# Enhancing respiratory comfort with fan respirators: computational analysis
of carbon dioxide reduction, temperature regulation, and humidity control

Hana Salati  Patrick Warfield-McAlpine  David F Fletcher  Kiao Inthavong
Mechanical & Automotive Engineering, School of Engineering, RMIT University,
Bundoora, VIC, 3083, Australia  School of Chemical and Biomolecular
Engineering, The University of Sydney, Sydney, NSW, 2006 Australia

###### Abstract

Respirators provide protection from inhalation exposure to dangerous
substances, such as chemicals and infectious particles, including SARS-Covid-
laden droplets and aerosols. However, they are prone to exposure to stale air
as the masks create a microclimate influenced by the exhaled air. As a result,
exhaled air from the lungs accumulating in the mask produce a warm and humid
environment that has a high concentration of carbon dioxide (CO2), unsuitable
for re-inhalation. Fans are a favourable option for respirators to ventilate
the mask and remove the stale air. This study utilized computational fluid
dynamics simulation consisting of a hybrid Reynolds-averaged Navier-Stokes
(RANS)-large eddy simulation (LES) turbulence method to compare the inhalation
flow properties for different fan locations (bottom, top, and side) with
regular respirator breathing. Three mask positions, top, side, and bottom,
were evaluated under two breathing cycles (approximately 9.65s of breathing
time). The results demonstrated that adding a fan respirator significantly
decreased internal mask temperature, humidity, and CO2 concentration. The
average CO2 concentration decreased by 87%, 67% and 73% for locations bottom,
top and side respectively. Whilst the top and side fan locations enhanced the
removal of the exhaled gas mixture, the bottom-fan respirator was more
efficient in removing the nostril jet gas mixture and therefore provided the
least barrier to respiratory function. The results provide valuable insights
into the benefits of fan respirators for long-term use for reducing CO2
concentration, mask temperature, and humidity, improving wearer safety and
comfort in hazardous environments, especially during the COVID-19 pandemic.

###### keywords:

N95 , respirator , CFD , computational fluid dynamics , fan , SBES

††journal: Physics of Fluids

##  1 Introduction

Although over two years have passed since the start of the COVID-19 pandemic,
masks and respirators are still mandatory in some indoor settings such as
public transport and hospitals. Face mask requirements vary between different
countries and transmission peak times. Sense of nasal stuffiness, dry nose,
overheating, and nasal obstruction are the feelings experienced by healthcare
workers [Purushothaman et al., 2021]. Exposure to CO2 results in various
health effects, including increased respiratory rate, brain blood flow,
metabolic stress, headache, dizziness, confusion, and dyspnea [Atangana and
Atangana, 2020, Azuma et al., 2018, Raub et al., 2000]. Salati et al. [2021]
reported that a respirator could cause excessive CO2 inhalation by
approximately $7\times$ greater per breath compared with normal breathing.
Fan-powered masks have been proposed to resolve respiratory-related
complications and provide breathing comfort for an extended period.

Previous studies have investigated mask-related topics including filtration
[Dbouk and Drikakis, 2020, Verma et al., 2020, Arumuru et al., 2020], mask
leakage [Solano et al., 2022, Kang et al., 2021], and filter material [Xi et
al., 2022, Hossain et al., 2020]. Limited studies focused on how fresh air can
be supplied to the dead-space of the mask for the user and how exhalation
valves change the flow characteristics inside the mask. Bhattacharyya et al.
[2022] proposed a battery-powered active respirator design that uses an
exhalation valve on the side of an N95 respirator to provide fresh air with
every breath. The CO2 concentration for the conventional N95 and modified N95
were compared with normal breathing (control). The relative CO2 concentration
of the control case was 100$\%$, and modified N95 reduced it to 16.67$\%$.
Zhang et al. [2016b] measured the temperature and CO2 concentration within the
dead-space of an N95 mask using a fan installed at the middle front region of
the mask. CFD simulations and experimental measurements showed that adding the
ventilation fan reduced the dead-space temperature by 2∘C. Staymates [2020]
analyzed the flow characteristics of an N95 respirator with and without an
exhalation valve using Schlieren imaging. The results demonstrated that N95
respirators with exhalation valves are not suitable for reducing the
proliferation of infectious diseases. Despite the increasing use of fan
respirators in various hazardous settings, there is a lack of comprehensive
studies investigating the effects of different fan locations on inhalation
flow properties, highlighting the need for further research in this area.

This paper aims to understand the breathing airflow characteristics inside an
N95 respirator with different fan locations (bottom, top, and side of the
respirator). This study compares the re-inhalation air quality (air CO2 and
H2O concentration and temperature) from different fan locations with N95
respirator breathing (without a fan) and normal breathing (without a
respirator). To achieve these objectives, we applied a hybrid RANS-LES
turbulence model to investigate the fluid characteristics of the nasal airflow
jets inside a fan-powered N95 respirator during two consecutive breathing
cycles (8 seconds).

##  2 Method

###  2.1 Geometry model

A 3D model of the N95 respirator was imported into the modelling CAD software
(Ansys SpaceClaim), then positioned and attached to the human face. The
respirator was a non-fully sealed mask representing the natural leakage of a
respirator [Salati et al., 2021]. A gap was considered between the respirator
and the face, ranging from 1mm to 2mm due to the curvature of the face. This
study investigated the effects of adding a fan to existing N95 respirators
used in clinical practice. As such masks are designed to be lightweight and
disposable, they do not contain the necessary materials to provide an airtight
seal between the mask and the wearer’s face. Subsequently, a leakage threshold
was applied to best represent real-world applications. Further investigations
into more advanced sealable respirators may be investigated in future studies.

The nasal vestibules from a healthy human subject were taken from a CT scan
and imported to 3D Slicer® segmentation software. A micro fan (Pelonis
Technologies, AGA178) was represented with a cylindrical volume of $1.25\times
10^{-6}$ m3 (diameter of 20 mm) on the respirator in three different locations
top, bottom, and side as shown in Figure 1 (geometrical details of the fan
were neglected).

Figure 1: (a) Computational domain and applied boundary conditions for
ambient, fan, and respirator. (b) Nasal vestibule and breathing boundary
condition surfaces. (c) Different fan locations include the bottom-fan, side-
fan, and top-fan respirator. (d) Cross-sectional view of the volume mesh and
sizing.

###  2.2 Boundary conditions and meshing

An external domain outside of the nose was created in front of the face where
the exterior surface of the domain was set to atmospheric pressure. The
respiration was initiated by defining a mass flow rate at the flow exit
located at the internal region of each nostril (i.e., nasal vestibule). The
respiratory cycle was simplified to pure sine waves based on the measured
physiology data from Benchetrit et al. [1989] and used in Calmet et al.
[2020], allowing a simple method for describing the tidal volume, breathing
periods, and periodicity:

$\dot{m}=A\sin{\left(\frac{\pi(t-C)}{B}\right)}$ (1)

For inhalation the amplitude was $A=5.832\times 10^{-4}$ kg/s; period $B=1.65$
s; while the periodicity for multiple breathing cycles was $C=4(n-1)$ s, where
$n$ is the respiration cycle number. For exhalation $A=4.0945\times 10^{-4}$
kg/s; $B=2.35$ s; and $C=(1.65+4(n-1))$ s. This expression produces a tidal
volume of 500 mL, where the inhalation period is 1.65 s and the exhalation
period is 2.35 s. The mass flow rate was divided between the nostril outlets
evenly, although in reality, most nasal cavities exhibit some asymmetric flow
distribution [Tan et al., 2012, Wang et al., 2016, Kahana-Zweig et al., 2016].
The solution was initialized at time $t=0$ s with steady-state settings of
ambient conditions and then continued for 9.65 seconds (Figure 2).

Figure 2: Breathing cycle mass flow rate profile assigned to the nostrils. The
mass flow rate was divided equally between the nostrils and converted to the
velocity profile. The simulation was performed for two complete respiration
cycles with an additional inhalation phase which was discarded to avoid the
startup effects on the results.

The simulation time included three inhalation and two exhalation periods. The
first inhalation phase was excluded to avoid startup effects from the analysis
and to ensure the respiration results represented continuous breathing. The
ambient air temperature, relative humidity, and CO2 concentration were set to
20∘C, 30%, and 385 ppm, respectively. The exhaled air temperature, relative
humidity, and CO2 concentrations were assumed to be 36∘C, 100%, and 36000 ppm
[Salati et al., 2021].

The model was meshed following the method used in Van Strien et al. [2021]
where the LES component of the SBES model had a sufficiently fine mesh to
resolve the large eddies. The model was meshed with poly-hexcore elements
using Ansys Fluent 2021R1. A minimum surface sizing of 0.1 mm and 0.4 mm was
applied to all the regions. A body of influence was applied to the front of
the mask. Local mesh element refinements for regions of rapid flow changes
were created through ‘body of influence’ mesh zones with a sizing of 0.8 mm
was applied. Five prism layers were applied with a last ratio of 28 $\%$ and a
first layer height of 0.05 mm (Fig. 1d) ensuring the normalized wall distance
y+ < 1 on all walls. The strict requirements of normalized wall distances x+
and z+ were not required for the SBES model. We selected an iterative meshing
strategy to have a sufficiently fine mesh to resolve the energy-containing
eddies. The first step meshed the model using 1.5 million poly-hexcore
elements with five inflation layers on all walls. Then the model was run in a
pre-cursor steady state simulation, and the ratio of the smallest resolved
length scale to the turbulence integral length scale was checked to see if it
resolved 80% of the turbulence kinetic energy. Finally, we refined the region
of interest using a body of influence, with the final mesh consisting of 2.4
million cells.

The N95 mask surface was set as a porous zone with a porosity of 0.88, and a
viscous resistance coefficient of $1.12\times 10^{10}$ m-2 based on Zhang et
al. [2016a], and Salati et al. [2021]. The flow rate of the fan was 0.8 CFM. A
fan boundary condition was used, with a 5 Pa pressure jump applied to
reproduce the fan flow rate. The pressure jump was regulated to reach the
targeted flow rate of this specific fan. The wall temperature was set to 34∘C
in this region, and the wall thickness was assumed to be 6 mm [Senanayake et
al., 2021, Lindemann et al., 2002, 2004, Salati et al., 2021, Keck et al.,
2000].

###  2.3 Numerical setup

This study used the Stress-Blended Eddy Simulation (SBES) turbulence model.
SBES is a hybrid RANS-LES model that applies a Large Eddy Simulation (LES) in
the bulk flow region to resolve large eddies and an $k-\omega-$SST model was
used in Reynolds-Averaged Navier Stokes (RANS) region near the wall [Vara
Almirall et al., 2022, Bradshaw et al., 2022, Menter, 2018, Wickramarachchi et
al., 2023]. Recent studies have adopted the SBES model to simulate the
respiratory flow field [Vara Almirall et al., 2022, Bradshaw et al., 2022].
Similarly, in our previous study, we verified the SBES model to be reliable in
depicting airflow through a nasal cavity airway by comparing the pressure
variation at different regions of the airway with experimental measurements in
a 3D-printed model in Van Strien et al. [2021]. The study demonstrated that
applying best practices including sensitivity analyses for mesh size and time
steps ensured the accuracy and reliability of the numerical simulations.

The incompressible flow equations describing the conservation for mass and
momentum are expressed as:

$\displaystyle\frac{\partial}{\partial x_{i}}\left(\bar{u}_{i}\right)$
$\displaystyle=0$ (2) $\displaystyle\frac{\partial}{\partial
t}\left(\rho\bar{u}_{i}\right)+\frac{\partial}{\partial
x_{j}}\left(\rho\bar{u}_{i}\bar{u}_{j}\right)$
$\displaystyle=-\frac{\partial\bar{p}}{\partial
x_{i}}+\frac{\partial\sigma_{ij}}{\partial
x_{j}}+\frac{\partial\tau_{ij}}{\partial x_{j}}$ (3)

where $u_{i}$ is the flow velocity vector, $\rho$ is the fluid density and $p$
is the pressure, and $\sigma_{ij}$ is the stress tensor due to molecular
viscosity. The overbar $\bar{\varphi}$ on the scalar quantity $\varphi$
denotes a Reynolds-averaging operation in the RANS formulation and a spatio-
temporal filtering operation in the LES formulation. The turbulence stress
tensor
$\tau_{ij}\equiv\rho\left(\overline{u_{i}u_{j}}-\bar{u}_{i}\bar{u}_{j}\right)$
for the SBES formulation is defined to blend between the Reynolds stress
tensor $\tau_{ij}^{\text{RANS}}$ for the RANS formulation and the subgrid-
scale stress tensor $\tau_{ij}^{\text{LES}}$ for the LES formulation,
according to the blending function:

$\tau_{ij}=f_{s}\tau_{ij}^{\text{RANS}}+\left(1-f_{s}\right)\tau_{ij}^{\text{LES}}$
(4)

where $0\leq f_{s}\leq 1$ is the shielding function.

The time-step size chosen aimed to resolve the energy-containing eddies. An
estimate of the time resolution of the model was calculated based on the time
scale. To maintain stability whilst minimising simulation time a variable
time-step was employed to ensure a local Courant–Friedrichs–Lewy (CFL) number
< 1\. At peak inhalation and exhalation, a minimum time step of 1 $\times
10^{-}5$ was used. As the flow rate decrease the time step was increased to 1
$\times 10^{-}4$ for computational efficiency.

The SIMPLE scheme was selected for pressure-velocity coupling. The convective
terms of the momentum equations were resolved using second-order bounded
schemes. The gradients were calculated using the least-squares cell-based
methodology and time stepping was discretised using a second-order bounded
implicit scheme. Default under-relaxation factors were employed. Residual
monitors for energy were set to 1 $\times 10^{-}6$ and the remainder set to 1
$\times 10^{-}4$.

##  3 Results

The ambient and exhalation CO2 level is 385 ppm and 36,000 ppm, respectively
(the exhalation CO2 values is 93 times more than the ambient level). Exposure
to a CO2 concentration of 10,000 ppm for 30 minutes or more in healthy adults
results in respiratory acidosis, and excessive CO2 can cause increased
respiratory rate and metabolic stress. The CO2 distribution in the mid-
sagittal plane of the right nasal passage during the first respiration cycle
($t=1.65$ to 4.00 s) due to different fan locations is shown in Figure 3.
There were three fan locations (top, side, and bottom) which are shown in
Figure 1c.

Figure 3: CO2 distribution at the mid-sagittal plane of the right nasal
passage for different fan locations.

At $t=2$ s, the nostril jet formed due to the fan located in the ’top’
position containing a high level of CO2 (36,000 ppm) was directed towards the
bottom of the respirator. When the jet impinged on the interior surface of the
fan, the jet dispersed with some of the gas moving towards the fan at the top-
position. In the side-fan position, the redirection was toward the right-hand
side of the mask where the fan was located, and this caused a circulation in
the respirator. The fan’s location at the bottom of the respirator is parallel
to the nostril jets, which reduced the flow circulation inside the respirator.
At $t=3.6$ s, the top-fan position is filled with high CO2 concentration
(>28,000 ppm), while the bottom-fan position maintained a lower CO2
concentration during exhalation.

Inhalation occurs from $t=4-5.65$ s, and it is observed that during this
period, the residual CO2 inside the respirator during exhalation was
discharged for the bottom-fan respirator. Therefore, fresh air containing a
low level of CO2 (385 ppm) entered the mask region, and high CO2 concentration
build-up is avoided during inhalation. However, for the top-fan position, high
CO2 concentration can be seen inside the respirator during inhalation.

Figure 4 illustrates the air temperature distribution during the first
breathing cycle. During exhalation, the high-temperature air (36∘C) exited the
nostrils and accumulated in the mask and the temperature inside the respirator
increased. At $t=3.6$ s, the high-temperature air filled the entire top-fan
position respirator compared to the other locations. For the bottom-fan
position, the high-temperature air accumulated only around the bottom of the
mask. During inhalation, the high-temperature air inside the respirator was
inhaled back into the nasal cavity, and the temperature in the respirator
decreased for different fan locations. At $t=4.4$ s the air temperature
reached the ambient temperature (20∘C) for the bottom-fan respirator.

Figure 4: Air temperature distribution for the mid-sagittal plane of the right
nasal passage for different fan locations

Figure 5 shows the water vapour mass fraction in the mid-sagittal plane of the
right nasal nostril. The water vapour distribution follows that of the
temperature distribution. The airflow from the lung during exhalation was 100%
humid, and the respirator trapped the moist, exhaled air, and the water vapour
level within the respirator increased during exhalation. At $t=4.4$ s, the
bottom-fan respirator expelled all the humid air and the water vapour mass
fraction reached the ambient level for the inhalation.

Figure 5: Water vapor mass fraction distribution at the mid-sagittal plane of
the right nasal passage at different fan locations. The H2O mass fraction is
the specific humidity in kg of water vapour per kg of dry air. As a reference,
the peak H2O mass fraction of 0.039 provides 100% relative humidity at 36∘C.

Figure 6 shows the CO2 concentration, water vapour mass fraction, and
temperature during inhalation, and these results were compared with reported
data of a respirator mask without a fan taken from Salati et al. [2021].
Breathing without a fan respirator (labelled ‘no-fan’) increased all
parameters, i.e., an increase in the moisture and CO2 content and temperature.
The bottom-fan position produced the lowest inhaled CO2 concentration (580
ppm) compared with other fan locations. A slight increase in the CO2 level
occurred in the second breathing cycle, which demonstrates the residual build-
up of the gas-mixtures from exhalation. The bottom-fan position also produced
the greatest temperature decrease to 20∘C.

Figure 6: CO2 concentration, water vapour mass fraction, and air temperature
variation during the first and second breathing cycles. The results of the
fan-powered respirator with different fan locations are compared with the no-
fan respirator taken from Salati et al. [2021].

##  4 Discussion

N95 respirator masks are essential personal protective equipment (PPE) to
filter inhaled air to protect individuals from environmental pollutants like
viruses, pathogens, and allergens. Healthcare workers have the highest risk of
exposure to COVID-19 virus, therefore must wear PPEs to ensure their safety.
However, due to the high flow resistance and buildup of temperature and CO2,
while wearing the mask, healthcare workers prefer to use a less restrictive
respirator while working in high-risk environments. Despite the compulsory
regulations for the use of respirators within a healthcare environment Goh et
al. [2020], people refuse to wear respirators due to physical and breathing
discomfort Feng et al. [2020]. A modified N95 respirator can produce fresher
air and improve breathing conditions.

This study investigated the effect of different fan locations on an N95
respirator and analysed the airflow parameters of CO2, temperature, and
relative humidity inside the respirator. Salati et al. [2021] demonstrated
that respirator breathing increases the time-averaged inhaled air temperature
by 7% compared with normal breathing, which increases mucosal temperature,
especially in the turbinate region, that reduced the air conditioning capacity
of the nasal cavity. According to [Sullivan et al., 2014, Zhao et al., 2011,
Lindemann et al., 2008], nasal mucosal cooling (air conditioning function)
affects the sensation and perception of ample nasal airflow. Figure 6
demonstrated that a fan-respirator decreased (i.e. cooled) the inhaled air
temperature significantly where a bottom-fan position respiratory refreshed
the air sufficiently to the ambient temperature of 20∘C. This finding
demonstrates that using a fan-respirator eliminates the nasal fullness and
stuffiness sensation, particularly when the fan is located at the bottom of
the mask. A fan-less N95 respirator restricts mixing of the exhaled air with
the ambient air, causing an accumulation of CO2 inside this closed space area
[Salati et al., 2021]. This study showed that adding a fan to the existing
respirators can significantly reduce the accumulation of CO2 inside the
respirator.

Figure 7 demonstrated the time-average inhaled CO2 concentration rate during
the first and second inhalation cycles of breathing in a respirator mask with
the three positioned fans and without a fan. The data for the inhaled CO2
concentration rate without a fan was obtained from Salati et al. [2021]. The
effect of fan respirators was a significant decrease in CO2 exposure as the
fan allowed a portal for exhaled air to escape. The reduction percentage of
the CO2 concentration for each fan position positioned respirator mask
relative to the no-fan respirator is labelled above each bar in Figure 7.

Figure 7: Time-averaged inhaled CO2 using a fan-respirator for different fan
locations. The results of the fan-powered respirator with different fan
locations are compared with the no-fan respirator taken from Salati et al.
[2021], and the reduction percentage are written on top of each bar.

The bottom-fan respirator produced the greatest decrease in the average CO2
concentration 87% compared with the no-fan respirator breathing. The
performance of the top- and side-fan positions also reduced the average CO2
concentration but not as well as the bottom-fan position, where the reduction
ranged from 67% to 78%. Including a fan in a respirator provided significant
refresh of the air within the respirator by allowing the exhaled air to
escape. The fans provided a reduction in the CO2 concentration within the
respirator to a non-hazardous level, thus allowing the user to wear the
respirator for a prolonged period time safely, and avoiding the need to remove
the respirator.

Although our study investigated the effects of different fan locations in a
scenario with a gap between the respirator and face, it is worth noting that
fan respirators in a sealed respirator would be more beneficial in reducing
CO2 concentration, internal mask temperature, and humidity. Sealed respirators
removes the additional gas exchange occurring through the gaps, thus there is
a greater reliance on the fans to provide the gas exchange. Traditional
respirators, such as N95 masks, have limitations in terms of breathability,
comfort, and efficacy, particularly in high-risk environments with hazardous
substances or infectious particles. These limitations can result in poor
compliance with respiratory protection protocols, particularly in non-
occupational settings where respiratory protection is not typically required.
By contrast, the fan respirator offers improved ventilation and reduced CO2
concentration, mask temperature, and humidity, which could increase wearer
comfort and compliance with respiratory protection protocols. This could be
particularly important in the context of the COVID-19 pandemic, where the use
of respiratory protection has been widely recommended but may be met with
resistance or non-compliance due to discomfort or other factors.

Furthermore, the presence of a large temperature difference between the
ambient air at 20∘C and the human body at 34∘C, will induce a thermal plume
that can influence the flow field in the breathing zone by countering the
exhaled air to move upward. The presence of a thermal plume produces a
vertical flow field where experimental and computational studies showed
velocities of approximately 0.2 to 0.4 m/s [Tacutu et al., 2022, Zong et al.,
2022]. These low velocities are typical of natural convection flows that are
much smaller than forced convection flows. Salati et al. [2022] modelled
exhalation flows incorporating a thermal plume, where despite the thermal
plume the exhalation jet dominated the natural convective forces during most
of the breathing period of the sinusoidal breathing waveform where both
inhalation and exhalation are greater than 0.4 m/s. However, there is a small
window of time during breathing when the flow is less than 0.4 m/s which
occurs during the transition between inhalation and exhalation. In this period
the thermal plume may have an influence on the flow field.

##  5 Conclusion

Inhaled gas-mixture properties, including temperature, moisture, and CO2 were
monitored for fan respirator breathing with different fan locations using CFD
simulations. The results were compared with a no-fan respirator during
breathing. The inhaled air was sufficiently refreshed when a bottom-fan
position was used, where the air temperature in the mask for inhalation
returned to the ambient temperature, eliminating the nasal stuffiness
sensation associated with fan-less respirators. The average CO2 concentration
decreased significantly for all fan locations. The bottom-fan, side-fan, and
top-fan positions reduced the average CO2 concentration to 87%, 73%, and 67%,
respectively, compared with the no-fan respirator. The CO2 level exposure
using any fan-position respirator was within the non-hazardous level. The
nostril jets during exhalation move towards the bottom of the mask. Therefore,
the top-fan and side-fan respirators could not discharge the exhaled gas
quickly. Overall, the bottom-fan respirator was more efficient in removing the
nostril jet gas mixture with exhalation properties than the other fan
respirators.

## References

  * Arumuru et al. [2020] Arumuru, V., Pasa, J., Samantaray, S.S., 2020.  Experimental visualization of sneezing and efficacy of face masks and shields.  Physics of Fluids 32, 115129\.
  * Atangana and Atangana [2020] Atangana, E., Atangana, A., 2020\.  Facemasks simple but powerful weapons to protect against covid-19 spread: Can they have sides effects?  Results in physics , 103425.
  * Azuma et al. [2018] Azuma, K., Kagi, N., Yanagi, U., Osawa, H., 2018\.  Effects of low-level inhalation exposure to carbon dioxide in indoor environments: A short review on human health and psychomotor performance.  Environment international 121, 51–56.
  * Benchetrit et al. [1989] Benchetrit, G., Shea, S., Dinh, T.P., Bodocco, S., Baconnier, P., Guz, A., 1989\.  Individuality of breathing patterns in adults assessed over time.  Respiration physiology 75, 199–209.
  * Bhattacharyya et al. [2022] Bhattacharyya, N., Singh, S., Ghosh, R., Banerjee, A., Adhikari, A., Halder, A., Goswami, M., Chattopadhyay, A., Mondal, P., Natto, S., Ahmed, S., Mallick, A., Pal, S., 2022.  Development of a smart active respirator for comfortable and hygienic breathing.  Physics of Fluids 34, 051901\.  doi:10.1063/5.0091456.
  * Bradshaw et al. [2022] Bradshaw, K.R., Warfield-McAlpine, P., Vahaji, S., Emmerling, J., Salati, H., Sacks, R., Fletcher, D.F., Singh, N., Inthavong, K., 2022.  New insights into the breathing physiology from transient respiratory nasal simulation.  Physics of Fluids .
  * Calmet et al. [2020] Calmet, H., Inthavong, K., Owen, H., Dosimont, D., Lehmkuhl, O., Houzeaux, G., Vázquez, M., 2020.  Computational modelling of nasal respiratory flow.  Computer Methods in Biomechanics and Biomedical Engineering , 1–19.
  * Dbouk and Drikakis [2020] Dbouk, T., Drikakis, D., 2020\.  On respiratory droplets and face masks.  Physics of Fluids 32, 063303\.
  * Feng et al. [2020] Feng, S., Shen, C., Xia, N., Song, W., Fan, M., Cowling, B.J., 2020.  Rational use of face masks in the covid-19 pandemic.  The Lancet Respiratory Medicine 8, 434–436.
  * Goh et al. [2020] Goh, Y., Tan, B.Y., Bhartendu, C., Ong, J.J., Sharma, V.K., 2020.  The face mask: How a real protection becomes a psychological symbol during covid-19?  Brain, behavior, and immunity 88, 1–5.
  * Hossain et al. [2020] Hossain, E., Bhadra, S., Jain, H., Das, S., Bhattacharya, A., Ghosh, S., Levine, D., 2020.  Recharging and rejuvenation of decontaminated n95 masks.  Physics of Fluids 32, 093304\.
  * Kahana-Zweig et al. [2016] Kahana-Zweig, R., Geva-Sagiv, M., Weissbrod, A., Secundo, L., Soroker, N., Sobel, N., 2016\.  Measuring and characterizing the human nasal cycle.  PLoS One 11, e0162918.
  * Kang et al. [2021] Kang, Y., Park, J., Park, H., 2021.  Particle leakage through the exhalation valve on a face mask under flow conditions mimicking human breathing: A critical assessment.  Physics of Fluids 33, 103326\.
  * Keck et al. [2000] Keck, T., Leiacker, R., Heinrich, A., Kühnemann, S., Rettinger, G., 2000.  Humidity and temperature profile in the nasal cavity.  Rhinology 38, 167–171.
  * Lindemann et al. [2004] Lindemann, J., Keck, T., Wiesmiller, K., Sander, B., Brambs, H.J., Rettinger, G., Pless, D., 2004.  A numerical simulation of intranasal air temperature during inspiration.  Laryngoscope 114, 1037–1041.
  * Lindemann et al. [2002] Lindemann, J., Leiacker, R., Rettinger, G., Keck, T., 2002\.  Nasal mucosal temperature during respiration.  Clinical Otolaryngology & Allied Sciences 27, 135–139.
  * Lindemann et al. [2008] Lindemann, J., Tsakiropoulou, E., Scheithauer, M.O., Konstantinidis, I., Wiesmiller, K.M., 2008.  Impact of menthol inhalation on nasal mucosal temperature and nasal patency.  American journal of rhinology 22, 402–405.
  * Menter [2018] Menter, F., 2018.  Stress-blended eddy simulation (sbes)—a new paradigm in hybrid rans-les modeling, in: Progress in Hybrid RANS-LES Modelling: Papers Contributed to the 6th Symposium on Hybrid RANS-LES Methods, 26-28 September 2016, Strasbourg, France 6, Springer. pp. 27–37.
  * Purushothaman et al. [2021] Purushothaman, P., Priyangha, E., Vaidhyswaran, R., 2021.  Effects of prolonged use of facemask on healthcare workers in tertiary care hospital during covid-19 pandemic.  Indian Journal of Otolaryngology and Head & Neck Surgery 73, 59–65.
  * Raub et al. [2000] Raub, J.A., Mathieu-Nolf, M., Hampson, N.B., Thom, S.R., 2000\.  Carbon monoxide poisoning—a public health perspective.  Toxicology 145, 1–14.
  * Salati et al. [2022] Salati, H., Fletcher, D.F., Khamooshi, M., Dong, J., Ito, K., Vahaji, S., Inthavong, K., et al., 2022.  Exhaled jet and viral-laden aerosol transport from nasal sneezing.  Aerosol and Air Quality Research 22, 210338.
  * Salati et al. [2021] Salati, H., Khamooshi, M., Vahaji, S., Christo, F.C., Fletcher, D.F., Inthavong, K., 2021\.  N95 respirator mask breathing leads to excessive carbon dioxide inhalation and reduced heat transfer in a human nasal cavity.  Physics of Fluids 33, 081913\.
  * Senanayake et al. [2021] Senanayake, P., Salati, H., Wong, E., Bradshaw, K., Shang, Y., Singh, N., Inthavong, K., 2021.  The impact of nasal adhesions on airflow and mucosal cooling–a computational fluid dynamics analysis.  Respiratory Physiology & Neurobiology 293, 103719.
  * Solano et al. [2022] Solano, T., Ni, C., Mittal, R., Shoele, K., 2022\.  Perimeter leakage of face masks and its effect on the mask’s efficacy.  Physics of Fluids 34, 051902\.
  * Staymates [2020] Staymates, M., 2020.  Flow visualization of an n95 respirator with and without an exhalation valve using schlieren imaging and light scattering.  Physics of Fluids 32, 111703\.
  * Sullivan et al. [2014] Sullivan, C.D., Garcia, G.J., Frank-Ito, D.O., Kimbell, J.S., Rhee, J.S., 2014.  Perception of better nasal patency correlates with increased mucosal cooling after surgery for nasal obstruction.  Otolaryngology–Head and Neck Surgery 150, 139–147.
  * Tacutu et al. [2022] Tacutu, L., Bode, F., Năstase, I., Croitoru, C., Dogeanu, A., 2022.  Experimental and numerical study on the thermal plumes of a standing and lying human in an operating room.  Science and Technology for the Built Environment 28, 2–20.
  * Tan et al. [2012] Tan, J., Han, D., Wang, J., Liu, T., Wang, T., Zang, H., Li, Y., Wang, X., 2012.  Numerical simulation of normal nasal cavity airflow in chinese adult: a computational flow dynamics model.  European archives of oto-rhino-laryngology 269, 881–889.
  * Van Strien et al. [2021] Van Strien, J., Shrestha, K., Gabriel, S., Lappas, P., Fletcher, D.F., Singh, N., Inthavong, K., 2021.  Pressure distribution and flow dynamics in a nasal airway using a scale resolving simulation.  Physics of Fluids 33, 011907\.
  * Vara Almirall et al. [2022] Vara Almirall, B., Inthavong, K., Bradshaw, K., Singh, N., Johnson, A., Storey, P., Salati, H., 2022.  Flow patterns and particle residence times in the oral cavity during inhaled drug delivery.  Pharmaceuticals 15.  URL: https://www.mdpi.com/1424-8247/15/10/1259, doi:10.3390/ph15101259.
  * Verma et al. [2020] Verma, S., Dhanak, M., Frankenfield, J., 2020.  Visualizing droplet dispersal for face shields and masks with exhalation valves.  Physics of Fluids 32, 091701\.
  * Wang et al. [2016] Wang, T., Chen, D., Wang, P., Chen, J., Deng, J., 2016\.  Investigation on the nasal airflow characteristics of anterior nasal cavity stenosis.  Brazilian Journal of Medical and Biological Research 49.
  * Wickramarachchi et al. [2023] Wickramarachchi, A., Gregory, S.D., Khamooshi, M., 2023.  Comparison of single-stage and multi-stage drainage cannula flow characteristics during venoarterial extracorporeal membrane oxygenation.  Physics of Fluids 35, 021905\.
  * Xi et al. [2022] Xi, J., Barari, K., Si, X.A., Abdollahzadeh Jamalabadi, M.Y., Park, J.H., Rein, M., 2022\.  Inspiratory leakage flow fraction for surgical masks with varying gaps and filter materials.  Physics of Fluids .
  * Zhang et al. [2016a] Zhang, X., Li, H., Shen, S., Cai, M., 2016a.  Investigation of the flow-field in the upper respiratory system when wearing n95 filtering facepiece respirator.  Journal of occupational and environmental hygiene 13, 372–382.
  * Zhang et al. [2016b] Zhang, X., Li, H., Shen, S., Rao, Y., Chen, F., 2016b.  An improved ffr design with a ventilation fan: Cfd simulation and validation.  PloS one 11, e0159848.
  * Zhao et al. [2011] Zhao, K., Blacker, K., Luo, Y., Bryant, B., Jiang, J., 2011.  Perceiving nasal patency through mucosal cooling rather than air temperature or nasal resistance.  PLOS ONE 6, e24618.
  * Zong et al. [2022] Zong, J., Liu, J., Ai, Z., Kim, M.K., 2022.  A review of human thermal plume and its influence on the inhalation exposure to particulate matter.  Indoor and Built Environment 31, 1758–1774.


# Deep Bayesian Active Learning, A Brief Survey on Recent Advances

Salman Mohamadi  Computer Science and Electrical Engineering
West Virginia University
Morgantown, WV, USA
<EMAIL_ADDRESS>Hamidreza Amindavar  Electrical Engineering
Amirkabir University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
###### Abstract

Active learning frameworks offer efficient data annotation without remarkable
accuracy degradation. In other words, active learning starts training the
model with a small size of labeled data while exploring the space of unlabeled
data in order to select most informative samples to be labeled. Generally
speaking, representing the uncertainty is crucial in any active learning
framework, however, deep learning methods are not capable of either
representing or manipulating model uncertainty. On the other hand, from the
real world application perspective, uncertainty representation is getting more
and more attention in the machine learning community. Deep Bayesian active
learning frameworks and generally any Bayesian active learning settings,
provide practical consideration in the model which allows training with small
data while representing the model uncertainty for further efficient training.
In this paper, we briefly survey recent advances in Bayesian active learning
and in particular deep Bayesian active learning frameworks.

###### Index Terms:

Bayesian Active Learning, Deep learning, Posterior estimation, Bayesian
inference, Semi-supervised learning

##  I Introduction

Active learning is a framework in the area of machine learning in which the
model starts training by small amount of labeled data and then, in a
sequential process asks for more data samples from a pool of unlabeled data to
be labeled. In fact, the key idea behind this framework is to achieve desired
accuracy while lowering the cost of labeling by efficiently asking for more
labeled data. Therefore, compared to many other frameworks, active learning
tries to achieve the same or higher accuracy by using smaller amount of
labeled data [1, 2]. In contrast, semi-supervised learning addresses
relatively similar problem domain, however, there is differences between their
learning paradigm. In more detain, semi-supervised learning frameworks use the
unlabeled data for feature representations in order to better model the
labeled data [3, 4, 5]. Classical methods and tools in signal processing area,
with an emphasis on parametric modeling, have been used in many areas with
different type of data [7, 9, 8]. However recent advances in machine learning
and in particular, artificial neural network, have shown that non-parametric
models, are capable of almost modeling any type of data at the cost of higher
complexity. In this line, deep learning could be incorporated with classical
tools and frameworks of machine learning such as active learning in order to
improve the performance.

Representing the uncertainty of either embedding space or output probability
space is challenging especially when we are going to use deep learning tools
and concepts. It would show up in multiple scenarios in which we need to
measure the model uncertainty such as problems addressed by classical active
learning or its various versions. In practice, we are interested in forming a
desired output probability space in a typical active learning framework, which
necessitates the model uncertainty measurement and representation. In recent
years, multiple efforts have been performed to introduce deep learning tools
to active learning framework. Authors of [6], in their work as a pioneered
effort, discussed that bringing deep learning tools into active learning
setting poses two major problem; uncertainty representation and the amount of
data needed to train the model. In next two sections, in order have a tast of
basic concepts and some theoretical concepts, a brief overview of the learning
paradigm of active learning and Bayesian neural network will be presented.

Figure 1: Learning paradigm of active learning; as it is shown, at every
iteration the training starts from scratch on modified training data.

##  II Learning Paradigm in Active Learning

Simply put, The goal of active learning is to minimize the cost of data
annotation or labeling by efficiently selecting the unlabeled data to be
labeled. In more detail, in every iteration of active learning, a new labeled
data sample (or even batch of data) will be added to the training data, and
the training process starts from scratch. This sequential training will
continue until the accuracy reaches to the desired level. The overview of the
learning cycle of active learning is shown in Fig.1.

In each iteration, all of the unlabeled data samples will be evaluated in the
sense of uncertainty, and the best one will be selected by a function, namely
acquisition function. Generally speaking,acquisition function performs as a
function for uncertainty sampling, or diversity sampling or both of them.
While random acquisition is considered baseline, depending on data setting,
there are several acquisition functions, some of them for image data are
presented and approximately formulated in [6].

##  III Convolutional Neural Networks with Bayesian Prior

Deep learning algorithms mostly rely on training a convolutional neural
networks (CNNs). In fact with the recent advancement of CNNs, one of the main
advantages of CNNs is that they enable capturing spatial information of the
data [10]. Bayesian CNNs are capable of learning from small amount of data
while at the same time allow model uncertainty representation using an
acquisition function, which essentially makes them the key tool for active
learning on big data with high dimensional samples. Accordingly, authors of
[11] proposed a new version of CNNs with Bayesian prior on a set of
weights,i.e., Gaussian prior ${p(w);w}$ is ${\\{W_{1},W_{2},...W_{N}\\}}$.
Bayesian CNNs for classification tasks with a softmax layer, would be
formulated with a likelihood model:

$\displaystyle p(y=c|x,w)=softmax(f^{w}(x)).$ (1)

However, in order to practically implement this model, the Gal et.al [6]
suggest approximate inference using stochastic regularization techniques, and
perform it by applying dropout during the training as well as test process (to
estimating the posterior). In more detail, they do this by finding a
distribution, namely ${q_{\theta}^{*}(w)}$ which given a set of training data
${D}$, minimizes the Kullback-Leibler (KL) divergence between estimated
posterior and exact posterior ${p(w|D)}$. Finally using Mont Carlo integration
we will have:

$\displaystyle p(y=c|x,D)=\int p(y=c|x,w)p(w|D)dw$ (2)
$\displaystyle\approx\int p(y=c|x,w)q_{\theta}^{*}(w)dw$ (3)
$\displaystyle\approx\frac{1}{T}\sum_{t=1}^{T}p(y=c|x,\hat{w}_{t}),$ (4)

with ${q_{\theta}(w)}$ as dropout distribution and ${\hat{w}_{t}}$ as
estimation of ${q_{\theta}^{*}}$ [6].

##  IV Review on Recent advances

Bayesian inference methods allow the introduction of probabilistic framework
to machine learning and deep learning. The notion behind the introduction of
these kind of frameworks to machine learning is that learning from data would
be treated as inferring optimal or near optimal models for data
representation, such as automatic model discovery. In this sense, Bayesian
methods and here, specifically Bayesian active learning methods gain attention
due to their ability for uncertainty representation and even better
generalization on small amount of data [20]. One of the main work on
introduction of model uncertainty measurement and manipulation to active
learning is done by Gal and Ghahramani [6]. In fact the major contribution of
this paper is special introduction of Bayesian uncertainty estimation to
active learning in order to form a deep active learning framework. In more
detail, deep learning tools are data hungry while active learning tends to use
small amount of data, moreover, generally speaking deep learning is no
suitable for uncertainty representation while active learning relies on model
uncertainty measurement or even manipulation. Understanding these big natural
differences, authors of this paper found the Bayesian approach to be the
solution. In fact they refine the active learning general framework, which
usually work with SVM and small amount of data, to be well scaled to high
dimensional data such as images in the case of big data. in contrast to small
data. It practice, the authors put a Bayesian prior on the kernels of a
convolutional neural network as the training engine of active learning
framework. They refer to their previous work [21] suggesting that in order to
have a practical Bayesian CNN, the Bayesian inference could be done through
approximate inference in the Bayesian CNN, which makes the solution
computationally tractable. The interesting point is that they empirically
showed that dropout is a Bayesian approximation which can be used as a way to
introduce uncertainty to deep learning [22]. Here the point is that dropout is
not only used in the training process, i.e., they do inference applying
dropout before every weight layer during training, and also during test to
sample from the approximate posterior. This framework compared to other active
learning methods addressing big data for image, such as those using RBF,
performs better.

In this line, Jedoui et. al. [23] even go further in the level of uncertainty
of model by assuming that the output space is no longer mutually exclusive,
for instance we have more that one output for a single input. They empirically
show that classical uncertainty sampling does not perform better than random
sampling at these sort of tasks such as Visual Question Answering, therefore
they refer to [6, 18, 21, 19] take a similar strategy by using Bayesian
uncertainty in a semantically structured embedding space rather than modeling
uncertainty of the output probability space. Referring to Gal and Ghahramani’s
works, they mention that dropout can be interpreted as a variational Bayesian
approximation [21, 22], where the approximating distribution is a mixture of
two Gaussians with small variances which the mean of one of the Gaussians is
zero. The prediction uncertainty caused by uncertainty in the weights which
could be measured by approximate posterior using Monte Carlo integration.

Authors of reference [24] poses another similar problem by introducing deep
learning with relatively very large amount of data and big network into active
learning; and suggesting the necessity of systematic request for labeling in
the form of batch active learning (batch rather than sample in each active
learning iteration). They offer batch active learning in order to address the
problem that existing greedy algorithms become computationally costly and
sensitive to the model slightest changes. The authors propose a model aimed at
efficiently scaled active learning by well estimating data posterior. They
suggest scenarios in which more efficiency comes with one batch rather than
one data sample at each iteration. In this paper, authors take multiple active
learning methods, different acquisition functions, into account for their
objective of efficient batch selection in the sense of sparsity, or sparse
subset approximation. Moreover, they claim that based on their experiments,
that reference [6], as a Bayesian approach, outperforms others in many problem
setting. More specifically, with the same Bayesian active learning framework
proposed by [6] for capturing uncertainty, they target the most optimum batch
selection by finding data posterior, however as active learning setting does
not provide access to the labels before querying the pool set, they take
expectation w.r.t. the current predictive posterior distribution. This work
represents a closed-form solution consistent with basic theoretical setting of
reference [6].

Gal and Ghahramani [11] suggest a Bernoulli approximate variational inference
method, which prevents from CNNs over-fitting, i.e., by considering a Bayesian
prior on the weights of the network, it will become capable of learning from
small data, with no over-fitting or higher computational complexity. This work
can be considered as one of the basics of developing deep Bayesian active
learning. Authors of [12] with an emphasis on the fact that the nature of
active learning does not allows thorough comparison of models and acquisition
functions, explore more than 4 experiments of different models and acquisition
functions for multiple tasks of natural language processing, and finally show
that deep Bayesian active learning consistently provides the best performance.
Kandasamy et. al[13] underscore the fact that classical methods for posterior
estimation are query inefficient in the sense of estimating likelihood. They
suggest that a query efficient approach would be posterior approximation using
Bayesian active learning framework. Considering a Gaussian prior, a utility
function as a measure of divergence between probabilities (here densities)is
formed and at each time step, the estimated most informative query would be
sent to an oracle. Then the posterior will be updated. Their experiment
confirms the query efficiency of the approach. Reference [14] suggests that
since most of the methods of distance metric learning are sensitive to the
size of the data and randomly select the training pairs, they could not return
satisfactory results in many real world problems. The authors address this
problem by firstly introducing Bayesian approach to distance metric learning,
and then developing this framework by uncertainty modeling, which ends up in a
Bayesian framework for active distance metric learning. This framework enables
efficient pair selection for training as well as posterior probability
approximation. Reference [15] tries to combine the benefits of Bayesian active
learning and semi-supervised learning by introducing a active expectation
maximization framework with pseudo-labels. Authors use Mont Carlo dropout
introduce in [6] to compute the probability outputs.

Zeng et. al [16] address the question that is it possible to measure the model
uncertainty without fully Bayesian CNNs. Their results on several Bayesian
CNNs confirm that in order to represent the model uncertainty, one needs to
apply the Bayesian prior on only a few last layer before the output. With this
setting, the model would enjoy the benefits of both deterministic and Bayesian
CNNs. Lewenberg et. al [17] address the problem of active surveying using a
Bayesian active learner. In fact they use dimensionality reduction in an
active learning framework by applying Bayesian prior in order to design a
system to predict the answers to unasked question using a limited sequential
active question asking. Their framework outperform several state of-the-art
frameworks based on enhanced linear regression in terms of prediction accuracy
and response to missing data.

Houlsby et. al.[18] propose a measurement of predictive entropy which later is
used in a classification framework based on Gaussian Process. Their method
performs well compared to several similar classification frameworks while the
computational complexity is not greater than other methods. Finally they
develop their framework to a Gaussian process preference learning by extension
of binary preference learning to classification setting. One of the main
advantages of this method is that it provides desired accuracy at a relatively
low computational complexity cost.

##  V Conclusion

In this paper, we surveyed recent advances in Bayesian active learning, with
an emphasis on deep Bayesian active learning. Our main focus in on the works
contributing to the theory of this problem domain, however some interesting
works on the application of Bayesian active learning are surveyed. Bayesian
inference approaches hold very important place in machine learning and
recently, many attentions have shifted toward data, model and even network
structure uncertainty representation using these approaches. Bayesian active
learning and its intersection with deep learning concepts provide very
interesting frameworks in terms of theroy and application.

## References

  * [1] Settles, B. (2009). Active learning literature survey. University of Wisconsin-Madison Department of Computer Sciences.
  * [2] Sener, O., & Savarese, S. (2017). Active learning for convolutional neural networks: A core-set approach. arXiv preprint arXiv:1708.00489.
  * [3] Taherkhani, Fariborz, Nasser M. Nasrabadi, and Jeremy Dawson. ”A deep face identification network enhanced by facial attributes prediction.” In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, pp. 553-560. 2018.
  * [4] Taherkhani, Fariborz, Hadi Kazemi, and Nasser M. Nasrabadi. ”Matrix completion for graph-based deep semi-supervised learning.” In Proceedings of the AAAI Conference on Artificial Intelligence, vol. 33, pp. 5058-5065. 2019.
  * [5] Taherkhani, Fariborz, Hadi Kazemi, Ali Dabouei, Jeremy Dawson, and Nasser M. Nasrabadi. ”A weakly supervised fine label classifier enhanced by coarse supervision.” In Proceedings of the IEEE International Conference on Computer Vision, pp. 6459-6468. 2019.
  * [6] Gal, Yarin, Riashat Islam, and Zoubin Ghahramani. ”Deep bayesian active learning with image data.” arXiv preprint arXiv:1703.02910 (2017).
  * [7] Mohamadi, Salman, Hamidreza Amindavar, and SM Ali Tayaranian Hosseini. ”Arima-garch modeling for epileptic seizure prediction.” In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 994-998. IEEE, 2017.
  * [8] Mohamadi, Salman, Farhang Yeganegi, and Nasser M. Nasrabadi. ”Detection and Statistical Modeling of Birth-Death Anomaly.” arXiv preprint arXiv:1906.11788 (2019).
  * [9] Mohamadi, Salman, Farhang Yeganegi, and Hamidreza Amindavar. ”A New Framework For Spatial Modeling And Synthesis of Genome Sequence.” arXiv preprint arXiv:1908.03342 (2019).
  * [10] Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. ”Imagenet classification with deep convolutional neural networks.” Communications of the ACM 60, no. 6 (2017): 84-90.
  * [11] Gal, Yarin, and Zoubin Ghahramani. ”Bayesian convolutional neural networks with Bernoulli approximate variational inference.” arXiv preprint arXiv:1506.02158 (2015).
  * [12] Siddhant, Aditya, and Zachary C. Lipton. ”Deep bayesian active learning for natural language processing: Results of a large-scale empirical study.” arXiv preprint arXiv:1808.05697 (2018).
  * [13] Kandasamy, Kirthevasan, Jeff Schneider, and Barnabás Póczos. ”Bayesian active learning for posterior estimation.” In Proceedings of the 24th International Conference on Artificial Intelligence, pp. 3605-3611. 2015.
  * [14] Yang, Liu, Rong Jin, and Rahul Sukthankar. ”Bayesian active distance metric learning.” arXiv preprint arXiv:1206.5283 (2012).
  * [15] Matthias, Rottmann, Kahl Karsten, and Gottschalk Hanno. ”Deep bayesian active semi-supervised learning.” In 2018 17th IEEE International Conference on Machine Learning and Applications (ICMLA), pp. 158-164. IEEE, 2018.
  * [16] Zeng, Jiaming, Adam Lesnikowski, and Jose M. Alvarez. ”The relevance of Bayesian layer positioning to model uncertainty in deep Bayesian active learning.” arXiv preprint arXiv:1811.12535 (2018).
  * [17] Lewenberg, Yoad, Yoram Bachrach, Ulrich Paquet, and Jeffrey S. Rosenschein. ”Knowing What to Ask: A Bayesian Active Learning Approach to the Surveying Problem.” In AAAI, pp. 1396-1402. 2017.
  * [18] Houlsby, Neil, Ferenc Huszár, Zoubin Ghahramani, and Máté Lengyel. ”Bayesian active learning for classification and preference learning.” arXiv preprint arXiv:1112.5745 (2011).
  * [19] Karpathy, Andrej, Armand Joulin, and Li F. Fei-Fei. ”Deep fragment embeddings for bidirectional image sentence mapping.” In Advances in neural information processing systems, pp. 1889-1897. 2014.
  * [20] Ghahramani, Zoubin. ”Probabilistic machine learning and artificial intelligence.” Nature 521, no. 7553 (2015): 452-459.
  * [21] Gal, Yarin, and Zoubin Ghahramani. ”Dropout as a bayesian approximation: Insights and applications.” In Deep Learning Workshop, ICML, vol. 1, p. 2. 2015.
  * [22] Gal, Yarin, and Zoubin Ghahramani. ”Dropout as a bayesian approximation: Representing model uncertainty in deep learning.” In international conference on machine learning, pp. 1050-1059. 2016.
  * [23] Jedoui, Khaled, Ranjay Krishna, Michael Bernstein, and Li Fei-Fei. ”Deep Bayesian Active Learning for Multiple Correct Outputs.” arXiv preprint arXiv:1912.01119 (2019).
  * [24] Pinsler, Robert, Jonathan Gordon, Eric Nalisnick, and José Miguel Hernández-Lobato. ”Bayesian batch active learning as sparse subset approximation.” In Advances in Neural Information Processing Systems, pp. 6359-6370. 2019.