T<EMAIL_ADDRESS>XXXXXXX/PAR.20XX.00000 1 Comprehensive
Analysis of the Open Cluster Collinder 74–Comprehensive Analysis of the Open
Cluster Collinder 74

# Comprehensive Analysis of the Open Cluster Collinder 74

T. Yontan${}^{1\cc}$0000-0002-5657-6194  and R. Canbay2 0000-0003-2575-9892
1Istanbul University  Faculty of Science  Department of Astronomy and Space
Sciences  34119  Beyazıt  Istanbul  Turkey
2Istanbul University  Institute of Graduate Studies in Science  Programme of
Astronomy and Space Sciences  34116  Beyazıt  Istanbul  Turkey

(2023; XX.XX.XXXX XX.XX.XXXX XX.XX.XXXX XX.XX.XXXX XX.XX.XXXX0 )

###### Abstract

In this study, we have used the Gaia Third Data Release (Gaia DR3) to
investigate an intermediate-age open cluster Collinder 74. Taking into account
the stars with membership probabilities over 0.5 and inside the limiting
radius of the cluster, we identified 102 most likely cluster members. The mean
proper-motion components of Collinder 74 are estimated as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(0.960\pm 0.005,-1.526\pm 0.004$) mas
yr-1. We detected previously confirmed four blue straggler stars which show
flat radial distribution. Colour excess, distance, and age of the cluster were
estimated simultaneously by fitting PARSEC isochrones to the observational
data on Gaia based colour magnitude diagram. These values were derived as
$E(G_{\rm BP}-G_{\rm RP})=0.425\pm 0.046$ mag, $d=2831\pm 118$ pc and
$t=1800\pm 200$ Myr, respectively. The mass function slope was estimated as
$\Gamma=1.34\pm 0.21$ within the mass range $0.65\leq M/M_{\odot}\leq 1.58$
which is well matched with that of Salpeter. Stellar mass distribution
indicated that the massive and most likely stars are concentrated around the
cluster center. The total mass of the cluster was found to be 365
$M/M_{\odot}$ for the stars with probabilities $P>0$. Galactic orbit
integration shows that the Collinder 74 follows a boxy pattern outside the
solar circle and is a member of the thin-disc component of the Galaxy.

###### keywords:

Galaxy: open clusters and associations; individual: Collinder 74, stars:
Hertzsprung Russell (HR) diagram, Galaxy: Stellar kinematics,

††volume: 0

##  1 Introduction

Open star clusters (OCs), also known as Galactic clusters, are loosely bound
groups of stars that emerged from the same molecular cloud, sharing a common
origin and age. As relatively young and dynamically active systems, OCs
typically contain hundreds to thousands of stars that share similar chemical
composition, age, and distance. The formation origin of OCs plays a crucial
role in our understanding of stellar formation and evolution and makes them
ideal laboratories for studying stellar properties, such as temperature,
luminosity, and mass (Lada & Lada, 2003; Kim et al., 2017). Also, member stars
of OCs have similar movement directions in the sky which knowledge makes
proper-motion components useful tools to separate physical members from the
field star contamination (Sariya et al., 2021). This method provides a
reliable sample of member stars for the estimation of astrophysical parameters
for OCs (Bisht et al., 2020; Sariya et al., 2021).

In this study, we estimated the structural, kinematic, and astrophysical
properties of Collinder 74 (Coll 74) open cluster. The cluster is positioned
at $\alpha=05^{\rm h}48^{\rm m}40^{\rm
s}.8,\delta=+07^{\circ}22^{\rm{}^{\prime}}26^{\rm{}^{\prime\prime}}.4$ (J2000)
corresponding to Galactic coordinates $l=199^{\circ}.019,b=-10^{\circ}.379$
according to Cantat-Gaudin et al. (2020). Coll 74 is a centrally concentrated
old open cluster located in the third Galactic quadrant toward the Galactic
anti-center region. Ann et al. (1999) afforded UBVI CCD photometry and
suggested that the age of the cluster is $1300\pm 200$ Myr and it is located
at $2511\pm 245$ pc from the Sun. Tadross (2001) performed UBV CCD
observations and defined colour excess, distance and age of the cluster as
$E(B-V)=0.38$ mag, $d=2254$ pc and $t=1600$ Myr, respectively. Dias et al.
(2006) presented the kinematics of the cluster using UCAC2 Catalogue positions
and proper motions. They derived proper-motion components as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(-0.49,-3.49)$ mas yr-1. Carraro &
Costa (2007) investigated CCD photometry in the $V$ and $I$-bands and obtained
colour excess, distance and age of the cluster as $E(B-V)=0.28\pm 0.04$ mag,
$d=1500$ pc and $t=3000$ Myr, respectively. From the analyses of $BVI$
photometry Hasegawa et al. (2008) concluded that the Coll 74 is 3680 pc
distant from the Sun and 1400 Myr old cluster. Dias et al. (2014) derived the
mean proper-motion components by taking into account the U.S. Naval
Observatory CCD Astrograph Catalogue (UCAC4; Zacharias et al., 2013) as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(1.92\pm 0.55,-3.00\pm 0.10$) mas yr-1.
Loktin & Popova (2017) redetermined main parameters using published
photometric measurements provided by 2MASS catalogue for 959 clusters
including Coll 74. They estimated colour excess, distance and age as
$E(B-V)=0.418$ mag, $d=3293$ pc and $t=1400$ Myr, respectively. Also, from
RAVE catalogue Loktin & Popova (2017) estimated proper-motion components in
equatorial coordinate system as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(0.294\pm 0.181,-1.091\pm 0.172$) mas
yr-1, respectively.

Table 1: Basic parameters of the Collinder 74 open cluster collected from the
literature.

$E(B-V)$ | $\mu_{\rm V}$ | $d$ | [Fe/H] | $t$ | $\langle\mu_{\alpha}\cos\delta\rangle$ | $\langle\mu_{\delta}\rangle$ | $V_{\gamma}$ | Ref
---|---|---|---|---|---|---|---|---
(mag) | (mag) | (pc) | (dex) | (Myr) |  (mas yr-1) |  (mas yr-1) |  (km s${}^{-1})$ |
0.38$\pm$0.04 |  13.08$\pm$0.25 |  2511$\pm$245 | 0.07 |  1300$\pm$200 | $-$ | $-$ | $-$ | (01)
0.38 | 13.00 | 2254 | 0.054 | 1600 | $-$ | $-$ | $-$ | (02)
0.38 | 13.21 | 2549 | $-$ | 1300 | $-$ | $-$ | $-$ | (03)
$-$ | $-$ | $-$ | $-$ | $-$ | -0.49 | -3.49 | $-$ | (04)
0.28$\pm$0.04 |  11.75$\pm$0.10 | 1500 | $-$ | 3000 | $-$ | $-$ | $-$ | (05)
0.36 | 13.95 | 3680 | -0.38 | 1400 | $-$ | $-$ | $-$ | (06)
0.604 | 13.98 | 2637 | $-$ | 1230 | 0.91 | -3.86 | $-$ | (07)
$-$ | $-$ | $-$ | $-$ | $-$ |  1.92$\pm$0.55 |  -3.00$\pm$0.10 | $-$ | (08)
$-$ | $-$ | 2510 | 0.050 | 1288 | $-$ | $-$ | $-$ | (09)
0.418 | 12.588 | 3293 | $-$ | 1400 |  0.294$\pm$0.181 |  -1.091$\pm$0.172 | $-$ | (10)
$-$ | $-$ | $-$ | $-$ | $-$ |  0.77$\pm$2.15 |  0.47$\pm$3.54 | $-$ | (11)
$-$ | $-$ | $2453_{-484}^{+797}$ | $-$ | $-$ |  1.011$\pm$0.016 |  -1.512$\pm$0.016 | $-$ | (12)
$-$ | $-$ | $2453_{-484}^{+797}$ | $-$ | $-$ |  1.011$\pm$0.016 |  -1.512$\pm$0.016 |  20.18$\pm$0.39 | (13)
$-$ | $-$ |  2747$\pm$332 | $-$ |  2190$\pm$131 |  0.981$\pm$0.200 |  -1.497$\pm$0.199 | $-$ | (14)
$-$ | $-$ | $2453_{-484}^{+797}$ |  -0.05$\pm$0.03 | $-$ |  1.011$\pm$0.121 |  -1.512$\pm$0.122 |  15.94$\pm$17.57 | (15)
0.274 | 11.99 |  2498$\pm$494 | $-$ | 1900 |  1.011$\pm$0.121 |  -1.512$\pm$0.122 | $-$ | (16)
0.391$\pm$0.076 | $-$ |  2153$\pm$144 |  -0.083$\pm$0.084 | 2760 |  0.995$\pm$0.170 |  -1.528$\pm$0.175 |  20.20$\pm$0.80 | (17)
$-$ | $-$ | 2356 | $-$ | 2100 |  1.011$\pm$0.121 |  -1.512$\pm$0.122 |  20.18$\pm$0.39 | (18)
0.511$\pm$0.074 | $-$ |  2466$\pm$22 | $-$ |  627$\pm$348 |  0.964$\pm$0.007 |  -1.546$\pm$0.007 |  20.93$\pm$4.10 | (19)
0.301$\pm$0.033 |  13.052$\pm$0.088 |  2831$\pm$118 |  0.052$\pm$ 0.034 |  1800$\pm$200 |  0.960$\pm$0.005 |  -1.526$\pm$0.004 |  20.55$\pm$0.41 | (20)

(01) Ann et al. (1999), (02) Tadross (2001), (03) Lata et al. (2002), (04)
Dias et al. (2006), (05) Carraro & Costa (2007), (06) Hasegawa et al. (2008),
(07) Kharchenko et al. (2013), (08) Dias et al. (2014), (09) Marsakov et al.
(2016), (10) Loktin & Popova (2017), (11) Dias et al. (2018), (12) Cantat-
Gaudin et al. (2018), (13) Soubiran et al. (2018), (14) Liu & Pang (2019),
(15) Zhong et al. (2020), (16) Cantat-Gaudin et al. (2020), (17) Dias et al.
(2021), (18) Tarricq et al. (2021), (19) Hunt & Reffert (2023), (20) This
study

With the first data release of the Gaia (Gaia Collaboration et al., 2016),
many researchers investigated the astrometric and kinematic properties of the
Coll 74. According to literature studies performed with Gaia data, values of
the mean radial velocity of the cluster differs from $15.94\pm 17.57$ km s-1
(Zhong et al., 2020) to $20.20\pm 0.80$ km s-1 (Dias et al., 2021) and
distance to the Sun changes between 1500 pc (Carraro & Costa, 2007) and 3680
pc (Hasegawa et al., 2008). Also, the age of the cluster varies from 1230 Myr
(Kharchenko et al., 2013) to 3000 Myr (Carraro & Costa, 2007). The literature
results are listed in Table 1 for detailed comparison. The main purpose of
this study is to find out structural, astrophysical, and kinematic properties
of the Coll 74 open cluster.

##  2 Data

The astrometric, photometric, and spectroscopic data for Coll 74 open cluster
was taken from Gaia’s third data release (Gaia DR3, Gaia Collaboration et al.,
2023). To do this, we used the central equatorial coordinates of Cantat-Gaudin
et al. (2020) $\langle\alpha,\delta\rangle=(05^{\rm h}48^{\rm m}40^{\rm
s}.8,+07^{\circ}22^{\rm{}^{\prime}}26^{\rm{}^{\prime\prime}}.4$) and gathered
the detected stars in the direction of the cluster for
35${}^{\rm{}^{\prime}}$-radius field. Hence, we reached 73,326 stars within
the applied radius. The finding chart of the Coll 74 ($35^{\prime}\times
35^{\prime}$) is shown in Figure 1. The main cluster catalogue contains each
stars’ position ($\alpha$, $\delta$), photometric magnitude and colour index
($G$, $G_{\rm BP}-G_{\rm RP}$), trigonometric parallax ($\varpi$), proper-
motion components ($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$), radial velocity
($V_{\gamma}$) and their errors within the $8<G\leq 22$ mag.

To achieve reliable structural and astrophysical parameters for Coll 74, we
obtained a faint limited magnitude of the used data. For this, we calculated
the number of stars that correspond to the $G$ magnitude intervals. The
histogram of a number of stars versus $G$ magnitudes is shown in Figure 2,
where a number of stars rise towards the fainter $G$ magnitudes and declines
after a certain limit. This limit value is $G=20.5$ mag for the Coll 74 and in
the following analyses, we used only the stars brighter than $G=20.5$ mag. We
calculated the mean photometric errors of the stars for $G$ magnitude
intervals. The mean errors for $G$ and $G_{\rm BP}-G_{\rm RP}$ colour indices
reach up 0.011 and 0.228 mag for $G=20.5$ limiting magnitude, respectively.
The photometric errors for $G$ magnitudes and $G_{\rm BP}-G_{\rm RP}$ colour
indices versus $G$ magnitude intervals are shown in Figure 3.

Figure 1:  Finding chart of the Coll 74 for $35^{\prime}\times 35^{\prime}$
region. Up and left directions represent North and East, respectively.
Figure 2: Distribution of the stars in the direction of Coll 74 for $G$
magnitude intervals. The photometric completeness limit is indicated by a red
dashed line. Figure 3:  Distribution of mean photometric errors obtained for
$G$ apparent magnitude (a) and $G_{\rm BP}-G_{\rm RP}$ colour index versus $G$
magnitude intervals.

##  3 Results

###  3.1 Spatial structure of Collinder 74

To interpret stellar distribution within the cluster we constructed the radial
density profile (RDP) considering adopted central equatorial coordinates
presented by Cantat-Gaudin et al. (2020). We divided the $35^{\prime}$ cluster
region into many concentric rings surrounding the cluster center and
calculated stellar densities ($\rho(r)$) from the stars within $G\leq 20.5$
mag. The stellar densities in $\it i^{\rm th}$ ring were calculated by the
equation of $R_{i}=N_{i}/A_{i}$, where, $N_{i}$ and $A_{i}$ indicate the
number of stars and area of related ring, respectively. To visualise RDP, we
plotted stellar density distribution versus distance from cluster center and
fitted the empirical King (1962) model identified by the following equation:

$\rho(r)=f_{\rm bg}+\frac{f_{0}}{1+(r/r_{\rm c})^{2}}$ (1)

Here, $r$ is the radius of the cluster. The $f_{\rm bg}$, $f_{0}$, $r_{\rm c}$
are the background stellar density, the central stellar density and the core
radius, respectively. We used the $\chi^{2}$ minimisation technique for RDP
analyses and estimated $f_{\rm bg}$, $f_{0}$ and $r_{\rm c}$. In Figure 4 we
showed the best-fit result of RDP which is demonstrated by the black solid
line. It can be seen from the figure that stellar density is higher near the
cluster center and it flattens toward the outer region of the cluster and at a
point merges with the field star density. This point is described as limiting
radius ($r_{\rm lim}$) and visually adopted as $10^{\prime}$. The best-fit
solution of RDP analyses resulted that the structural parameters to be
$f_{0}=8.42\pm 0.35$ stars arcmin-2, $f_{\rm bg}=5.45\pm 0.16$ stars arcmin-2
and $r_{\rm c}=1.38\pm 0.12$ arcmin for the Coll 74.

Figure 4: The RDP of King (1962) for Coll 74. Stellar density errors were
determined from Poisson statistics $1/\sqrt{N}$, where $N$ is the number of
stars. The fitted black curve and horizontal grey shaded area show the best-
fitted RDP and background stellar density, respectively. Also, red-shaded area
indicates the $1\sigma$ uncertainty of the fit.

###  3.2 Membership Analyses of Collinder 74

Working with a sample of stars that are part of the cluster itself is crucial
to accurately characterise its properties, such as its age, mass, luminosity
function, and dynamics. Field star contamination may contribute to noise and
bias during the analyses and can cause inaccurate results. Therefore, field
star separation is necessary to understand the astrophysical, astrometric, and
kinematic properties of the studied cluster. As their physical formation
processes, cluster members are gravitationally bound to the cluster.
Therefore, they exhibit similar vectorial movements across the sky relative to
the background field stars. Through the analysis of proper-motion components,
stars that share the motion of the cluster can be identified as cluster
members. Hence, proper-motion components are useful tools to separate cluster
members and calculate their membership probabilities (Sariya et al., 2021;
Bisht et al., 2020). The precise astrometric data from Gaia DR3 data (Gaia
Collaboration et al., 2023) provide a crucial role for membership analyses.

We used the method of Photometric Membership Assignment in Stellar Clusters
(UPMASK, Krone-Martins & Moitinho, 2014) considering Gaia DR3 astrometric data
to calculate membership probabilities of stars in the region of the Coll 74.
UPMASK uses a clustering algorithm to group stars that have similar positions,
proper-motion components, trigonometric parallaxes and are close to each other
in space. The algorithm then assigns membership probabilities to each star
based on its likelihood of belonging to a particular cluster. We utilised the
method in five-dimensional astrometric space considering astrometric
measurements ($\alpha$, $\delta$, $\mu_{\alpha}\cos\delta$, $\mu_{\delta}$,
$\varpi$), also their uncertainties, of each star. To determine stars’
membership probabilities ($P$) we run the program with 100 iterations. As a
result, taking into account the stars within both the estimated limiting
radius ($r_{\rm lim}$=10${}^{\rm{}^{\prime}}$) and the completeness limit
$G=20.5$ mag, for the open cluster Coll 74 we obtained 102 most likely member
stars with membership probabilities of $P\geq 0.5$. These stars are used in
the estimation of astrometric and astrophysical parameters of Coll 74.

Figure 5: VPD of Coll 74. The colour scale on the right panel indicates the
membership probabilities of the stars for the cluster. The enlarged panel
inset shows the cluster’s denser region in VPD. The intersection of the dashed
blue lines represents the mean proper-motion value for Coll 74.
Figure 6: Histogram of star count of the most likely members ($P\geq 0.5$) in
trigonometric parallaxes. The red dashed line indicates the fitted Gaussian
function.

To visualise the clustering of the most likely member stars, we plotted
vector-point diagram (VPD) by using proper-motion components of the stars and
showed it in Figure 5. It is evident from the VPD that Coll 74 is embedded in
the field stars. Even if this is the case, the cluster structure can be
distinguished by investigation of the probability values of the stars. We
estimated mean proper-motion values from the stars with membership
probabilities greater than 0.5 and found the values as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(0.960\pm 0.005,-1.526\pm 0.004$) mas
yr-1. The trigonometric parallax histogram of the most likely member stars is
shown in Figure 6. By fitting the Gaussian function to the histogram, we
estimated the mean trigonometric parallax as $\varpi=0.363\pm 0.043$ mas for
the Coll 74 and corresponding distance value (with the linear equation $d({\rm
pc})=1000/\varpi$) mas as $d_{\varpi}=2755\pm 326$ pc. This value of distance
is close to the values estimated in Gaia era, as listed in Table 1.

###  3.3 The Blue Straggler Stars of Collinder 74

Blue straggler stars (BSSs) in open clusters defy the natural aging process by
appearing younger and bluer than their surrounding companions. While most
stars in open clusters follow common evolutionary processes, BSSs defy the
laws of stellar evolution in the cluster by reversing this trend. Interactions
between stars in binary systems or stellar collisions within the dense cluster
environment are among the leading formation mechanisms for BSSs (Sandage,
1953; Zinn & Dahn, 1976; Hills & Day, 1976).

We identified four BSSs in Coll 74. These stars are confirmed as cluster
members with membership probabilities $P\geq 0.9$. The positions of BSSs are
shown in Figure 7. Rain et al. (2021) defined five BSSs by using Gaia DR2
(Gaia Collaboration et al., 2018) photometric and astrometric data. Due to the
membership analyses being based on Gaia DR3 data and we took into account the
stars within limiting radius ($r_{\rm lim}\leq 10^{\prime}$), one of the Rain
et al. (2021) BSS was not considered. Jadhav & Subramaniam (2021) using Gaia
DR2 data and identified BSSs in 1246 open clusters. They found four BSSs
members in Coll 74 which are the same star sample as presented in this study.

Ferraro et al. (2012) considering the radial distribution of BSSs in the
clusters, defined three classes for BSSs. Three of BSSs in Coll 74 are located
at a radial distance 0.42, 0.88, and 0.98 arcmin, whereas one star is located
at 6.25 arcmin. According to their radial distribution and the criterion of
Ferraro et al. (2012), we can conclude that the BSSs of Coll 74 show flat
distribution and cluster belongs to the family I. The formation mechanisms of
blue stragglers in family I clusters are thought to be dominated by stellar
collisions and mass transfer in close binary systems.

###  3.4 Astrophysical Parameters of Collinder 74

To derive age, distance modulus and colour excess of Coll 74, we fitted
theoretical PARSEC isochrones of Bressan et al. (2012) to the observed CMD
constructed from the most likely cluster members. The age, distance modulus
and colour excess of the cluster were estimated simultaneously, while the
metallicity of Coll 74 was taken from Zhong et al. (2020) as [Fe/H]=$-0.052\pm
0.034$ dex. We fitted PARSEC models by taking into account the morphology of
the cluster and reached the best fit. For the selection of the isochrones and
estimation of astrophysical parameters, we transformed the assumed metallicity
([Fe/H]=$-0.052\pm 0.034$ dex) to the mass fraction $z$. To do this, we
applied the equation of
Bovy111https://github.com/jobovy/isodist/blob/master/isodist/Isochrone.py that
are available for PARSEC models (Bressan et al., 2012). The equations are
given as follow:

$z_{\rm x}={10^{{\rm[Fe/H]}+\log\left(\frac{z_{\odot}}{1-0.248-2.78\times
z_{\odot}}\right)}}$ (2)

and

$z=\frac{(z_{\rm x}-0.2485\times z_{\rm x})}{(2.78\times z_{\rm x}+1)}.$ (3)

where $z_{\rm x}$ and $z_{\odot}$ are intermediate values where solar
metallicity $z_{\odot}$ was adopted as 0.0152 (Bressan et al., 2012). The
calculated mass fraction is $z=0.0136$ for Coll 74.

We plotted $G\times(G_{\rm BP}-G_{\rm RP})$ and superimposed isochrones,
scaled to the $z=0.0136$, of different ages ($\log t$=9.20, 9.25 and 9.30 yr)
by visual inspection to the most likely cluster main-sequence, turn-off and
giant members with probabilities over $P\geq 0.5$ as shown in Figure 7. The
best fit supports the isochrone with $\log t$=9.25 yr to the cluster
morphology, this isochrone corresponding to $t=1800\pm 200$ Myr. The estimated
age is comparable with the values of Tadross (2001) and Cantat-Gaudin et al.
(2020). Also, good isochrone fitting result supplies the distance modulus and
colour excess of the Coll 74 to be $\mu_{\rm G}=13.052\pm 0.088$ mag,
corresponding to isochrone distance $d_{\rm iso}=2831\pm 118$ pc and $E(G_{\rm
BP}-G_{\rm RP})=0.425\pm 0.046$ mag, respectively. We used the relations of
Carraro et al. (2017) to estimate the errors of distance modulus and isochrone
distance. Our derived isochrone distance is compatible with most of the
studies presented by different researchers (see Table 1) as well as the
trigonometric parallax distance $d_{\rm\varpi}=2755\pm 326$ pc estimated in
this study. For a more accurate comparison with literature studies, we
converted this value to the $U\\!BV$-based colour excess $E(B-V)$. We utilised
the equation $E(G_{\rm BP}-G_{\rm RP})=1.41\times E(B-V)$ given by Sun et al.
(2021) and determined the value as $E(B-V)=0.301\pm 0.033$ mag. This result is
close to the values given by Tadross (2001), Lata et al. (2002), Carraro &
Costa (2007) and Hasegawa et al. (2008) within the errors (see Table 1).

Figure 7: Colour-magnitude diagram for the studied cluster Coll 74. Different
colour and colourbar scales show the membership probabilities of stars with
$P\geq 0.5$. Stars with probabilities $P<0.5$ are demonstrated with filled
grey circles. BSSs of the cluster are shown in a blue dashed-lined box. The
best solution of fitted isochrones and their errors are inferred as the blue
and green lines, respectively. The age of the blue-lined isochrone matches
1800 Myr for the cluster.

We also estimated heliocentric Galactic coordinates $(X,Y,Z)_{\odot}$ of Coll
74. Here, $X$ is the distance from the Galactic center in the Galactic plane
($l=0^{\rm o}$, $b=0^{\rm o}$), $Y$ is the distance in the direction of
Galactic rotation ($l=90^{\rm o}$, $b=0^{\rm o}$) and $Z$ is the vertical
distance from Galactic plane to the North Galactic Pole ($l=0^{\rm o}$,
$b=90^{\rm o}$). Galactocentric coordinates provide a convenient way to
describe the positions of celestial objects relative to the Galactic center,
Sun, and Galactic plane. By considering isochrone distance, Galactic
longitude, and latitude of the cluster, we derived these distances as
$(X,Y,Z)_{\odot}=(-2633,-907,-510)$ pc.

##  4 Galactic Orbit Study of the Collinder 74

The Galactic orbits of open clusters are important for understanding how these
celestial objects dynamically evolve within the Milky Way (Taşdemir & Yontan,
2023). We derived the orbits and orbital parameters of Coll 74 with the help
of the python based Galactic dynamics library galpy222See also
https://galpy.readthedocs.io/en/v1.5.0/ of Bovy (2015). This library
implements MWPotential2014 model, which commonly uses a potential model in
Galactic dynamics. The MWPotential2014 model is based on a combination of
different components that represent the various structures within the Milky
Way, including the bulge, disc, and halo. These components are parameterised
to approximate the observed properties of the Galaxy: the bulge is modelled as
a power-law density profile as described in Bovy (2015), the disc is typically
modelled as an exponential disk with a specified scale length and scale height
as defined by Miyamoto & Nagai (1975) and the halo is often modelled as a
spherical or ellipsoidal distribution with a specified density profile as
defined by Navarro et al. (1996). We accepted Sun’s Galactocentric distance
and orbital velocity as $R_{\rm gc}=8$ kpc and $V_{\rm rot}=220$ km s-1,
respectively (Bovy, 2015; Bovy & Tremaine, 2012), as well as Sun’s distance
from the Galactic plane was adopted as $25\pm 5$ pc (Jurić et al., 2008).

The mean radial velocity ($V_{\gamma}$) of the cluster was calculated from
available Gaia DR3 radial velocity measurements of the stars. We considered
the most likely member stars with probabilities over $P\geq 0.5$ whose number
are 16. We used the equations of Soubiran et al. (2018) which are based on the
weighted average of the radial velocities of the stars. Hence, the mean radial
velocity of the Coll 74 was determined as $V_{\gamma}=20.55\pm 0.41$ km s-1
which is in good agreement with mean radial velocity findings presented by
Soubiran et al. (2018), Dias et al. (2021), Tarricq et al. (2021) and Hunt &
Reffert (2023). To estimate orbital parameters, we used equatorial coordinates
($\alpha=05^{\rm h}48^{\rm m}40^{\rm s}.8$, $\delta=+07^{\rm
o}22^{\rm{}^{\prime}}26^{\rm{}^{\prime\prime}}.4$) taken from Cantat-Gaudin et
al. (2020), mean proper-motion components ($\mu_{\alpha}\cos\delta=0.960\pm
0.005$, $\mu_{\delta}=-1.526\pm 0.004$ mas yr-1), isochrone distance ($d_{\rm
iso}=2831\pm 118$ pc) and the radial velocity ($V_{\gamma}=20.55\pm 0.41$ km
s-1) calculated in the study (see also Table 2) for Coll 74 as input
parameters.

The orbit integration was applied for the forward with an integration step of
1 Myr up to 2.5 Gyr to estimate the possible current position of Coll 74. The
resultant orbit is shown in Figure 8a. The figure pictures the path followed
by the cluster in $Z\times R_{\rm gc}$ plane, which represents the side view
of the orbit. Here, $Z$ and $R_{\rm gc}$ are the distance from the Galactic
plane and the Galactic center, respectively. Also, the orbit analyses were
carried out for the past epoch across a time equal to $t=1800\pm 200$ Myr
cluster’s age. Figure 8b shows the cluster’s distance variation in time on the
$R_{\rm gc}\times t$ plane. The figure also represents the influence of errors
in the input parameters on the orbit of Coll 74. Orbit analyses stated that
the Coll 74 was formed outside the solar vicinity with a birth radius of
$R_{\rm gc}=10.97\pm 0.32$ kpc.

Figure 8: The Galactic orbits and birth radii of Coll 74 in the $Z\times
R_{\rm gc}$ (a) and $R_{\rm gc}\times t$ (b) planes. The filled yellow circles
and triangles show the present-day and birth positions, respectively. The red
arrow is the motion vector of Collinder 74. The green and pink dotted lines
show the orbit when errors in input parameters are considered, while the green
and pink filled triangles represent the birth locations of the open cluster
based on the lower and upper error estimates.

From the orbit integration we derived the following parameters for Coll 74:
apogalactic ($R_{\rm a}=10987\pm 112$ pc) and perigalactic ($R_{\rm p}=9337\pm
20$ pc) distances, eccentricity ($e=0.081\pm 0.004$), maximum vertical
distance from Galactic plane ($Z_{\rm max}=506\pm 22$ pc), space velocity
components ($U,V,W=-11.43\pm 0.79$, $-29.50\pm 1.08$, $-2.56\pm 0.05$ km s-1),
and orbital period ($P_{\rm orb}=291\pm 2$ Myr). The Local Standard of Rest
(LSR) correction was applied to the $(U,V,W)$ components of the Coll 74. To do
this, we considered the space velocity component values
$(U,V,W)_{\odot}=(8.83\pm 0.24,14.19\pm 0.34,6.57\pm 0.21$) km s-1 of
Coşkunoǧlu et al. (2011). Hence, LSR corrected space velocity components were
found to be $(U,V,W)_{\rm LSR}$ = ($-2.60\pm 0.25$, $-15.31\pm 1.13$, $4.01\pm
0.22$) km s-1. Total space velocity was estimated as $S_{\rm LSR}=16.04\pm
1.18$ km s-1, which is compatible with the velocity value given for thin-disc
objects (Leggett, 1992). We interpreted from the perigalactic and apogalactic
distances that Coll 74 is completely outside the solar circle (Figure 8a). The
cluster reaches a maximum distance above the Galactic plane at $Z_{\rm
max}=506\pm 22$ pc, shows that Coll 74 belongs to the thin-disc component of
the Milky Way (Bilir et al., 2006a, b, 2008).

##  5 Luminosity and Mass Functions

The luminosity function (LF) of an open cluster represents the number of stars
at different brightness within the cluster. The LF and mass function (MF) of
open clusters are related because the luminosity of a star is generally
correlated with its mass. This correlation is also defined as mass-luminosity
relation and provides transform LF into the MF (Bisht et al., 2019).

For LF analyses of Coll 74, first, we selected the main-sequence stars with
membership probabilities $P>0$ and located inside the limiting radius obtained
in the study ($r_{\rm lim}^{\rm obs}=10^{\prime}$). Hence, we reached 324
stars within the $15\leq G\leq 20.5$ magnitude interval. Then considering
distance modulus ($M_{\rm G}=G-5\times\log d+A_{\rm G}$) with apparent
magnitude ($G$), isochrone distance ($d_{\rm iso}$) and $G$ band absorption
($A_{\rm G}$) estimated in the study, we transformed apparent $G$ magnitudes
into the absolute $M_{\rm G}$ magnitudes. The histogram of LF for the cluster
that constructed an interval of 1 mag is shown in Figure 9. This figure shows
that the number of main-sequence stars rises up to $M_{\rm G}$=6 mag properly,
after this limit the counts drop gradually.

Figure 9:  The luminosity function of Coll 74.

For MF estimation, we used PARSEC isochrones (Bressan et al., 2012) that
scaled to the derived age and adopted metallicity fraction ($z$) for the
cluster. From this isochrone, we produced a high-degree polynomial equation
between the $G$-band absolute magnitudes and masses. By applying this equation
to the selected main-sequence stars ($P>0$), we transformed their absolute
magnitudes $M_{\rm G}$ into masses. Hence, we found a mass range of the 324
stars within the $0.65\leq M/M_{\odot}\leq 1.58$. The MF slope was derived
from the following equation:

$\displaystyle{\rm log(dN/dM)}=-(1+\Gamma)\times\log M+C.$ (4)

In the equation $dN$ is the number of stars per unit mass $dM$, $M$ is the
central mass, $C$ denotes the constant for the equation, and $\Gamma$
represents the slope of the MF. The estimated MF slope for Coll 74 is
$\Gamma=1.34\pm 0.21$, which is in good agreement with the value of Salpeter
(1955). The resulting MF is shown in Figure 10.

Figure 10:  Derived mass function for Coll 74. The blue line represents the
MF, whereas the green lines indicate the $\pm 1\sigma$ standard deviations.
The black dashed line represented Salpeter (1955)’s slope.

The masses of stars in Coll 74 were derived as a function of stars’ membership
probabilities. The number of stars with probabilities $P>0$ and $P\geq 0.5$
was determined as 324 and 102, respectively. Hence, the total mass of the
cluster for these probabilities is to be 365$M_{\odot}$ and 132$M_{\odot}$,
respectively. We interpreted that the total mass of the cluster estimated from
the stars with probabilities $P\geq 0.5$ corresponds to about 36$\%$ of the
total mass for the stars with all probabilities. To investigate the mass
distribution of the stars in Coll 74, 324 stars with $P>0$ were plotted
according to their equatorial coordinates and membership probabilities as
shown in Figure 11. It can be interpreted from the figure that the stars with
probabilities over 0.8 and massive ones are mostly concentrated central region
of the cluster, whereas low-mass stars with probabilities under 0.8 are
distributed beyond the cluster center. This case shows that the Coll 74 is a
mass segregated open cluster.

Figure 11:  Mass distribution of the stars in the Coll 74. The radius sizes of
the stars indicate the masses, and the different colours show the membership
probabilities of the stars. The intersection of the red dashed lines indicates
the central position of the cluster in the equatorial coordinate system.

##  6 Conclusion

We performed a detailed Gaia DR3 data-based study of open cluster Collinder
74. The number of member stars with probabilities over 0.5 were 102.
Considering these stars, we calculated structural and fundamental
astrophysical parameters, investigated luminosity and mass functions, and
estimated the orbit of the cluster. All parameters obtained in the study are
listed in Table 2. The main results of the study are summarized as follows:

Table 2:  Fundamental parameters of Coll 74. Parameter | Value
---|---
($\alpha,~{}\delta)_{\rm J2000}$ (Sexagesimal) | 05:48:40.8, $+$07:22:26.4
($l,b)_{\rm J2000}$ (Decimal) | 199.0189, $-10.3791$
$f_{0}$ (stars arcmin-2) | $8.42\pm 0.35$
$f_{\rm bg}$ (stars arcmin-2) | $5.45\pm 0.16$
$r_{\rm c}$ (arcmin) | $1.38\pm 0.12$
$r_{\rm lim}$ (arcmin) | 10
$r$ (pc) | 8.24
Cluster members ($P\geq 0.5$) | 102
$\mu_{\alpha}\cos\delta$ (mas yr-1) | $0.960\pm 0.005$
$\mu_{\delta}$ (mas yr-1) | $-1.526\pm 0.004$
$\varpi$ (mas) | $0.363\pm 0.043$
$d_{\varpi}$ (pc) | $2755\pm 326$
$E(B-V)$ (mag) | $0.301\pm 0.033$
$E(G_{\rm BP}-G_{\rm RP})$ (mag) | $0.425\pm 0.046$
$A_{\rm G}$ (mag) | $0.792\pm 0.086$
$[{\rm Fe/H}]$ (dex)∗ | $-0.052\pm 0.034$
Age (Myr) | $1800\pm 200$
Distance modulus (mag) | $13.052\pm 0.088$
Isochrone distance (pc) | $2831\pm 118$
$(X,Y,Z)_{\odot}$ (pc) | ($-2633$, $-907$, $-510$)
$R_{\rm gc}$ (kpc) | 10.67
MF slope | $1.34\pm 0.21$
Total mass ($M/M_{\odot})$ ($P>0$) | 365
$V_{\gamma}$ (km s-1) | $20.55\pm 0.41$
$U_{\rm LSR}$ (km s-1) | $-2.60\pm 0.25$
$V_{\rm LSR}$ (kms-1) | $-15.31\pm 1.13$
$W_{\rm LSR}$ (kms-1) | $4.01\pm 0.22$
$S_{{}_{\rm LSR}}$ (kms-1) | $16.04\pm 1.18$
$R_{\rm a}$ (pc) | $10987\pm 112$
$R_{\rm p}$ (pc) | $9337\pm 20$
$z_{\rm max}$ (pc) | $506\pm 22$
$e$ | $0.081\pm 0.004$
$P_{\rm orb}$ (Myr) | $291\pm 2$
Birthplace (kpc) | $10.97\pm 0.32$
∗Zhong et al. (2020) |

  1. 1.

From the RDP analyses, we determined the limiting radius by visual inspection
as $r_{\rm lim}^{\rm obs}=10^{{}^{\prime}}$.

  2. 2.

Considering results of photometric completeness limit, membership probability
analyses, and limiting radius, we identified 102 most likely members with
probabilities $P\geq 0.5$ for Coll 74. These stars were used in the cluster
analyses.

  3. 3.

The mean proper-motion components were obtained as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(0.960\pm 0.005,-1.526\pm 0.004$) mas
yr-1.

  4. 4.

Four most probable BSS members were identified within the limiting radius of
the cluster. We concluded that the Coll 74 belongs to a family I according to
the radial distribution of its BSSs.

  5. 5.

The metallicity value for the cluster was adopted as $[{\rm Fe/H}]=-0.052\pm
0.034$ dex which is presented by Zhong et al. (2020). We transformed this
value into the mass fraction $z=0.0136$ and kept it as a constant parameter
for the age and distance modulus estimation.

  6. 6.

By fitting PARSEC isochrone (Bressan et al., 2012) to the $G$ versus ($G_{\rm
BP}-G_{\rm RP}$) colour-magnitude diagram, we estimated colour excess of the
Coll 74 as $E(G_{\rm BP}-G_{\rm RP})=0.425\pm 0.046$ mag, which corresponds to
a colour excess in $U\\!BV$ system to be $E(B-V)=0.301\pm 0.033$ mag. We
estimated this value by using the equation $E(G_{\rm BP}-G_{\rm
RP})=1.41\times E(B-V)$ as given by Sun et al. (2021).

  7. 7.

The isochrone fitting distance of Coll 74 was determined as $d_{\rm
iso}=2831\pm 118$ pc. This value is supported by the distance $d_{\varpi}$=
$2755\pm 326$ pc that derived from mean trigonometric parallax.

  8. 8.

PARSEC isochrone of Bressan et al. (2012) provides the age of the cluster to
be $t=1800\pm 200$ Myr.

  9. 9.

The LF and MF were investigated from the main-sequence stars with
probabilities $P>0$. The MF slope was found as $\Gamma=1.34\pm 0.21$ which is
in good agreement with the value of Salpeter (1955).

  10. 10.

Orbit integration was performed via MWPotential2014 model. We concluded that
Coll 74 orbits in a boxy pattern outside the solar circle, as well as the
cluster, is a member of the thin-disc component of the Milky Way. Moreover,
the birth radius ($10.97\pm 0.32$ kpc) shows that the forming region of the
cluster is outside the solar circle.

## Acknowledgements

This study has been supported in part by the Scientific and Technological
Research Council (TÜBİTAK) 122F109. This research has made use of the WEBDA
database, operated at the Department of Theoretical Physics and Astrophysics
of the Masaryk University. We also made use of NASA’s Astrophysics Data System
as well as the VizieR and Simbad databases at CDS, Strasbourg, France and data
from the European Space Agency (ESA) mission _Gaia_
333https://www.cosmos.esa.int/gaia, processed by the _Gaia_ Data Processing
and Analysis Consortium
(DPAC)444https://www.cosmos.esa.int/web/gaia/dpac/consortium. Funding for DPAC
has been provided by national institutions, in particular, the institutions
participating in the _Gaia_ Multilateral Agreement.

## References

  * Ann et al. (1999) Ann H. B., Lee M. G., Chun M. Y., Kim S. L., Jeon Y. B., Park B. G., 1999, Journal of Korean Astronomical Society, 32, 7
  * Bilir et al. (2006a) Bilir S., Karaali S., Ak S., Yaz E., Hamzaoğlu E., 2006a, New A, 12, 234
  * Bilir et al. (2006b) Bilir S., Karaali S., Gilmore G., 2006b, MNRAS, 366, 1295
  * Bilir et al. (2008) Bilir S., Cabrera-Lavers A., Karaali S., Ak S., Yaz E., López-Corredoira M., 2008, PASA, 25, 69
  * Bisht et al. (2019) Bisht D., Yadav R. K. S., Ganesh S., Durgapal A. K., Rangwal G., Fynbo J. P. U., 2019, MNRAS, 482, 1471
  * Bisht et al. (2020) Bisht D., Zhu Q., Yadav R. K. S., Durgapal A., Rangwal G., 2020, MNRAS, 494, 607
  * Bovy (2015) Bovy J., 2015, ApJS, 216, 29
  * Bovy & Tremaine (2012) Bovy J., Tremaine S., 2012, ApJ, 756, 89
  * Bressan et al. (2012) Bressan A., Marigo P., Girardi L., Salasnich B., Dal Cero C., Rubele S., Nanni A., 2012, MNRAS, 427, 127
  * Cantat-Gaudin et al. (2018) Cantat-Gaudin T., et al., 2018, A&A, 618, A93
  * Cantat-Gaudin et al. (2020) Cantat-Gaudin T., et al., 2020, A&A, 640, A1
  * Carraro & Costa (2007) Carraro G., Costa E., 2007, A&A, 464, 573
  * Carraro et al. (2017) Carraro G., Sales Silva J. V., Moni Bidin C., Vazquez R. A., 2017, AJ, 153, 99
  * Coşkunoǧlu et al. (2011) Coşkunoǧlu B., et al., 2011, MNRAS, 412, 1237
  * Dias et al. (2006) Dias W. S., Assafin M., Flório V., Alessi B. S., Líbero V., 2006, A&A, 446, 949
  * Dias et al. (2014) Dias W. S., Monteiro H., Caetano T. C., Lépine J. R. D., Assafin M., Oliveira A. F., 2014, A&A, 564, A79
  * Dias et al. (2018) Dias W. S., Monteiro H., Assafin M., 2018, MNRAS, 478, 5184
  * Dias et al. (2021) Dias W. S., Monteir H., Moitinho A., Lépine J. R. D., Carraro G., Paunzen E., Alessi B., Villela L., 2021, MNRAS, 504, 356
  * Ferraro et al. (2012) Ferraro F. R., et al., 2012, Nature, 492, 393
  * Gaia Collaboration et al. (2016) Gaia Collaboration et al., 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2018) Gaia Collaboration et al., 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2023) Gaia Collaboration et al., 2023, A&A, 674, A1
  * Hasegawa et al. (2008) Hasegawa T., Sakamoto T., Malasan H. L., 2008, PASJ, 60, 1267
  * Hills & Day (1976) Hills J. G., Day C. A., 1976, Astrophys. Lett., 17, 87
  * Hunt & Reffert (2023) Hunt E. L., Reffert S., 2023, A&A, 673, A114
  * Jadhav & Subramaniam (2021) Jadhav V. V., Subramaniam A., 2021, MNRAS, 507, 1699
  * Jurić et al. (2008) Jurić M., et al., 2008, ApJ, 673, 864
  * Kharchenko et al. (2013) Kharchenko N. V., Piskunov A. E., Schilbach E., Röser S., Scholz R. D., 2013, A&A, 558, A53
  * Kim et al. (2017) Kim S. C., Kyeong J., Park H. S., Han I., Lee J. H., Moon D.-S., Lee Y., Kim S., 2017, Journal of Korean Astronomical Society, 50, 79
  * King (1962) King I., 1962, AJ, 67, 471
  * Krone-Martins & Moitinho (2014) Krone-Martins A., Moitinho A., 2014, A&A, 561, A57
  * Lada & Lada (2003) Lada C. J., Lada E. A., 2003, ARA&A, 41, 57
  * Lata et al. (2002) Lata S., Pandey A. K., Sagar R., Mohan V., 2002, A&A, 388, 158
  * Leggett (1992) Leggett S. K., 1992, ApJS, 82, 351
  * Liu & Pang (2019) Liu L., Pang X., 2019, ApJS, 245, 32
  * Loktin & Popova (2017) Loktin A. V., Popova M. E., 2017, Astrophysical Bulletin, 72, 257
  * Marsakov et al. (2016) Marsakov V. A., Gozha M. L., Koval’ V. V., Shpigel’ L. V., 2016, Astronomy Reports, 60, 43
  * Miyamoto & Nagai (1975) Miyamoto M., Nagai R., 1975, PASJ, 27, 533
  * Navarro et al. (1996) Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
  * Rain et al. (2021) Rain M. J., Ahumada J. A., Carraro G., 2021, A&A, 650, A67
  * Salpeter (1955) Salpeter E. E., 1955, ApJ, 121, 161
  * Sandage (1953) Sandage A. R., 1953, AJ, 58, 61
  * Sariya et al. (2021) Sariya D. P., et al., 2021, AJ, 161, 101
  * Soubiran et al. (2018) Soubiran C., et al., 2018, A&A, 619, A155
  * Sun et al. (2021) Sun M., Jiang B., Yuan H., Li J., 2021, ApJS, 254, 38
  * Taşdemir & Yontan (2023) Taşdemir S., Yontan T., 2023, Physics and Astronomy Reports, 1, 1
  * Tadross (2001) Tadross A. L., 2001, New A, 6, 293
  * Tarricq et al. (2021) Tarricq Y., et al., 2021, A&A, 647, A19
  * Zacharias et al. (2013) Zacharias N., Finch C. T., Girard T. M., Henden A., Bartlett J. L., Monet D. G., Zacharias M. I., 2013, AJ, 145, 44
  * Zhong et al. (2020) Zhong J., Chen L., Wu D., Li L., Bai L., Hou J., 2020, A&A, 640, A127
  * Zinn & Dahn (1976) Zinn R., Dahn C. C., 1976, AJ, 81, 527


# Inverse resolution of spatially varying diffusion coefficient using Physics-
Informed neural networks

Sukirt Thakur  School of Mechanical Engineering,Purdue University, West
Lafayette, 47907, Indiana, USA Ehsan Esmaili School of Mechanical
Engineering,Purdue University, West Lafayette, 47907, Indiana, USA Sarah
Libring Weldon School of Biomedical Engineering,Purdue University, West
Lafayette, 47907, Indiana, USA Luis Solorio Weldon School of Biomedical
Engineering,Purdue University, West Lafayette, 47907, Indiana, USA Arezoo M.
Ardekani School of Mechanical Engineering,Purdue University, West Lafayette,
47907, Indiana, USA

###### Abstract

Resolving the diffusion coefficient is a key element in many biological and
engineering systems, including pharmacological drug transport and fluid
mechanics analyses. Additionally, these systems often have spatial variation
in the diffusion coefficient which must be determined, such as for injectable
drug-eluting implants into heterogeneous tissues. Unfortunately, obtaining the
diffusion coefficient from images in such cases is an inverse problem with
only discrete data points. The development of a robust method that can work
with such noisy and ill-posed datasets to accurately determine spatially-
varying diffusion coefficients is of great value across a large range of
disciplines. Here, we developed an inverse solver that uses physics informed
neural networks (PINNs) to calculate spatially-varying diffusion coefficients
from numerical and experimental image data in varying biological and
engineering applications. The residual of the transient diffusion equation for
a concentration field is minimized to find the diffusion coefficient. The
robustness of the method as an inverse solver was tested using both numerical
and experimental datasets. The predictions show good agreement with both the
numerical and experimental benchmarks; an error of less than 6.31% was
obtained against all numerical benchmarks, while the diffusion coefficient
calculated in experimental datasets matches the appropriate ranges of other
reported literature values. Our work demonstrates the potential of using PINNs
to resolve spatially-varying diffusion coefficients, which may aid a wide-
range of applications, such as enabling better-designed drug-eluting implants
for regenerative medicine or oncology fields.

Keywords— Physics-Informed neural networks, Deep learning, Inverse modelling

##  1 Introduction

Understanding the spatial variation of the diffusion coefficient is a key
element in many biological and engineering systems (e.g., drug transport,
biomedical transport, petroleum engineering, and hydrogeology) [1, 2]. The
diffusion coefficient is a proportionality constant which relates the gradient
in concentration of a species with the mass flux and gives a measure of the
rate of diffusion. However, there are numerous hurdles when attempting to
resolve the diffusion coefficient from experimental data. First, the diffusion
coefficient often varies spatially, sometimes discontinuously, due to
heterogeneity in biological tissues, porous media, and soil. Second,
determining the diffusion coefficient from experimental images is an inverse
problem, where the parameters of the system need to be determined from the
given observations. Numerical instability is a challenge for such problems,
which are fundamentally ill-posed in the Hadamard sense [3]. Even a small
perturbation in experiments can significantly change the estimated diffusion
coefficient [4]. Additionally, data is always collected at discrete points,
such that any errors in observation will be amplified in the parameters to be
determined [5]. Moreover, multiple models can fit the data and uniqueness of
the solution for the inverse problem is not guaranteed. A robust method which
can work with noisy data to accurately determine the diffusion coefficient has
wide-ranging applications and is of great value across a large range of
disciplines. Many numerical methods replace the original inverse problem by
linearization around constant coefficients in order to solve a simpler problem
[6]. For example, numerical methods such as finite differences in combination
with least squares [7], genetic algorithm with sinc-Galerkin method [8], as
well as Bayesian methods [9, 10] have been used to estimate the diffusion
coefficient for the observed data. However, obtaining numerical solutions that
are satisfactory using such methods is often difficult [6].

Neural networks are powerful computational tools that have the capacity to
represent complex linear and non-linear systems [11]. There is a growing
interest in using neural networks to extract latent information from data in
engineering fields like fluid mechanics [12] and material engineering [13].
Machine learning techniques have been used to obtain corrective terms for
improving closure models in computational physics through inverse modelling of
high-fidelity simulation and experimental data [14, 15]. There is a growing
interest in using neural networks as a tool for better understanding
biological systems such as studying membranes in fluorescence microscopy [16]
and analysis of molecular images [17]. Although traditional neural networks
thrive on data, high-fidelity computational and experimental data is scarce
and expensive when it comes to many engineering and biological systems. Raissi
et al. [18, 19, 20] have introduced physics informed neural networks (PINNs),
which are neural networks that respect physical laws described by partial
differential equations (PDEs) while performing supervised learning tasks.
PINNs are extremely data efficient neural networks that can be used to solve
PDEs (forward problems) or to discover the coefficients of terms in the PDE
given the solution (inverse problems) [21]. PINNs have offered a novel
solution methodology to diverse problems like brittle fracture mechanics [22],
learning constitutive relations for a non-stationary bioreactor [23],
modelling the thermochemical curing process of composite-tool systems during
manufacture [24], predicting arterial blood pressure from 4D MRI data [25],
and modelling soft tissues [26]. The methodology has also been used to solve
inverse problems like reconstructing the velocity and pressure fields from
flow visualizations [27] and discovering turbulence models from noisy spatio-
temporal data [28]. The ability of PINNs to work with noisy data makes it a
promising option to solve inverse problems using experimental observations.

Our focus in this work was to employ PINNs to solve the inverse problem of
discovering the spatially varying diffusion coefficient from spatio-temporal
information of the diffused passive scalar. Our method was subjected to both
numerical and experimental data sets. We first used clean numerical data for
verification of our method and then worked with three distinct experimental
datasets which contained noise. These results demonstrated that we were able
to accurately resolve the diffusion coefficient in different liquids and
hydrogels. Additionally, the framework presented is robust and the method can
be modified to tackle inverse problems of similar nature with minor
modifications. In total, we conclude that this method may be an ideal platform
to predict the properties of many substrates in biological systems based on
spatially-varying diffusivity, with numerous engineering applications.

##  2 Problem definition and methodology

Our objective was to solve the inverse problem of determining the spatially
varying diffusion coefficient which best described a given set of spatio-
temporal data of the concentration of a passive scalar using PINNs. To do so,
we employed the use of fully connected neural networks. Fully connected neural
networks, or feed-forward neural networks, have the simplest architecture
where the output of one layer of the neural network is used as the input to
the next layer. For a network with an input layer, an output layer, and H
hidden layers, the $k^{th}$ hidden layer will receive an output from the
previous layer ($x^{k-1}$) and perform the affine transform first

$\mathcal{N}_{k}(x^{k-1})=w^{k}x^{k-1}+b^{k},$ (1)

where $w^{k}$ and $b^{k}$ are the weights and biases associated with the
$k^{th}$ hidden layer. Before sending this output as the input to the next
layer, a nonlinear activation function $\sigma$ is applied. For an input $x$,
a fully connected neural network can be mathematically written as

$\mathcal{NN}_{\theta}(x)=(\mathcal{N}_{H}\circ\sigma\circ\mathcal{N}_{H-1}\circ...\circ\sigma\mathcal{N}_{1})(x),$
(2)

where $\circ$ is the composition operator and $\theta$ represents the
trainable weights and biases of the network which are optimized. The key idea
of physics informed networks is to constrain the neural network such that the
residual of the pertinent partial differential equation is minimized. Consider
the transient diffusion equation for a concentration field of a passive scalar
$c(x,y,t)$ in two dimensions with a spatially varying diffusion coefficient
$D(x,y)$

$c_{t}-c_{x}D_{x}-c_{y}D_{y}-D(c_{xx}+c_{yy})=0.$ (3)

This is a parabolic equation, analogous to the equations pertinent to problems
in heat transfer and the same framework can be used to determine thermal
properties using temperature measurements. We approximate the functions
$\tilde{c}(x,y,t)$ and $\tilde{D}(x,y)$ using two deep fully connected neural
networks. The only known data are point clouds inside the domain
$\\{{x},{y},{t},{c(x,y,t)}\\}$. Given this spatio-temporal data, we want to
infer the spatially varying diffusion coefficient $D(x,y)$. To solve this
inverse problem, we designed a PINN to approximate the diffusion coefficient
$\tilde{D}(x,y)$ which best describes the concentration field $c(x,y,t)$ data.
Now, considering (3), we define

$f(c,D)={c}_{x}{D}_{x}-{c}_{y}{D}_{y}-{D}({c}_{xx}+{c}_{yy}),$ (4)

where the subscripts denote the derivatives. We now create a physics-informed
neural network using backward Euler discretization. We define

$c^{pi}(t,x,y;\Delta t,\theta,\phi)=\tilde{c}(t+\Delta t,x,y;\theta)-\Delta
tf(\tilde{c}(t+\Delta t,x,y;\theta),\tilde{D}(x,y;\phi)),$ (5)

where $\theta$ and $\phi$ are the parameters for the neural networks used to
approximate $\tilde{c}(x,y,t)$ and $\tilde{D}(x,y)$, respectively. As shown in
Fig. 1, to create our PINN we obtained $c^{pi}$ corresponding to equation (3)
using the two neural networks defined above. The identity operator is
represented using $I$ and the differential operators are represented using
$\partial_{x}$, $\partial_{y}$ and $\partial_{t}$ in Fig. 1. The derivatives
which do not appear in the equations will not be a part of the computational
graph and will not contribute to any computational costs. Another challenge
associated with this inverse problem worth emphasizing is that regions where
the concentration field does not vary, or the gradient is zero, gives no
information for us to learn and determine the diffusion coefficient. It is
noted that automatic differentiation provides us with derivatives calculated
with machine precision. We are working with point clouds and just one global
function and there is no discretization involved. This is significant as a
lattice like distribution of points is not a requirement and we can work with
any distribution of points in the spatio-temporal domain. The popular open-
source library Tensorflow [29] was used to create and compile the
computational graphs for our physics-informed neural network. The parameters
for neural networks $\tilde{c}(x,y,t)$ and $\tilde{D}(x,y)$ were optimized by
minimizing the following mean squared error $(MSE_{Diff})$ function:

$\displaystyle MSE_{Diff}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\left(|\frac{\tilde{c}(x^{i},y^{i},t^{i})-c(x^{i},y^{i},t^{i})}{\sigma_{c}}|^{2}\right)$
(6)
$\displaystyle+\frac{1}{N^{e}}\sum_{i=1}^{N^{e}}\left(|\frac{{c}^{pi}(x^{i},y^{i},t^{i})-\tilde{c}(x^{i},y^{i},t^{i})}{\sigma_{c}}|^{2}\right).$

Here $\sigma_{c}$ is the standard deviation of the concentration,
$\\{x^{i},y^{i},t^{i}\\}_{i=1}^{N}$ denote the points where we have
information on $c(x,y,t)$ and the number of such points is $N$,
$\\{x^{i},y^{i},t^{i}\\}_{i=1}^{N^{e}}$ denote the points where the equation
is evaluated on and the number of such points is $N^{e}$. The second term in
eq. (6) is the consistency loss, which ensures that there is consistency in
the regressed concentration and the physics informed concentration [30]. Both
neural networks $\tilde{c}(x,y,t)$ and $\tilde{D}(x,y)$ were densely connected
with 8 hidden layers each. The network (Fig. 1) was trained with a decaying
schedule for the learning rate using the Adam optimizer [31].

Figure 1: We use two densely connected neural networks which are 8 layers deep
for both the concentration ($\tilde{c}$) and diffusion
coefficient($\tilde{D}$). For the purpose of this schematics, the networks are
shown to have 2 hidden layers and 5 neurons per hidden layer.

##  3 Results and discussion

To evaluate the accuracy of our method, the predicted values for
concentrations and diffusion coefficients were compared with both data
generated from numerical simulations (Section 3.1) and experimentation
(Section 3.2).

###  3.1 Numerical Simulations Data

We considered four different spatially varying diffusion coefficients
(referred to as $D_{true}$) as our test cases and numerically solved the
diffusion equation with a spatially varying diffusion coefficient (Eq. (3)) to
get the concentration field of the passive scalar (referred to as $c_{true}$).
The PINN framework described in section 2 was then used to determine the
spatially varying diffusion coefficient that best described the concentration
field distribution. Numerical simulations were carried out using the open-
source CFD toolbox OpenFOAM [32] with a 500$\times$500 mesh $(x,y\in(0,1))$
and 100 time steps $(t\in(0,1))$. The mesh point generated from the simulation
were used as training points for the PINNs. For all four cases, 5000 points
were randomly sampled across time and space from inside the domain for each
iteration using the data iterator provided by Tensorflow [29].

For all cases, we considered a square domain and defined the side of the
square to be the length scale L and the total time to be the time scale T. The
concentration field was scaled using the maximum value of the concentration,
hence the dimensionless field $c\in[0,1]$. We defined
$x=\frac{x^{\prime}}{L}$, $y=\frac{y^{\prime}}{L}$ and
$t=\frac{t^{\prime}}{T}$ where $x^{\prime}$, $y^{\prime}$ and $t^{\prime}$ are
the dimensional x-coordinate, y-coordinate and time, respectively while $x$,
$y$ and $t$ are the dimensionless x-coordinate, y-coordinate and time,
respectively. The diffusion coefficient distribution was scaled by
$\frac{T}{L^{2}}$ to get a dimensionless diffusion coefficient distribution
$D(x,y)$. The parameters of the networks were updated using the gradient of
the loss with respect to the network parameters to get the optimum set of
parameters for the mapping $(x,y,t)\longmapsto(\tilde{c})$ and
$(x,y)\longmapsto(\tilde{D})$. For the given concentration distribution, the
networks used regression to learn the concentration distribution
$\tilde{c}(x,y,t)$ and minimized the residual defined in equation (5) to
determine the diffusion coefficient distribution $\tilde{D}(x,y)$ which best
described the concentration data.

Table 1: Numerically simulated data for the distribution of diffusion coefficient, ${D_{true}}(x,y)$, and the relative error of the predicted to true concentration field and diffusion coefficient, respectively, for each test case. Case | $D_{true}(x,y)$ | $\mathcal{L}(c_{true},\tilde{c}(x,y,t))$ | $\mathcal{L}(D_{true},\tilde{D}(x,y))$
---|---|---|---
I | $0.05+0.1[x(1-x)+y(1-y)]$ | $1.8\times 10^{-4}$ | $4.8\times 10^{-2}$
II | $0.25+0.1[sin(2\pi x)+sin(2\pi y)]$ | $2.6\times 10^{-4}$ | $4.6\times 10^{-2}$
III | $0.25+0.1[sin(4\pi x)+sin(4\pi y)]$ | $6.78\times 10^{-4}$ | $6.31\times 10^{-2}$
IV | $0.3+0.15[tanh(-10+20x)]+$ |  |
| $0.1[tanh(-7+20y)]$ | $4.81\times 10^{-4}$ | $4.58\times 10^{-2}$
Table 2: The effect of reducing the number of spatial points on the diffusion coefficient for case 1. Samples | $250000$ | $25000$ | $2500$ | $250$ | $25$
---|---|---|---|---|---
Error | $4.89\times 10^{-2}$ | $5.9\times 10^{-2}$ | $6.0\times 10^{-2}$ | $6.45\times 10^{-2}$ | $3.9\times 10^{-1}$
Figure 2: Results for the numerically simulated data. The first column shows
the exact concentration distribution, the second column the predicted
concentration distribution, the third column shows the exact distribution of
the diffusion coefficient and the fourth column shows the predicted
distribution of the diffusion coefficient for (a) Case I (b) Case II (c) Case
III and (d) Case IV. Figure 3: Comparison of the correct (exact) diffusion
coefficient and the diffusion coefficient predicted by the neural network
along the horizontal cross-section at the midpoint for (a) Case I (b) Case II
(c) Case III and (d) Case IV. The exact value is represented by a solid red
line while the prediction of the network is shown by a dotted blue line.

In case I, we simulated the concentration of a passive scalar corresponding to
the spatially varying diffusion coefficient
$D_{true}(x,y):=0.05+0.1[x(1-x)+y(1-y)]$. This resulted in a smooth function
akin to a two-dimensional parabola. To compare the results predicted by the
neural networks to the true value and simulation results, we defined the
relative error to be

$\mathcal{L}(a,a_{pred})=\sqrt{\frac{\overline{(a-a_{pred})^{2}}}{\overline{(a-\bar{a})^{2}}}},$
(7)

where the bar denotes the mean value. This definition for error is used so
that multiplication or addition of a constant does not change the error. In
case II, we considered a more complicated two dimensional sinusoidal
distribution $D_{true}(x,y):=0.25+0.1[\sin(2\pi x)+\sin(2\pi y)]$ and for case
III, we increased the frequency of the sinusoidal function in both directions
$D_{true}(x,y):=0.25+0.1[\sin(4\pi x)+\sin(4\pi y)]$. In case IV, we used
tangent hyperbolic functions to split the domain into four parts with
different diffusion coefficients and continuous boundaries. The diffusion
coefficient was defined by
$D_{true}(x,y):=0.3+0.15[\tanh(20(-0.5+x))]+0.1[\tanh(20(-0.35+y))]$ for case
IV. Such distributions can occur in biological tissues where the underlying
tissue microstructure can spatially vary [2]. To choose the penalization
parameter, we use the backward Euler discretization and the standard deviation
of the concentration field. The concentration network is eight layers deep and
each layer has 128 neurons. The same hyperparameters were used for all cases.
Figure 2 shows a snapshot of the results from the network at $t=0.5T$ for all
the four cases. The first column shows the exact value of the concentration
(${c}(x,y,0.5T)$) and the second column shows the value predicted by the
neural network ($\tilde{c}(x,y,0.5T)$). The third column in figure 2 shows the
true value of the diffusion coefficient (${D}(x,y)$) while the fourth column
shows the value predicted by the neural network ($\tilde{D}(x,y)$). The
relative error between the learned concentration and diffusion coefficient
distributions and the true values from the simulation are summarized in Table
1. Our method learned the diffusion coefficient reliably in all cases, with
the relative error being below $6.31\times 10^{-2}$ in all cases. Figure 3
compares the exact diffusion coefficient and the diffusion coefficient
predicted by the neural network in all the four cases along the horizontal
cross-section at the midpoint. To test the robustness of our method to sparse
data, we reduce the number of spatial points as observations to the
concentration field systematically for case 1. We look at 250000, 25000, 2500,
250 and 25 spatial data points and the results are shared in table 2. The
error in the diffusion coefficient only marginally increases till the number
of points are reduced to 250. The solver finally breaks down when only 25
spatial points or “pixels” are considered. This demonstrates that our
framework is not sensitive to resolution of data. With the method performing
well with noiseless data, we next considered images from experiments.

###  3.2 Experimental Data

To verify our algorithm on experimental data, we obtained time lapse images of
Rhodamine 6G dye diffusing through (1) two liquids (Section 3.2.1), (2)
agarose, and (3) gelatin hydrogels (Section 3.2.2). The intensity of the
images was assumed to be proportional to the concentration of the dye for our
calculations. Working with experimental data is significantly more challenging
compared to numerical data as various sources of noise contribute to
perturbations in the observations. These perturbations can significantly
change the estimated diffusion coefficient [4]. Neural networks can be
effective tools for processing the images and mitigating noise effects. A
neural network can be trained on a regression task over the concentration data
and automatic differentiation can be used to visualize the temporal and
spatial derivatives. This helps in identification and removal of aberrations
and helps in understanding the data better. As part of processing the images,
the space and the time domains were non-dimensionalized using an appropriate
length and time scale.

####  3.2.1 Diffusion through Liquid

The setup, shown in Fig. 4(a), was designed to investigate the diffusion
coefficients of different liquid solutions. A square cuvette [Polymethyl
methacrylate (PMMA) cuvette] with dimensions of 1.25$\times$1.25$\times$4.5 cm
was filled with two layers of solutions: DI water (liquid 1) as an upper layer
and a glycerine-water mixture (20$\%$ wt, liquid 2) in a lower layer. The
water solution on top contained 100 $\mu$M Rhodamine 6g dye (Sigma-Aldrich,
Rhodamine 6G, dye content 99%, $C_{28}H_{31}N_{2}O_{3}Cl$, molecular weight
479.01 g/mol, SKU 252433) and over time, diffusion occurred between the two
layers. A camera (Imperx, CLM-B6640M-TF000) was used to capture the diffusion
every minute with a resolution of 6640$\times$4400 pixels. The images were
analyzed using a custom Matlab algorithm to measure the dye intensity in each
frame. The setup was placed inside a photobox to prevent ambient light
interference and the temperature was maintained at 23∘ Celsius.

Different values have been reported for the diffusion coefficient of Rhodamine
6G [33, 34]. Here we used the following viscosity values of
$\mu_{1}=9.35\times 10^{-4}$ Ns/m2 and $\mu_{2}=1.60\times 10^{-3}$ Ns/m2 for
the respective solutions and calculated the diffusion coefficient at room
temperature (using Stokes-Einstein equation) as $D_{1}=4.05\times 10^{-10}$
m2/s and $D_{{2}}=2.36\times 10^{-10}$ m2/s [33, 34, 35]. For this case, the
neural network prediction for the spatially varying diffusion coefficient is
shown in Fig. 4 (b). We have used a cuvette to look at the diffusion between
liquids. The images were then captured using a camera and then processed
through a solver. We believe that the curvature has been caused as the
interface between the two fluids isn’t perfectly flat and due to the
distortion effects of approximating the three-dimensional cuvette using two-
dimensional images. Two distinct regions are visible in the plot and the
diffusion coefficient from the neural network has the value $4.0\times
10^{-10}$ m2/s and $2.4\times 10^{-10}$ m2/s for the water and glycerine-water
regions, respectively, which are in good agreement with the calculated ground
truth values and with previously-reported literature using numerous techniques
[33, 34, 35].

Figure 4: Experimental setup and results for diffusion of Rhodamine 6G dye. In
each setup, the approximate areas used for diffusion analyses are denoted with
red boxes. (a) Different liquid solutions in a square cuvette (DI water and
glycerine-water mixture). (b) The neural network prediction for the diffusion
coefficient for water and water-Glycerine mixture. (c) Initial source
concentrations of Rhodamine 6G dye in PBS to diffuse into a 2.5% and 0.5%
agarose gel chamber. (d) The neural network prediction for the diffusion
coefficient for diffusion through the agarose gel. (e) Initial source
concentrations of Rhodamine 6G dye in PBS to diffuse into a 5% and 15% gelatin
chamber. (f) The neural network prediction for the diffusion coefficient for
diffusion through gelatin the gel. The x and y coordinates were made
dimensionless using appropriate length scales.

####  3.2.2 Diffusion through Gels

Rectangular prisms of either gelatin or agarose hydrogels were fabricated such
that varied concentrations of Rhodamine 6G dye in phosphate buffered saline
(PBS) would diffuse inward from each side, mimicking the computational setup.
A custom computer-aided design (CAD) part was drawn in Autodesk Inventor and
fabricated using Flexible Resin V1 (RS-F2-FLGR-02) on the Formlabs Form 3 SLA
3D printer. The part was made to fit the diameter of the CELLSTAR 35x10mm cell
culture petri dish (Cat. No. 627-160), segmenting the dish into four equal
reservoirs with an open container in the center where the hydrogel was
situated. In both experiments, the hydrogels were 4 mm2 by 5 mm tall
rectangular prisms, dictated by the container size of the 3D printed part. The
windows on the sides of the gel that contacted the PBS/dye baths were 3 mm2
squares. Directly before the kinetic experiment began, different volumes of a
1 mM Rhodamine 6G stock solution were added into the different PBS reservoirs,
resulting in the source dye concentrations shown in Fig. 4(c,e). Brightfield
and red fluorescence images (BioTek Cytation 5 RFP, 531, 593 filter cube and
LED cube set) were captured on the BioTek Cytation 5 using the Gen5 software
with the BioTek 4x Plan Fluorite phase objective (BioTek, 1320515).

In the agarose experiments, 2.5% wt/vol agarose (Fisher BioReagents, BP160500)
in PBS was flooded into the hydrogel container while a plastic cylinder, 1.5
mm in diameter, was positioned in the center of the container as a place-
holder. Once the 2.5% agarose gelled completely, the cylinder was removed and
0.5% wt/vol agarose was transferred into the resulting hole and allowed to
solidify (Fig. 4(c)). Brightfield and red fluorescent images were obtained
every minute.

For gelatin experiments, gelatin from porcine skin, gel strength 300, Type A
(Sigma Life Sciences, G2500-500G) was used at 5% and 15% wt/vol. Because of
the nature of gelatin, the same fabrication technique could not be used as
above. Instead, 15% gelatin in PBS was drawn up into the luer slip tip of a 1
mL syringe (approximately 2.2 mm in diameter) (Henke Sass Wolf, 4010-200V0)
and allowed to gel. The gelatin cylinder was then placed into the 3D printed
part and 5% gelatin was flooded around the rest of the container and allowed
to gel (Fig. 4(e)). Due to the slower diffusion through gelatin which required
longer kinetic runs, brightfield and red fluorescent images were obtained
every 2.5 minutes.

In both hydrogel cases, the neural network prediction was able to identify two
distinct regions of spatially-varying diffusion coefficients corresponding to
the various formulations of the hydrogels. Within the agarose gel, values of
approximately $2.0\times 10^{-10}$ m2/s and $9.0\times 10^{-10}$ m2/s were
obtained for the 2.5% and 0.5% wt/vol agarose regions, respectively (Fig.
4(d)). Similarly, the values for the diffusion coefficients in the two gelatin
regions were roughly $2.4\times 10^{-10}$ m2/s and $3.7\times 10^{-10}$ m2/s
for the 15% and 5% wt/vol gelatin regions, respectively (Fig. 4(f)). Because
our experimental setup introduces a varied concentration gradient of Rhodamine
6G at each interface of the hydrogel, the obtained diffusion coefficients will
not be identical to literature values that use saturated gels, such as the
common Fluorescence Recovery After Photobleaching (FRAP) technique. However,
we observed the expected trend where both the 2.5% and 15% gels had lower
diffusion coefficient values than their less-dense counterparts. Additionally,
the range of values obtained is consistent with those found in the literature
for similar setups. Golmohamadi et al. found the diffusion coefficient of
Rhodamine 6G to be approximately $2.5-3\times 10^{-10}$ m2/s at neutral pH in
1.5% agarose gels [36]. Samprovalaki et al. similarly found the diffusion
coefficient of Rhodamine 6G to be around $2.89-4.88\times 10^{-10}$ m2/s at 30
degree Celsius in 1.5% agar gel depending on the concentration of Rhodamine 6G
used [37]. Within gelatin, Stucchi et al. observed a diffusion coefficient for
Rhodamine 6G of $4\times 10^{-11}$ m2/s for 5% gelatin hydrogels. However,
these hydrogels were crosslinked using diethyl squarate to increase the gel
density and other mechanical properties, which would result in a slower rate
of diffusion than the unmodified gelatin used here [38]. Additionally, the
diffusion maps in Fig. 4(d,f) were able to capture the varied boundary
qualities between the agarose and gelatin samples. The agarose results
indicate a more jagged boundary between the gel concentrations, likely
resulted from the removal of the plastic cylinder after the 2.5% agarose had
gelled around it. This is not apparent with the gelatin results (Fig. 4(d))
after the fabrication method was altered such that the cylinder of 15% gelatin
was extruded separately, placed within the custom-printed holder, and flooded
with 5% gelatin. The jagged lines that we observe are due to the nature of the
experiment, where distinct gels were artificially connected to mimic spatially
varying diffusion. Hence, the boundaries are not smooth and lead to sudden
variation in the diffusion coefficient. We don’t expect to see such variations
where anomalous diffusion is naturally present and the underlying structure is
inherently heterogenous. Taken together, these results provide strong evidence
that the algorithm presented is accurate for several types of experimental
data that may utilize distinct materials and hardware to capture rates of
diffusion.

##  4 Conclusion

The diffusion coefficient often varies spatially in biological tissues, porous
media, and soil. In this work, we used a physics informed neural network
(PINN) to solve the inverse problem of discovering the spatially varying
diffusion coefficient from spatio-temporal information on the diffused passive
scalar. The results of the framework were first validated and the accuracy of
the method was tested using noiseless, numerically-generated data. Inverse
problems are notoriously difficult to solve as they are often ill-posed and
small perturbations can significantly change the estimated diffusion
coefficients. During experiments, multiple sources of noise create such
perturbations in the recorded data. To demonstrate the robustness of the
framework, we then considered experimental data as Rhodamine 6G dye diffused
through different liquids and hydrogels. Specifically, for diffusion through
liquids, diffusion of the dye was captured from a water layer into a layer of
a water-glycerine mixture. For diffusion through gels, Rhodamine 6G dye was
diffused through either 2.5% and 0.5% wt/vol heterogeneous agarose gels, or 5%
and 15% wt/vol heterogeneous gelatin hydrogels. It was shown that our
framework captured the spatial distribution of the diffusion coefficient and
the interface of distinct regions well in each test case. Using this
methodology can thus allow the prediction of the spatiotemporal distribution
and variation inside biological samples, where understanding the heterogeneity
is crucial. This has implications to aid numerous biological fields and could
allow for more accurate calculation of the diffusion of growth factors or
other therapeutics from implanted devices, or contribute to a diagnosis of
aberrations such as carcinogenic tumours in tissues using data acquired
through medical imaging, as two examples.

##  5 Author Contributions

ST performed computational research and analyzed data. EE, SL, and LS designed
and performed experimental research. ST, EE, SL, LS, and AMA wrote the paper.
ST and AMA designed research.

##  6 Declaration of Interests

The authors declare no competing interests.

##  7 Acknowledgements

The authors would like to thank Ángel Enríquez for operating the 3D printer to
fabricate the custom hydrogel divider and Andres Barrio-Zhang for helping with
the experimental setup. This research was supported in part through the
National Science Foundation (CBET-1700961 and CBET 1705371) to AMA and the
CTSI TL1 Predoctoral Training Fellowship (UL1TR002529) and the National Cancer
Institute Predoctoral to Postdoctoral Fellowship Transition Award
(F99CA264734) to SL.

## References

  * [1] T. O. Optimality, Diffusion in spatially varying porous media 25 (1) (2015) 76–101.
  * [2] L. Ning, E. Özarslan, C. F. Westin, Y. Rathi, Precise Inference and Characterization of Structural Organization (PICASO) of tissue from molecular diffusion, NeuroImage 146 (September 2016) (2017) 452–473.  doi:10.1016/j.neuroimage.2016.09.057.
URL http://dx.doi.org/10.1016/j.neuroimage.2016.09.057

  * [3] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, Journal of inverse and Ill-posed problems 16 (4) (2008) 317–357.
  * [4] L. Yang, J. N. Yu, Z. C. Deng, An inverse problem of identifying the coefficient of parabolic equation, Applied Mathematical Modelling 32 (10) (2008) 1984–1995.  doi:10.1016/j.apm.2007.06.025.
  * [5] M. Kern, Numerical Methods for Inverse Problems, John Wiley & Sons, 2016.
  * [6] V. Isakov, Inverse problems for partial differential equations, Vol. 127, Springer, 2006.
  * [7] A. Shidfar, R. Pourgholi, M. Ebrahimi, A numerical method for solving of a nonlinear inverse diffusion problem, Computers & Mathematics with Applications 52 (6-7) (2006) 1021–1030.
  * [8] H. D. Mazraeh, R. Pourgholi, T. Houlari, Combining genetic algorithm and sinc-galerkin method for solving an inverse diffusion problem 7 (1) (2017) 33–50.
  * [9] O. Fudym, H. R. Orlande, M. Bamford, J. C. Batsale, Bayesian approach for thermal diffusivity mapping from infrared images with spatially random heat pulse heating, Journal of Physics: Conference Series 135 (Mcmc) (2008).  doi:10.1088/1742-6596/135/1/012042.
  * [10] M. Boodaghi, S. Libring, L. Solorio, A. M. Ardekani, A Bayesian approach to estimate the diffusion coefficient of Rhodamine 6G in breast cancer spheroids, Journal of Controlled Release 340 (May) (2021) 60–71.  doi:10.1016/j.jconrel.2021.10.002.
URL https://doi.org/10.1016/j.jconrel.2021.10.002

  * [11] B. Lusch, J. N. Kutz, S. L. Brunton, Deep learning for universal linear embeddings of nonlinear dynamics, Nature Communications 9 (1) (2018).  doi:10.1038/s41467-018-07210-0.
  * [12] S. L. Brunton, B. R. Noack, P. Koumoutsakos, Machine Learning for Fluid Mechanics, Annual Review of Fluid Mechanics 52 (1) (2020) 477–508.  doi:10.1146/annurev-fluid-010719-060214.
  * [13] B. Sanchez-Lengeling, A. Aspuru-Guzik, Inverse molecular design using machine learning:Generative models for matter engineering, Science 361 (6400) (2018) 360–365.  doi:10.1126/science.aat2663.
  * [14] E. J. Parish, K. Duraisamy, A paradigm for data-driven predictive modeling using field inversion and machine learning, Journal of Computational Physics 305 (2016) 758–774.  doi:10.1016/j.jcp.2015.11.012.
URL http://dx.doi.org/10.1016/j.jcp.2015.11.012

  * [15] J. Ling, A. Kurzawski, J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, Journal of Fluid Mechanics 807 (2016) 155–166.  doi:10.1017/jfm.2016.615.
  * [16] D. Wirth, A. McCall, K. Hristova, Neural network strategies for plasma membrane selection in fluorescence microscopy images, Biophysical Journal 120 (12) (2021) 2374–2385.  doi:10.1016/j.bpj.2021.04.030.
URL https://doi.org/10.1016/j.bpj.2021.04.030

  * [17] M. Schneider, A. Al-Shaer, N. R. Forde, AutoSmarTrace: Automated chain tracing and flexibility analysis of biological filaments, Biophysical Journal 120 (13) (2021) 2599–2608.  doi:10.1016/j.bpj.2021.05.011.
URL https://doi.org/10.1016/j.bpj.2021.05.011

  * [18] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations (Part I) (2017) 1–22.
URL http://arxiv.org/abs/1711.10561

  * [19] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations (Part II) (2017) 1–19.
URL http://arxiv.org/abs/1711.10566

  * [20] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686–707.  doi:10.1016/j.jcp.2018.10.045.
URL https://doi.org/10.1016/j.jcp.2018.10.045

  * [21] L. Lu, X. Meng, Z. Mao, G. E. Karniadakis, DeepXDE: A deep learning library for solving differential equations, SIAM Review 63 (1) (2021) 208–228.  doi:10.1137/19M1274067.
  * [22] S. Goswami, C. Anitescu, S. Chakraborty, T. Rabczuk, Transfer learning enhanced physics informed neural network for phase-field modeling of fracture, Theoretical and Applied Fracture Mechanics 106 (July 2019) (2019) 102447.  doi:10.1016/j.tafmec.2019.102447.
URL https://doi.org/10.1016/j.tafmec.2019.102447

  * [23] R. Tipireddy, P. Perdikaris, P. Stinis, A. Tartakovsky, A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations (2019).
URL http://arxiv.org/abs/1904.04058

  * [24] S. Amini Niaki, E. Haghighat, T. Campbell, A. Poursartip, R. Vaziri, Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture, Computer Methods in Applied Mechanics and Engineering 384 (2021) 113959.  doi:10.1016/j.cma.2021.113959.
URL https://doi.org/10.1016/j.cma.2021.113959

  * [25] G. Kissas, Y. Yang, E. Hwuang, W. R. Witschey, J. A. Detre, P. Perdikaris, Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 358 (2020) 112623.  doi:10.1016/j.cma.2019.112623.
URL https://doi.org/10.1016/j.cma.2019.112623

  * [26] M. Liu, L. Liang, W. Sun, A generic physics-informed neural network-based constitutive model for soft biological tissues, Computer Methods in Applied Mechanics and Engineering 372 (2020) 113402.  doi:10.1016/j.cma.2020.113402.
URL https://doi.org/10.1016/j.cma.2020.113402

  * [27] M. Raissi, A. Yazdani, G. E. Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations (C) (2020) 1–5.  doi:10.1126/science.aaw4741.
URL https://science.sciencemag.org/content/367/6481/1026/tab-pdf

  * [28] M. Raissi, H. Babaee, P. Givi, Deep learning of turbulent scalar mixing, Physical Review Fluids 4 (12) (2019) 1–19.  doi:10.1103/PhysRevFluids.4.124501.
  * [29] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mane, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viegas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, X. Zheng, TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems (2016).
URL http://arxiv.org/abs/1603.04467

  * [30] S. Thakur, M. Raissi, H. Mitra, A. Ardekani, Temporal Consistency Loss for Physics-Informed Neural Networks (2023) 1–14.
URL http://arxiv.org/abs/2301.13262

  * [31] D. P. Kingma, J. L. Ba, Adam: A method for stochastic optimization, 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings (2015) 1–15.
  * [32] H. G. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to computational continuum mechanics using object-oriented techniques, Computers in Physics 12 (6) (1998) 620.  doi:10.1063/1.168744.
  * [33] P. O. Gendron, F. Avaltroni, K. J. Wilkinson, Diffusion coefficients of several rhodamine derivatives as determined by pulsed field gradient-nuclear magnetic resonance and fluorescence correlation spectroscopy, Journal of Fluorescence 18 (6) (2008) 1093–1101.  doi:10.1007/s10895-008-0357-7.
  * [34] G. Majer, J. P. Melchior, Characterization of the fluorescence correlation spectroscopy (FCS) standard Rhodamine 6G and calibration of its diffusion coefficient in aqueous solutions, Journal of Chemical Physics 140 (9) (2014).  doi:10.1063/1.4867096.
  * [35] C. T. Culbertson, S. C. Jacobson, J. Michael Ramsey, Diffusion coefficient measurements in microfluidic devices, Talanta 56 (2) (2002) 365–373.  doi:10.1016/S0039-9140(01)00602-6.
  * [36] M. Golmohamadi, T. A. Davis, K. J. Wilkinson, Diffusion and partitioning of cations in an agarose hydrogel, Journal of Physical Chemistry A 116 (25) (2012) 6505–6510.  doi:10.1021/jp212343g.
  * [37] K. Samprovalaki, P. T. Robbins, P. J. Fryer, Investigation of the diffusion of dyes in agar gels, Journal of Food Engineering 111 (4) (2012) 537–545.  doi:10.1016/j.jfoodeng.2012.03.024.
URL http://dx.doi.org/10.1016/j.jfoodeng.2012.03.024

  * [38] S. Stucchi, D. Colombo, R. Guizzardi, A. D’Aloia, M. Collini, M. Bouzin, B. Costa, M. Ceriani, A. Natalello, P. Pallavicini, L. Cipolla, Squarate Cross-Linked Gelatin Hydrogels as Three-Dimensional Scaffolds for Biomedical Applications, Langmuir 37 (48) (2021) 14050–14058.  doi:10.1021/acs.langmuir.1c02080.