan a navigation route,
manipulate objects, compute object recognition under challenging situations,
interact with people, among many others. However, granting autonomy of
decisions to robots based solely on technical advances without taking into
account the social consequences and analyzing the development process is
potentially dangerous. This risk is particularly concerning in a society still
affected by the historical discrimination and oppression of diverse
populations. Indeed, recently it has been demonstrated that algorithms and
robots can reflect and even amplify these social problems, particularly
affecting identities and communities that are usually the least represented
[1, 2, 3, 4, 5, 6].

Since there is a learning process for robots to operate and these robots
interact with diverse groups of people, it is important then to question:
Which learning techniques are utilized? What information is used in these
methodologies and who is involved in the development process? Robotics is
thought of as a human advance. Why not consider robot learning as an integral
process towards the emancipation of human biases so that technology can
disrupt traditional inequities? With these questions in mind, we argue that a
feminist perspective in the robot learning process can first facilitate the
identification of possible discriminating behaviors. Second, foresee biased
outcomes that affect certain people by understanding how the learning process
is susceptible to induce biased results. Third, promote changes in society by
a more conscious development of robots that influence adequate behaviours.

In this paper, we specifically consider intersectional feminism , community
feminism, decolonial feminism and pedagogy perspectives to provide an initial
analysis to the learning process. We also aim to promote the discussion about
robot behavior and feminism, by highlighting the relevance of a critical
analysis of their interaction with humans. In this paper we do not analyze the
gender that is attributed to robots. Instead, we explore how the behavior that
robots learn can produce disproportionate effects on identities and
communities. Additionally, we discuss how including a feminist perspective in
the learning process can be reflected in the human-robot interaction.
Therefore, to facilitate the analysis, we assume that robots are not perceived
with a specific gender.

Our work is comprised in the area of feminist robotics. We do not propose the
feminist perspective in robot design and development as the only viable path
to improve robots decisions and mitigate biased outputs. Nevertheless,
different feminists theories can provide substantial insights towards a future
with more robots coexisting with us. Furthermore, feminist perspectives can
guide us in creating robots that not only stop replicating historical
discrimination but also impact society to promote and accelerate equality.
Previously, cyberfeminism was perceived by Sadie Plant as an ”alliance between
women, machinery, and new technology” implying the conception of mechanisms,
tools, or strategies towards equality. In this sense, feminist robotics also
provides an encouraging field with the potential of structuring a better
future. Therefore, we expect more diverse identities to get involved in the
development of robots cohabiting with us.

##  II PRECEDENTS OF FEMINISM AS AN AGENDA FOR CHANGE IN AI AND ROBOTICS

Technologies are created to benefit society and positively change the world.
In addition, we expect their advantages to generate better living conditions
for us as humans (and hopefully, for all living beings that share this planet
with us). Likewise, technology should not at any time harm or unequally
benefit people. Technologies must be designed to have an explicitly positive
and uniform performance across the diversity of their users. Nevertheless, in
recent years, it has been demonstrated that the effects of technological
advances are not uniform. Particularly, it was shown that algorithms trained
with biased data generate biased outputs [1, 2, 3, 4]. What is more, it has
been pointed out that discrimination in technological developments also arises
from the lack of diverse identities involved in the development process [7],
set up of sensors [8], models definition, and evaluation procedure [9].
Different works exploring fairness can be found in areas such as natural
language processing [1, 10], computer vision [3], robotics [9], among others.
These studies and daily life experiences confirm that technological
developments can present a noticeable disadvantage for less represented
identities such as women, sexual-orientation minorities, people of color,
people of lower socioeconomic status, and people with disabilities [5, 6].
With the potential and dangerous risk of replicating and amplifying historical
discrimination in AI and robotics, it is not exceptional that different
scholars present feminist perspectives to criticize and propose structural,
social, and technological changes in recent decades.

In the multidisciplinary intersection of technology and feminism, we find
Haraway’s cyborg manifesto. Here, Haraway provides a feminist insight by means
of the fictional figure of a cyborg as the combination of animal and machine,
encouraging to overcome the limitations of the traditional roles of gender
[11]. Moreover, Suchman explores a set of feminist principles to consider when
designing technologies [12], while Noble presents the term algorithmic
oppression to define the negative effects of algorithms on specific groups of
people [13]. Based on black-intersectional feminism, she studies how the
output of algorithms can disproportionately affect people depending on their
gender and race. Recently, the Feminist Data Manifest-No presents a joint
statement rejecting the use of harmful data and engaging in creating new data
available to train fairer models [14]. In a similar direction, the book Data
Feminism is presented to point out data ethics considering ideas of
intersectional feminism [15]. More closely, in the Human-Robot Interaction
(HRI) field, feminist considerations have been proposed in the design of
robots [16, 17]. Additionally, numerous works address questions regarding the
genre-related attributes of robots and how robots are perceived according to
those attributes [18, 19]. All these guidelines and discussions are important
towards the development of more equitative technology.

How can we see this approach in technology, most directly in learned models
for robot decision-making? Let us think more directly about the perspective of
feminisms in the aforementioned cases. This provides a holistic and integral
perspective to problematic situations. Moreover, it not only demonstrates the
importance and scope of several critical perspectives, movements and
organizations in the world, but in the same sense, it also evidences the need
for an enhancement in technologies that are increasingly present in our lives.
Consequently, the process in which robots learn to perform tasks and the
effects on society must be constantly analyzed. For whom are robots created,
who are they benefiting and who is involved in their development? What
learning techniques are being used?

##  III LEARNING PROCESS FOR ROBOTS TO ACQUIRE INTERACTION ABILITIES

Currently, learning-based approaches play an essential role in the development
of robots able to perform complex tasks. These learning approaches are
composed of many stages to define the general procedure, data to use, task to
learn, training setup, testing environment, and evaluation metrics, to name a
few. From the technical point of view, Supervised Learning (SL) and Inverse
Reinforcement Learning (IRL) are two techniques that have gained particular
attention in the past years in the robot learning field. SL uses data and
annotations as learning guidance. The data is usually gathered from the real
world or simulations. In this case, SL uses both data and annotations during
the learning process to optimize a model with the aim of generating
predictions that are as similar as possible to the wanted output
(annotations). Therefore, SL requires a careful annotation process that
relates the input data and the correct result. On the other hand, IRL is a
technique that allows the recovery of a cost function that explains an
observed behavior. In this case, the model aims to imitate an ”expert”
behavior. This allows the model learning interaction abilities that clone the
demonstrations of the expert.

It is clear the human involvement in the different stages of robot
development. From the social and technical point of view, human decisions
directly influence the final model with which the robot is going to operate.
It has been studied before how multiple social and technical factors can lead
to bias while learning navigation models [9]. The reported factors include
data-related issues, bias induced in the model optimization, evaluation-
induced bias, and the complexity of defining a fairness metric. We explore in
more detail these factors in the following section.

##  IV THE IMPACT OF THE MODELS LEARNED BY THE ROBOT

In the past years, different situations have been highlighted in which AI
algorithms and robots have explicit discrimination outcomes. Such is the case
of RGB-based detection systems that disproportionately fail on pedestrians of
darker skin color [8], the race bias in the facial recognition applications
[3], and the gender bias found in the hiring system of amazon [20]. However,
other less explicit discrimination could be incorporated, especially in robots
operating around humans. These situations have been explored in the learning
algorithms that are fed with data that include systematic discrimination, when
discrimination factors are included through the learned model or biased test
methodologies are unable to properly detect the unwanted results [9]. In other
words, bias in robotics is a problem with great potential to amplify and
perpetuate problems in society, and although bias in terms of race and gender
is more evident, it is not the only bias that can be incorporated into robots.

In the specific case of robotics, critical analysis from different social
perspectives is important since many discriminations and injustices happen on
the physical spectrum. Unlike algorithms, robots operate and learn from
physical interactions. For this reason, it is crucial to question the
processes of creating a robot that will operate around humans. Not only
because of the risk of physical harm to people but also because of the
complexity of the learning models obtained that are susceptible to complex
social problems. Although algorithms can learn discrimination through
historical data, some discriminations can be included during the learning
process, especially the discrimination linked to the behavior and reactions we
have as people in social situations.

The relevance of analyzing from different perspectives robots with the
potential of influencing the lives of diverse people is exposing current
social problems that are at risk of being incorporated in the robot’s model.
In this case, understanding the factors of the learning process that are
susceptible to replicate social bias is important to prevent unwanted outcomes
that perpetuate structures of discrimination. Therefore, taking a step to
generate tools that allow maximizing the benefit of robots. Consequently,
instead of robot behavior reflecting problems in society, we can promote
better dynamics and relationships.

##  V FEMINIST PERSPECTIVE ON ROBOT LEARNING PROCESS

From feminisms, the ability to understand from the intersectional lens the
established dynamics has allowed us to diversify and reconfigure numerous
spaces. We understand that the goal of a critical perspective is not the
recognition of identities. We understand that the identification of these has
fallen into co-optation of discourses at the service of the market. The
primary purpose of this paper is to think of a world where people can abolish
traditional social categories that have discriminated against diverse people
across history. Robotics is an appropriate field to evaluate social
categories, roles, and dynamics if we develop robots as social agents and
provide them with social skills for human safety and comfort.

Within feminisms, the theoretical stream has developed several concepts to
explain how oppressions shape human relations and consequently people. One
such concept is intersectionality. This approach recognizes the
interconnectedness of different categories of inequality: gender, ethnicity
and class, age. Additionally, it incorporates the concept of subjectivities
and experiences of bodies situated in diverse historical contexts. It was
firstly introduced by Kimberle Crenshaw with the conceptualization of black
feminism and its political view of how antiracist politics are integral, since
racism is a structure, one that cements society [21]. This approach follows
the same critique developed by feminists against single essentialist thinking,
and it emphasizes the importance of diverse and multicultural historical
contexts.

Decolonial feminism states that oppressions are not a hierarchy but an
assemblage that privileges whiteness and the global north. For Viveros, social
relations are consubstantial and co-extensive [22]. They are consubstantial
inasmuch as they generate experiences that cannot be divided sequentially but
for analytical purposes, and they are co-extensive because they co-produce
each other [22]. Quijano [23], one of the precursors of decolonial theory in
Latin America explains how colonialism is institutionalized through knowledge
as he highlights how sciences are measured by the global north structure.
Paulo Freire wrote “Pedagogy of the oppressed”, in which he longs for an
emancipatory education, one where the liberation from structures can be
thought of. He reflects on the relationship between student and teacher and
how in order to think of an integral process, these two must be at the same
level [24]. Therefore, allowing them to learn from each other, dismantling the
hierarchies that are imposed on us from the institutional framework. Here,
Freire argues that the deconstruction and subsequent equitable relationship is
the path of an emancipatory guidance [24].

In community feminism, the theoretical basis is to think of the community as a
horizontal assembly of relationships that understand the colonial heritage
that tries to classify them. Community feminist thinkers such as Adriana
Guzma´n state that just as we learn in our communities that are mediated by
racism, classism and sexism to relate to each other we relate to nature.
Community feminism is a social theory that can discuss with other theories to
raise tools and proposals.

The aforementioned Global South theories, allow us to guide the initial
discussion of the complex process of thinking of an integral learning process
for robot decision-making within a feminist perspective. The learning
techniques we discuss in this work are often inspired or compared to hegemonic
learning processes in humans: such as traditional, technological and
conductive. In this frame of learning, different agents have different
importance. Differently, in another approach, students are not passive agents,
and can have equally importance in the process as the teacher, as remarked by
the South Global work in [22]. Learning is an ongoing process that includes
continuous research, adaptive pedagogic methods, evaluation of the
effectiveness in the current social environment, and analysis of context and
data. This is especially important in robot learning that is a complex process
with high involvement of human judgment.Taking that into account, how do we
analyze the robot learning techniques and information used? How can we make
sure it is, in fact, an integral process? Robot learning, and especially in
social environments, demand research and advanced tools to equip them to
confront complex situations and decisions, just as Humans face them on a daily
basis.

From a social perspective, groups of people lacking in diversity have a
limited understanding of the outcomes that can cause discrimination as a
result of the robot’s actions, behavior, and interactions. We have seen
similar results from learning models used in technologies deployed in daily
environment such as voice recognition that initially struggled to respond to
women’s voice commands [25]. In this case, the women’s interaction with this
technology presented a foreseeable systematic failure. From a technical
perspective, the human judgment to decide on the different stages of the
learning process as well as the society representation in data impact the
trained model. It is expected that while using biased data we obtain biased
outcomes. A less explored learning stage that can lead to discrimination is
the evaluation procedure [9]. As exposed in [9], such is the case of the
testing experiments for learned models in robot navigation where no woman was
included [26]. If a navigation model is trained and validated for safety and
comfort while excluding certain groups of people, we can foresee biased human-
robot interaction.

To make the robots’ decisions integral and not biased by our own historical
structures, interculturality and decoloniality offers a perspective for
change. This perspective encourages a strength initiative and ethical-moral
agency to question and disrupt the established [27]. With this perspective, it
is possible to question the current configuration of the methodologies,
creation of work teams, processes to record data, among others. It is
important to highlight that using a reduced sample of the population in the
process of learning social parameters in robotics is problematic, specially
with socio-political tensions explained in the decolonial theory. Similarly,
having non-diverse identities in the development team can lead to overall
biased judgment. Humanity is rich in diversity and social interactions are
extremely complex. In this sense, this work encourages nourishing ourselves
with research, proposals, and data from diverse parts of the globe, going
outside of the north perspective. As Freire exposes in the student-teacher
dynamic, robot learning can be confirmed through integral mutual growth,
allowing robots to adapt to novel environments and constantly learn from
experiences. Consequently, a feminist perspective in the learning process of
robot decision-making, emphasizes the importance of diverse and multicultural
historical contexts by remarking the learning process includes unlearning and
questioning the predefined standards in robot learning.

##  VI CONCLUSIONS

In this paper, we explored with a feminist perspective possible discriminating
behaviors that can be incorporated into robots in the learning process.
Besides questioning how these behaviors are represented in learned models to
make decisions, we explore why robots can learn biased behavior. It is not
unexpected how biased outcomes mainly affect certain communities that have
been historically oppressed. In this paper, we used theories that explain how
susceptible is the learning process to induce biased results while
understanding different historical contexts. We highlighted how robots operate
employing a learning process comparable to some extent with humans. Moreover,
we acknowledge the influence of human judgment in the stages of robot learning
processes. We recognized robotics as a field with potential influence on human
relationships. With our initial discussion, we encourage the conscious
development of robots that influence acceptable behaviors by researching and
nourishing novel methodologies for robot decision-making. Particularly,
feminisms from the global south give us tools to discuss towards a robotic
field in which interdisciplinary teams can respond to safety and comfort
requirements for robots to operate also under parameters for diversity and
inclusion.

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# Anomalous Directed Percolation on a Dynamic Network using Rydberg
Facilitation

Daniel Brady  Department of Physics and Research Center OPTIMAS, RPTU
Kaiserslautern, D-67663 Kaiserslautern, Germany  Simon Ohler  Department of
Physics and Research Center OPTIMAS, RPTU Kaiserslautern, D-67663
Kaiserslautern, Germany  Johannes Otterbach  Orthonogal Otter UG, Berlin,
Germany  Michael Fleischhauer  Department of Physics and Research Center
OPTIMAS, RPTU Kaiserslautern, D-67663 Kaiserslautern, Germany

###### Abstract

The facilitation of Rydberg excitations in a gas of atoms provides an ideal
model system to study epidemic evolution on (dynamic) networks and self
organization of complex systems to the critical point of a non-equilibrium
phase transition. Using Monte-Carlo simulations and a machine learning
algorithm we show that the universality class of this phase transition can be
tuned. The classes include directed percolation (DP), the most common class in
short-range spreading models, and mean-field (MF) behavior, but also different
types of anomalous directed percolation (ADP), characterized by rare long-
range excitation processes. In a frozen gas, ground state atoms that can
facilitate each other form a static network, for which we predict DP
universality. Atomic motion then turns the network into a dynamic one with
long-range (Lévy-flight type) excitations. This leads to continuously varying
critical exponents corresponding to the ADP universality class, eventually
reaching MF behavior. These findings also explain the recently observed
critical exponent of Rydberg facilitation in an ultra-cold gas experiment
[Helmrich et al., Nature 577, 481 (2020)], which was in between DP and MF
values.

Non-equilibrium phase transitions [1] and the dynamical self organisation of
complex systems to the corresponding critical point [2, 3] are key phenomena
believed to be the reason for the abundance of scale invariance in nature.
They are characteristic for a broad range of spreading processes ranging from
epidemic dynamics of diseases [4, 5], earthquakes [6], and forest fires [7],
to neural networks [8], electric circuits, and information spreading in the
internet [9]. The most relevant non-equilibrium phase transitions are those
between an active and an inactive phase (absorbing state) of dynamical
activity. In contrast to their equilibrium counterpart, they are much less
understood and the corresponding theoretical models can in general not be
solved exactly. However, in analogy to equilibrium statistical mechanics, the
behavior near the critical point shows universal features characterized by
different non-equilibrium universality classes [1].

One of the most prominent universality classes of non-equilibrium phase
transitions is directed percolation (DP) [1], originally describing the flow
of fluids through porous materials. Janssen and Grassberger conjectured that
non-equilibrium phase transitions in any classical system should belong to the
DP universality class if they: (i) exhibit a continuous phase transition
between an active and a unique absorbing state, (ii) the transition is
characterized by a positive one-component order parameter, (iii) the dynamical
rules involve only short-range interactions, and (iv) the system has no
special attributes such as additional symmetries or quenched randomness [10,
11]. To date no counterexamples to these criteria have been found [12], and DP
universality has even been predicted in more general systems, e.g. with
multiple absorbing states [13, 14].

Figure 1:  Critical exponent $\beta$ of the active density as a function of
the root mean square atom velocity for a percolating gas with ${\langle
k\rangle=1.25}$ (see text). The exponent displays two plateaus:
${\beta=0.993(6)}$ for high and ${\beta=0.808(5)}$ for low gas velocities -
error given by standard deviation. Mean field predicts
${\beta_{\text{MF}}=1.00}$ and 3D directed percolation
${\beta_{\text{DP}}\approx 0.813}$. Between these limits the critical exponent
changes continuously, characteristic for anomalous directed percolation (ADP).
Errors in $\beta$ result from fitting (see text). The black line shows a field
theoretical approximation of $\beta$ and continues as a grey dashed line in
the ADP II phase. The black ticks on the x-axis display relevant velocity
scales (see main text). Inset: Schematic of spreading dynamics in the gas.
Excited Rydberg atoms (red) shift atoms in a spherical shell with radius
$r_{\mathrm{f}}$ and width $\delta r_{\mathrm{f}}$ into resonance (orange
shells). All points use the parameters ${b=0.3}$, ${\Delta/\gamma=2000}$, with
varied ${\Omega/\gamma\in[1,10]}$ and ${n_{0}r_{\mathrm{f}}^{3}\in[20,30]}$.

In spite of the apparent generality of the directed percolation universality
class, only few experimental platforms are known for which DP behavior has
unambiguously been proven. In 2007 the first such platform was found in
turbulent liquid crystals and a full set of critical exponents in ${d=2+1}$
dimensions was measured [15, 16]. Since then, interacting many-body systems of
Rydberg atoms in the facilitation regime have been suggested as a platform to
study absorbing state phase transitions, for which DP universal behavior was
predicted on a lattice [17], and subsequently experimentally observed in a 1D
gas [18]. One important aspect, relevant for the emergence of scale invariance
in dynamical processes, which these model systems lack is the effect of atomic
losses from the system. In a number non-conserving regime the gas density
decreases over time which drives the system to its critical point [19, 20], a
phenomenon called self-organized criticality (SOC) [2, 3]. An experiment
investigating Rydberg facilitation in a 3D gas, performed in this number non-
conserving regime [19], showed signatures of SOC, but a deviation from DP
universality, which was attributed to the self-organization process.

Through numerical experiments and analytic considerations we show that this
deviation is neither due to the self-organization process [19] nor due to
heterogeneity [21], but results from a violation of the Jansen-Grassberger
conditions leading to a dependency of the universality class of the underlying
absorbing-state phase transition on the relative velocity of the atoms in the
gas. Tuning the parameters which set the reference scale of the atomic
velocity, the system can either display DP, mean-field (MF), or anomalous
directed percolation (ADP) universality. In Fig. 1 numerical predictions of
the critical scaling exponent of the active density around the critical point
can be seen as a function of the root mean square (RMS) thermal velocity of
the atoms.

Several relevant real-life spreading processes go beyond the Jansen-
Grassberger conjecture and display different universal behavior of the
absorbing-state phase transition than DP. An example is the spread of diseases
by flying insects in addition to the spread by direct contact, which violates
the condition of short-range excitations [22, 23]. Likewise spreading
processes often take place on dynamical rather than static networks [24]. For
example in social networks, no individual is in contact with all friends
simultaneously all the time. Contacts often change on a time scale comparable
with the one of the spreading process itself and thus real contact networks
are essentially dynamic [25].

Rydberg facilitation in a gas of cold atoms provides a powerful platform which
opens a way to experimentally study transport phenomena on resolvable length
and time scales, as well as in various static and dynamic network
configurations. It allows for the investigation of different types of non-
equilibrium phase transitions beyond DP universality as well as the self
organization of complex dynamical systems to the corresponding critical
points.

While so far most experimental work on Rydberg facilitation focused on the
dephasing dominated regime, in which the system largely behaves classically,
the system can also be studied in the quantum limit by tuning microscopic
parameters. Here, as argued in Ref. [26], universal features different from DP
or ADP could emerge when quantum effects become relevant. Furthermore, while
facilitation in a frozen gas of atoms (i.e. when the thermal movement is much
slower than the spreading dynamics) constitutes a spreading process on a
random network [27], using advanced trapping techniques (i.e. tweezer arrays)
various lattice geometries [28, 29, 30] can be investigated.

##  I Critical scaling at the absorbing-state phase transition

In Rydberg facilitation systems, atoms are continuously driven from the ground
to a high-lying Rydberg state by a laser with Rabi frequency $\Omega$ and
detuning from resonance $\Delta$, where ${\Delta\gg\Omega}$. As a result of
the strong detuning, off-resonant (seed) excitations are strongly suppressed.
However, in the presence of a Rydberg atom, other atoms with distance
${r\approx r_{\mathrm{f}}\equiv\sqrt[6]{\frac{C_{6}}{\Delta}}}$ are shifted
into resonance as a result of the van-der-Waals (vdW) interaction, where
$C_{6}$ is the vdW coefficient.

Consequently, atoms within a spherical shell with volume ${V_{s}\approx
4\pi\delta r_{\mathrm{f}}r_{\mathrm{f}}^{2}}$ around a Rydberg atom can be
facilitated (i.e. excited on much faster time scales). Here ${\delta
r_{\mathrm{f}}\approx\frac{\gamma_{\perp}}{2\Delta}r_{\mathrm{f}}}$ is the
width of the facilitation shell with $\gamma_{\perp}$ being the linewidth of
the transition. The rate of excitation for atoms within the facilitation shell
is given by ${\Gamma_{\mathrm{f}}=2\Omega^{2}/\gamma_{\perp}}$. These
facilitated excitations can also be interpreted as infection processes, with a
global spreading rate given by ${\kappa=\Gamma_{\mathrm{f}}nV_{s}}$, where $n$
is the gas density. Spontaneous decay of Rydberg atoms back to the ground
state then corresponds to recovery with rate $\gamma$. For more details about
the microscopic dynamics, we refer the reader to the Methods section.

These systems feature a non-equilibrium phase transition between an absorbing
phase, for ${\kappa<\gamma}$, with no excited atoms in the thermodynamic
limit, and an active phase, for ${\kappa>\gamma}$, featuring widespread and
infinitely long-lived activity. Near the critical driving strength
${\kappa\approx\gamma}$, there is universal behavior characterised by three
scaling relations for the Rydberg density $\rho$, as well as the temporal and
spatial correlation lengths, $\xi_{\parallel}$ and $\xi_{\perp}$ respectively

$\displaystyle\rho$ $\displaystyle\sim(p-p_{c})^{\beta},$ (1a)
$\displaystyle\xi_{\parallel}$
$\displaystyle\sim|p-p_{c}|^{-\nu_{\parallel}},$ (1b)
$\displaystyle\xi_{\perp}$ $\displaystyle\sim|p-p_{c}|^{-\nu_{\perp}}.$ (1c)

Here ${p-p_{c}}$ corresponds to the distance of the control parameter from the
critical point, and $\beta$, $\nu_{\parallel}$, $\nu_{\perp}$ are critical
exponents. Finally, while seed excitations are strongly suppressed, they still
occur for finite separation of time scales with the rate ${\tau\sim
1/\Delta^{2}}$.

In the following, we consider the system in the SOC regime, allowing Rydberg
atoms to additionally decay to an inert state, effectively removing them from
the system, with the rate ${b\gamma}$ (see inset of Fig.2a). As a consequence,
the system drives itself to the critical density given in MF approximation by

$\displaystyle n_{c}=\frac{\Delta\gamma}{4\pi\Omega^{2}}r_{\mathrm{f}}^{-3}.$
(2)

The SOC dynamics for different initial gas densities can be seen in Fig. 2a.
In the initial active phase there is a fast loss of atoms to inert states
until the critical point is reached where this loss slows down substantially.
To observe universal critical behavior these two time scales of atom loss must
be well separated [31].

First we consider the limit where the thermal movement of atoms occurs on a
much slower timescale than the internal dynamics, rendering them effectively
static. In this frozen-gas limit, the spreading of excitations is constrained
to a random Erdős–Rényi network with the average network degree $\langle
k\rangle$ given by [27]

$\displaystyle\langle k\rangle=nV_{s}.$ (3)

Figure 2:  (a) Time dynamics of density of ground and Rydberg atoms in the
high temperature limit for ${b=0.3}$, ${\Omega/\gamma=3.7}$,
${\Delta/\gamma=1000}$, and ${v/r_{\mathrm{f}}\gamma=100}$, with varied
$n_{0}$ showing self-organized criticality to a single density
$n_{\mathrm{crit}}$. Machine learning (ML) predictions of the critical density
of the phase transition for each trajectory (horizontal dashed lines). (b) ML
predictions of critical density $n_{c}$ normalized by initial density $n_{0}$
depending on the rescaled driving (see main text) for ${\langle k\rangle>1}$
and ${v=100\>r_{\mathrm{f}}\gamma}$ (red) and ${v=0\>r_{\mathrm{f}}\gamma}$
(blue), as well as for ${\langle k\rangle<1}$ (inset). The exponent $\alpha$
is tuned until all data points collapse.

At ${\langle k\rangle=1}$ a transition occurs between a non-percolating
network of ground state atoms with distance $r_{\mathrm{f}}$, composed of many
small disconnected clusters, and a percolating network with one large cluster
on the order of the size of the system [32]. For ${\langle k\rangle<1}$ this
gives rise to a heterogeneous, non-universal Griffiths phase of the Rydberg
facilitation, replacing the absorbing-state phase transition. Above the
percolation transition, however, i.e. ${\langle k\rangle>1}$, the absorbing-
state phase transition is recovered [27, 33]. (The SOC dynamics does not
change the Erdős–Rényi character of the network, but only leads to a reduction
of $\langle k\rangle$.)

At high gas temperatures the continuous mixing of atomic positions and
subsequent fast decay of spatial correlations leads to the expectation of mean
field behavior regardless of $\langle k\rangle$ [34].

An unambiguous signature of universal behavior and a precise method for the
classification of a given system into a certain universality class is the
collapse of data obtained over a large parameter range onto specific scaling
functions. Following Ref. [19], we consider the density of atoms in active
states (i.e. in the ground and Rydberg state but not in the inert state) at
the critical point, $n_{c}$, normalized to the initial density $n_{0}$ as a
function of the generalized driving strength $\Omega^{2}n_{0}^{1/\alpha}$,
with $\alpha$ being tuned until all data points collapse onto a single curve.
Scale invariance requires

$\frac{n_{c}}{n_{0}}=f\Bigl{(}\Omega^{2}n_{0}^{1/\alpha}\Bigr{)}$ (4)

to hold over the entire parameter range, with a scaling function $f(x)$, which
can be chosen as
${f(x)=x_{c}^{\beta}(x^{\mu\beta}+x_{c}^{\mu\beta})^{-1/\mu}}$ [19], where
$x_{c}$ and $\mu$ are free parameters defining the position and sharpness of
the critical point. Finally, $\beta$ corresponds to the critical exponent from
eq. (1a).

For the high temperature limit the result is plotted in Fig. 2b (orange dots).
For both ${\langle k\rangle>1}$ and ${\langle k\rangle<1}$ (inset) we receive
a collapse of all data points onto a single power-law using the tuning
exponent ${\alpha=1.26(1)}$ and ${\alpha=1.08(1)}$ respectively and we extract
the critical exponents ${\beta_{\text{high$\langle k\rangle$}}=0.996(18)}$ and
${\beta_{\text{low$\langle k\rangle$}}=1.049(19)}$ respectively, which both
fall in line with the expected mean field exponent
${\beta_{\mathrm{MF}}=1.00}$. Errors are calculated from the covariance matrix
of the respective fit parameters.

For the frozen gas regime, on the other hand, (blue dots in Fig. 2b) we find
no collapse of data below the percolation threshold, i.e. ${\langle
k\rangle<1}$, for values of ${\alpha\in[0.5,2.0]}$, indicating non-universal
behavior which is consistent with the observation of a heterogeneous Griffiths
phase [27]. For ${\langle k\rangle>1}$ however, the data collapses onto a
single power-law for ${\alpha=0.88(1)}$, with the slope of this power-law
clearly differing from the high temperature slope. Furthermore, when using the
above mentioned fit function we obtain the power-law exponent
${\beta_{\mathrm{frozen}}=0.809(13)}$, very close to the expected 3D DP
critical exponent ${\beta_{\mathrm{DP}}=0.813}$ [12].

To unambiguously confirm DP and MF universality in the frozen-gas and high-
temperature limits respectively, we also determine the critical exponent
$\nu_{\parallel}$. To this end we consider the decay of excitations from the
fully excited state ${\rho(t=0)=n(t=0)}$, in the number conserving case, i.e.
${b=\tau=0}$. For the case of weak driving, i.e. in the absorbing phase
${\kappa<\gamma}$, a pure exponential decay to ${\rho(t\to\infty)=0}$ is
expected, while for strong driving, i.e. in the active phase
${\kappa>\gamma}$, a non-zero steady state density emerges [12]
${\rho(t\to\infty)>0}$. At the critical driving strength, i.e.
${\kappa=\kappa_{c}=\gamma}$, there is a power-law decay in the active density
of the form ${\rho\sim t^{-\delta}}$ with the exponent
${\delta=\beta/\nu_{\parallel}}$. For contact processes on networks, the
critical driving strength is expected to be slightly larger however, i.e.
${\kappa_{c}\gtrsim\gamma}$ [33]. For MF ${\delta=\beta=\nu_{\parallel}=1}$,
whereas for 3D DP universality a less steep slope with exponent
${\delta\approx 0.732}$ has been predicted, since
${\nu_{\parallel}^{\mathrm{DP}}=1.11(1)}$, see e.g. Ref. [1].

In Fig. 3 we see this expected behavior with an intermediate power-law for the
driving strength ${\kappa(\Omega,n_{0})=\kappa_{c}}$, which is exponentially
truncated as a result of finite size effects. The behavior around the critical
driving strength is very sensitive to the Rabi frequency $\Omega$, which is
reflected in the rather large error margins in $\delta$. Still, we find a good
agreement with the DP and MF predictions of $\delta$ and a clear difference
between the values $\delta$ takes in the two limits.

Figure 3:  Decay of the density of excited atoms over time with all atoms
initially excited ${\rho(t=0)=n_{0}}$ with varying driving $\Omega$ with
${n_{0}r_{\mathrm{f}}^{3}=20}$ for high temperature
(${v/r_{\mathrm{f}}\gamma=100}$, left) and the frozen gas
(${v/r_{\mathrm{f}}\gamma=0}$, right). Both plots display an absorbing state
phase transition with exponential decay for
${\kappa<\kappa_{c}\approx\gamma}$, decay to a steady state for
${\kappa>\kappa_{c}}$ and power-law decay for ${\kappa=\kappa_{c}}$. The
dashed lines represent a power-law decay with exponent ${\delta=1.0(1)}$
(left) and ${\delta=0.65(10)}$ (right). Furthermore we use
${\Delta/\gamma=1000}$ and ${b=0}$.

##  II Anomalous directed percolation

From the above discussion one would naively expect that there is a critical
value of the RMS atom velocity where a phase transition between DP and MF
behavior takes place. Astonishingly we find however in Fig. 1 for gas
temperatures between the two limits (and ${\langle k\rangle>1}$) a universal
collapse of data points with a _monotonously changing critical exponent_
$\beta$ over multiple orders of magnitude in the RMS gas velocity.

Increasing the temperature the system leaves the DP regime at rather low
velocities corresponding to the (very small) width of the facilitation shell
per facilitation time, i.e. ${v_{-}=\delta
r_{\mathrm{f}}\langle\Gamma_{\mathrm{f}}\rangle}$ (left mark in Fig.1). This
is because the number of ground-state atoms that can be facilitated by a
single Rydberg atom starts to increase once this velocity is exceeded. On the
other hand the network character of ground state atoms gets completely washed
out leading to MF behavior if the RMS velocity of a Rydberg atom is so large
that it flies a distance larger than the facilitation distance in a
facilitation time, i.e.
${v_{+}=r_{\mathrm{f}}\langle\Gamma_{\mathrm{f}}\rangle}$ (right mark in
Fig.1).

In the following we show that the critical behavior with continuously varying
exponents $\beta$ in the velocity range between these two limiting values is a
signature of ADP universality, resulting from effective long-range spreading
processes and heavy-tailed waiting time distributions [35].

Absorbing-state phase transitions in complex systems where excitation
distances follow a Lévy flight distribution as

$\displaystyle P(r)\sim\frac{1}{r^{d+\sigma}},$ (5)

where $d$ is the dimension and $\sigma$ is a free parameter, no longer fulfill
the constraints of the Janssen-Grassberger conjecture. Such systems however,
still show universal behavior, albeit with continuously varying critical
exponents depending on the value of $\sigma$ [12, 35]. The same is true if the
distribution of time intervals between successive excitations (i.e. waiting
time distribution $P(\delta t)$) is heavy-tailed. In general terms, the
algebraic spatial and temporal distributions effectively reduce the upper
critical dimension, and the critical exponents approach the MF values.

In the frozen gas limit each atom is confined to a cluster and has $k$ atoms
in its facilitation shell, with $k$ given by a Poissonian distribution as
${P(k)=\frac{(nV_{s})^{k}}{k!}\mathrm{e}^{-nV_{s}}}$. With increasing thermal
velocity, the probability that an atom finds another connection outside of its
original cluster increases. Since the underlying network is a random network,
even small distances in real space can correspond to completely new
connections, i.e. very distant jumps in the network.

For an initially excited Rydberg atom with velocity $v$, the distribution of
distances to the next facilitated atom can be seen in Fig. 4a. Outside of the
facilitation shell (vertical black dashed line), we find that this probability
decays as a power-law with an exponent $\sigma$ decreasing with increasing
atom velocity. For large distances the excitation probability is exponentially
truncated, with the scale given by ${v/\Gamma_{\mathrm{f}}}$.

Figure 4:  (a) Distribution of distances between excitations in units of mean
free path $\Lambda$ for different RMS gas velocities $v$. Half of the width of
the facilitation shell is given by the vertical gray dashed line. The black
dashed lines are combined power-law/exponential fits and the black solid lines
are analytical predictions given by eq. (9). (b) Power-law exponent $\sigma$
(blue) of infection distance distribution and field theoretical limits of ADP
(horizontal dashed lines, see main text), as well as critical scaling exponent
$\beta$ (gray) continuously changing between MF and DP plateaus for gas
velocities where $\sigma$ is within ADP bounds. And distribution of times
between excitations ${P(\delta t)}$ for (c)
${v/r_{\mathrm{f}}\gamma=10^{-3}}$, (d) ${v/r_{\mathrm{f}}\gamma=10^{-1}}$,
and (e) ${v/r_{\mathrm{f}}\gamma=10^{1}}$. All plots use ${\Omega/\gamma=20}$,
${\Delta/\gamma=2000}$, ${n_{0}r_{\mathrm{f}}^{3}=20}$.

In 3D systems, MF behavior occurs for ${\sigma<1.5}$ (lower black dashed line
in Fig. 4b), while for ${\sigma>2.118(17)}$ regular DP behavior is expected
(upper black dashed line in Fig. 4b) [12]. In between these limits, the long
range interactions are prevelant enough to disrupt DP universality, but not
strong enough to suppress all correlations. Here, the system falls into the
ADP universality class with its critical exponents taking on values between
their MF and DP limits and which vary continuously with $\sigma$ [12].

Fitting the spatial distribution of excitation distances with an exponentially
truncated power-law ${f(x)=c_{1}x^{-c_{2}}\mathrm{e}^{-c_{3}x}}$, with
${c_{2}=\sigma-1}$ (as ${P(\vec{r})\,\mathrm{d}^{3}x=P(r)4\pi
r^{2}\,\mathrm{d}r}$), we receive very good agreement between the data and the
fit function (dashed lines in Fig. 4a). From this we can extract the power-law
slope ${\sigma(v)}$ governing the flight distance distribution depending on
the thermal gas velocity seen in Fig. 4b.

For the waiting time distribution (seen in Fig. 4c-e for different $v$), we
find an exponential decay for the high temperature and frozen gas limits.
However, for gas velocities in the interval
${v/r_{\mathrm{f}}\gamma\in[0.01,1.0]}$ we find a deviation from a pure
exponential decay in the form of a power-law for times on the order of the
facilitation time $\Gamma_{\mathrm{f}}^{-1}$. This additional power-law decay
in the time distribution gives rise to a regime where both spatial and
temporal long-range processes are relevant (ADP II in Fig. 1).

For gas velocities in the interval ${v/r_{\mathrm{f}}\gamma\in[1.0,3.0]}$, we
find an exponential waiting time distribution, but a spatial power-law
distribution with ${\sigma>1.5}$, giving rise to the ADP I regime [1]. Here
the critical exponent of the activity scaling $\beta$ can be field
theoretically approximated in perturbation theory to one-loop order, which
yields [12]

$\displaystyle\beta=1+2\frac{3-2\sigma}{7\sigma}.$ (6)

For gas velocities ${v/r_{\mathrm{f}}\gamma\gtrsim 1.0}$ we see a very good
agreement between the field theoretical approximation of $\beta(\sigma(v))$
and our simulation results (black line in Fig. 1). With decreasing velocity,
i.e. entering the ADP II regime, the field theoretical predictions begin to
diverge (gray dashed line in Fig. 1) resulting from the non-exponential
distribution in waiting times and the failure of the perturbation expansion.

##  III Lévy Flights

The heavy-tailed distribution of excitation distances $P(r)$ is caused by
atomic motion as we will show in the following. To this end we describe the
distribution considering the distance $z$ a Rydberg atom would cover before
facilitating another atom. By discretizing space into infinitesimal steps
${\delta z}$ we can write the probability that the excitation happens after at
least $J$ steps as ${P(X>J)=(1-p_{\mathrm{exc}})^{J}}$. Here
$p_{\mathrm{exc}}$ is the excitation probability in a given time interval
${\delta t=\delta z/v}$. As the number of atoms in the facilitation shell of a
Rydberg atom in ${t+\delta t}$ are given by the Poissonian distribution, the
excitation probability reads

$\displaystyle p_{\mathrm{exc}}$
$\displaystyle=\sum_{k=0}^{\infty}P(k)(1-(1-p_{\uparrow})^{k})$ (7a)
$\displaystyle=1-\exp\Bigl{\\{}-\langle k\rangle\frac{\delta z}{\delta
r_{\mathrm{f}}}(1-\mathrm{e}^{-\delta
r_{\mathrm{f}}\Gamma_{\mathrm{f}}/v})\Bigr{\\}}.$ (7b)

Inserting (7b) into ${P(X\geq J)}$ and taking the limit ${\delta z\to 0}$ for
fixed $z$, with ${J=z/\delta z}$ and ${J\delta z=\mathrm{const.}}$, we receive
the exponential distribution as the continuum limit of the geometric
distribution. The probability density function of the flight distance, for
${z\geq 0}$, reads

$\displaystyle f_{Z}(z)=\xi\mathrm{e}^{-\xi z},$ (8)

with ${\xi=\langle k\rangle/\delta r_{\mathrm{f}}(1-\mathrm{e}^{-\delta
r_{\mathrm{f}}\Gamma_{\mathrm{f}}/v})}$. After the Rydberg atom flies the
distance $z$, an atom is facilitated in a random position around it with
distance $r_{\mathrm{f}}$, i.e. in spherical coordinates with uniformly
distributed random variables ${\theta\in[0,\pi)}$ and ${\varphi\in[0,2\pi)}$,
as well as a fixed ${r\equiv r_{\mathrm{f}}}$. The probability distribution of
the distance from the initial position of the Rydberg to the position where
the first atom is facilitated is given by

$\displaystyle P(r)=$ $\displaystyle 2\pi\xi
r\int_{0}^{\pi}\\!\\!\\!\mathrm{d}\theta\,\,\frac{\mathrm{e}^{-\xi(\sqrt{\cos^{2}\theta+r^{2}-1}-\cos\theta})}{\sqrt{\cos^{2}\theta+r^{2}-1}},$
(9)

where ${\vec{r}=r_{\mathrm{f}}\hat{e}_{r}(\theta,\varphi)+(0,0,z)^{T}}$. In
eq. (9) the distribution is given up to a numerically solvable integral and
the solution can be seen as the black solid lines in Fig. 4a. We see an
excellent agreement between the distribution given by eq. (9) and the
numerical data.

##  IV Methods

###  IV.1 Microscopic Description of the Rydberg Gas

We consider a three dimensional gas of $N$ atoms coupled between a ground
$\ket{g}$ and a Rydberg $\ket{r}$ state with a laser with Rabi-frequency
$\Omega$ and detuning $\Delta$. The unitary dynamics are described by the
Hamiltonian
${\hat{H}=\sum_{i}\Omega\hat{\sigma}_{i}^{x}+\Delta\big{(}\hat{V}_{i}-1\big{)}\hat{\sigma}_{i}^{rr}}$,
where $\hat{\sigma}_{i}^{rr}$ is the projection operator of the $i$th atom
onto its Rydberg state and the Rydberg-Rydberg van-der-Waals interaction is
given by the potential ${\hat{V}_{i}=\sum_{j<i}\frac{C_{6}}{r_{ij}^{6}}}$ with
van-der-Waals coefficient $C_{6}$, and $r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$
being the distance between the $i$th and $j$th atom.

In addition to the unitary dynamics, we account for spontaneous decay of the
Rydberg state into the ground or an additional dark $\ket{0}$ state described
by the jump operators ${\hat{L}_{1,i}=\sqrt{(1-b)\gamma}\ket{g}_{ii}\bra{r}}$
and ${\hat{L}_{2,i}=\sqrt{b\gamma}\ket{0}_{ii}\bra{r}}$ respectively. Here the
parameter ${b\in[0,1]}$ describes the portion of atoms lost from the system
following decay into inert states. Finally, dephasing of the Rydberg state is
accounted for by ${\hat{L}_{3,i}=\sqrt{\gamma_{\perp}}\ket{r}_{ii}\bra{r}}$.
Typically in Rydberg many-body gases ${\gamma_{\perp}\gg\Omega}$, allowing
classical rate equations to describe these systems to high accuracy [36].

The evolution of the $N$-body density matrix is given by the Lindblad master
equation
${\frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho}=-i[\hat{H},\hat{\rho}]+\hat{\mathcal{L}}(\hat{\rho})}$,
with the Lindblad superoperator $\hat{\mathcal{L}}(\hat{\rho})$ [37]. After
adiabatic elimination of coherences a set of rate equations for the occupation
probabilities in Rydberg ($p_{r}$) and ground states ($p_{g}$) of each atom
can be derived. These read

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}p_{r}^{(i)}$
$\displaystyle=-\frac{2\Omega^{2}\gamma_{\perp}}{\gamma_{\perp}^{2}+V_{i}^{2}}(p_{r}^{(i)}-p_{g}^{(i)})-\gamma
p_{r}^{(i)}$ (10a) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}p_{g}^{(i)}$
$\displaystyle=+\frac{2\Omega^{2}\gamma_{\perp}}{\gamma_{\perp}^{2}+V_{i}^{2}}(p_{r}^{(i)}-p_{g}^{(i)})+\gamma
p_{r}^{(i)},$ (10b)

with
${V_{i}=\Delta\big{(}-1+\sum_{j\in\Sigma}\frac{r_{\mathrm{f}}^{6}}{r_{ij}^{6}}\big{)}}$,
where $\Sigma$ is the set of all other atoms in the Rydberg state.

For all simulations we initiate random positions in a 3D box with length
${L=7r_{\mathrm{f}}}$ and periodic boundary conditions. Atom velocities are
sampled from a Maxwell-Boltzmann distribution, i.e. Gaussian in each
direction, with the temperature given by the RMS velocity $v$. Furthermore, we
use ${\gamma_{\perp}=20}$ and a fixed time step Monte-Carlo algorithm with
time step ${\gamma\,dt=0.0025}$.

###  IV.2 Machine Learning the Critical Density

Upon the system reaching its critical density, the total density continues to
decay as a result of a finite separation of time-scales in the system (see
Fig. 2a), which makes the unambiguous determination of the true critical point
difficult. Therefore, we developed a machine learning algorithm which learns
the critical density based on the time dependent total density of the gas. For
the case of high gas temperatures, specifically an average thermal velocity
${v_{\mathrm{th}}=100\>r_{\mathrm{f}}\gamma}$, the dynamics of the system are
accurately described by mean field equations [34]. For the mean field limit
the critical density $n_{c}$ is given by eq. (2).

We then train the algorithm by passing it a sub-sampled vector $X$, consisting
of 100 density points from $n(t)$, which are equally spaced in time. The times
are spaced in the interval ${\gamma t\in[0,2000]}$ and have a spacing of
${\gamma\,\delta t=20}$. All training trajectories use
${v_{\mathrm{th}}=100r_{\mathrm{f}}\gamma}$, i.e. are in the mean field limit,
and are the average of 50 Monte-Carlo runs. As all training data is in the
percolating limit, the system consists of roughly $10^{4}$ atoms, resulting in
a fast convergence of Monte-Carlo simulations.

In total, the algorithm is trained using 5866 trajectories, each with a unique
combination of $\Omega$ and $n_{0}$. The algorithm outputs ${Y\in[0,1]}$ and
predicts ${\bar{Y}\equiv n_{c}(\Omega)/n_{0}}$ for ${n_{c}<n_{0}}$ and
${\bar{Y}\equiv 1}$ for ${n_{c}\geq n_{0}}$, where we found a modified Huber-
Loss function [38] to give the most accurate predictions. This is defined as

$\displaystyle
L_{\Delta}(Y-\bar{Y})=\begin{cases}100\times\frac{1}{2}\frac{|Y-\bar{Y}|^{2}}{\bar{Y}},\;\;|Y-\bar{Y}|\leq\Delta\\\
100\times\Delta\frac{|Y-\bar{Y}|}{\bar{Y}},\;\;|Y-\bar{Y}|>\Delta\end{cases}.$
(11)

Using ${\Delta=1}$ yields an average percentage error of 4 %. We then apply
the algorithm to determine ${n_{\mathrm{crit}}/n_{0}}$ for arbitrary
$v_{\mathrm{th}}$.

###  IV.3 Critical Density Scaling

It has been shown for this system that the critical gas density $n_{c}$
separating the absorbing from the active phase exhibits scale-invariant
behavior with respect to the driving strength and the initial density of the
SOC dynamics. Applying the scaling ansatz
${n_{c}=n_{0}f(\Omega^{2}n_{0}^{1/\alpha})}$ and tuning the exponent $\alpha$
until all curves collapse onto a single power-law, following [19], the
critical exponent $\beta$ of the directed percolation universality class can
be extracted.

In order to find the exponent $\alpha$ where all curves collapse, we choose
exponents in the interval ${\alpha\in[0.5,2.0]}$. For each exponent we fit the
data using the heuristic fit function defined in the main text and defining a
loss function as the mean squared error between the data points scaled with
$\alpha$ and a fit using
${f(x)=x_{c}^{\beta}(x^{\mu\beta}+x_{c}^{\mu\beta})^{-1/\mu}}$ and determine
the mean squared error between the data points and $f(x)$. For all percolating
datasets, as well as for the non-percolating high temperature dataset, we find
a convex loss function with a well defined minimum where the data points
collapse. Additionally, we find no universal collapse of data for the low
temperature, non-percolating case (see Fig. 2).

### Acknowledgement

Financial support from the DFG through SFB TR 185, project number 277625399,
is gratefully acknowledged.

The simulations were [partly] executed on the high performance cluster
“Elwetritsch” at the RPTU Kaiserslautern which is part of the “Alliance of
High Performance Computing Rheinland-Pfalz” (AHRP). We kindly acknowledge the
support of the RHRZ.

### Authors contributions

MF conceived and supervised the project. Monte-Carlo numerical simulations
were performed by DB with support from SO and JO. The machine learning
algorithm was conceived by DB with support from SO and JO and implemented
together by DB and SO. All authors discussed the results and developed a
physical understanding describing the numerical findings. DB and MF wrote the
initial version of the manuscript with support by SO.

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