MNRAS 000, 1–38 (2020) Preprint 22 September 2020 Compiled using
MNRAS LATEX style file v3.0
Stellar Parameter Determination from Photometry usingInvertible
Neural Networks
Victor F. Ksoll,1,2? Lynton Ardizzone,3 Ralf Klessen,1,2 Ullrich
Koethe,3
Elena Sabbi,5 Massimo Robberto,5 Dimitrios Gouliermis,1,4
Carsten Rother, 3
Peter Zeidler5,6 and Mario Gennaro51Universität Heidelberg,
Zentrum für Astronomie, Institut für Theoretische Astrophysik,
Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany2Universität
Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches
Rechnen, Im Neuenheimer Feld 205, 69120 Heidelberg,
Germany3Universität Heidelberg, Heidelberg Collaboratory for Image
Processing, Visual Learning Lab, Berliner Str. 43, 69120
Heidelberg, Germany4Max Planck Institute for Astronomy, Königstuhl
17, 69117 Heidelberg, Germany5Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA6Department of
Physics and Astronomy, Johns Hopkins University, Baltimore, MD
21218, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACTPhotometric surveys with the Hubble Space Telescope
(HST) allow us to study stellarpopulations with high resolution and
deep coverage, with estimates of the physicalparameters of the
constituent stars being typically obtained by comparing the
surveydata with adequate stellar evolutionary models. This is a
highly non-trivial task dueto effects such as differential
extinction, photometric errors, low filter coverage,
oruncertainties in the stellar evolution calculations. These
introduce degeneracies thatare difficult to detect and break. To
improve this situation, we introduce a novel deeplearning approach,
called conditional invertible neural network (cINN), to solve
theinverse problem of predicting physical parameters from
photometry on an individualstar basis and to obtain the full
posterior distributions. We build a carefully curatedsynthetic
training data set derived from the PARSEC stellar evolution models
to pre-dict stellar age, initial/current mass, luminosity,
effective temperature and surfacegravity. We perform tests on
synthetic data from the MIST and Dartmouth models,and benchmark our
approach on HST data of two well-studied stellar clusters,
West-erlund 2 and NGC 6397. For the synthetic data we find overall
excellent performance,and note that age is the most difficult
parameter to constrain. For the benchmark clus-ters we retrieve
reasonable results and confirm previous findings for Westerlund 2
oncluster age (1.04+8.48−0.90 Myr), mass segregation, and the
stellar initial mass function. ForNGC 6397 we recover plausible
estimates for masses, luminosities and temperatures,however,
discrepancies between stellar evolution models and observations
prevent anacceptable recovery of age for old stars.
Key words: methods: data analysis – methods: statistical –
stars: formation – stars:fundamental parameters – stars:
pre-main-sequence – galaxies: clusters: individual:Westerlund 2,
NGC6397.
1 INTRODUCTION
Machine learning (ML) employs statistical models to predictthe
characteristics of a dataset using samples of previouslycollected
data without relying on physical models of the sys-tem. The
introduction of ML for solving regression, clas-sification and
clustering problems has revolutionised scien-
? E-mail: [email protected]
tific research, and in particular has provided effective
meth-ods for analyzing big astronomical data (Feigelson &
Babu2012; Ivezic et al. 2014). In order to construct a model
fromobserved data, machine learning methods rely on human-defined
classifiers or ’feature extractors’ (Hastie et al. 2009).However,
complex problems require algorithms that auto-mate the creation of
feature extractors using large amountsof data. These algorithms
represent a family of ML tech-niques, named deep learning, and they
are based on the
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construction of artificial neural networks (NNs; Goodfellowet
al. 2016). While training NNs requires significant com-putational
power, they achieve far higher levels of accuracythan classic ML
for many non-linear problems. In this pi-lot study we employ
invertible NNs to infer stellar ages andmasses from Hubble Space
Telescope (HST) imaging of twowell-studied stellar clusters. Our
aim is to explore the effi-ciency of NNs in extracting stellar
physical parameters fromphotometry alone. We train our networks
using modeled-observable properties relations provided by
theoretical evo-lutionary models.
Star clusters, the building blocks of galaxies, are thesignposts
guiding our understanding of the formation andevolution of stars.
This understanding stems from the phys-ical properties of stars in
clusters, being deduced from de-tailed comparisons of photometric
observations to theo-retical evolutionary models. The interface
where observa-tions meet theory is often provided by the
observationalcolourâĂŞmagnitude diagram (CMD) and its
theoreticalcounterpart, the Hertzsprung-Russell diagram (HRD).
Inthe HRD two physical properties of stars, the effective
tem-perature and the luminosity, are compared to stellar
evolu-tionary models to determine fundamental stellar
parameters,the initial mass and the age of the star, which are not
directlyaccessible by observations alone. This comparison can be
di-rectly performed through fitting of isochronal
evolutionarymodels to the observed CMDs. This method, however,
lacksproper statistical basis because the relations between
observ-ables and physical properties may present degeneracies
thatneed to be accounted for. More advanced methods, basedon Bayes
statistics, derive probabilistically the cumulativeproperties of
stellar populations, such as the mean age, interms of posterior
probability distribution functions of theproperties of individual
stars, e.g. the age (see Valls-Gabaud2014, and references therein).
These methods provide a sig-nificant improvement by tackling the
intrinsic model degen-eracies through priors on the stellar initial
mass function,binary fraction, or extinction distribution (e.g.
Jørgensen,B. R. & Lindegren, L. 2005; Da Rio et al. 2010).
Bayesian inference encompasses a specific class of ma-chine
learning models, i.e. those based on strong prior intu-itions.
However, these priors do not add significant value inthe case of
big data, and are computationally expensive andslow. As a
consequence, other ML methods are employedto infer stellar physical
parameters from photometry. Themost successful techniques developed
so far are generallybased on time-domain observations, such as
light curves us-ing photometric-brightness variations (e.g. Miller
et al. 2015)or time-series asteroseismic observations (e.g.
Bellinger et al.2016). These methods make use of various instances
of eachspecific target star in time, a dataset which cannot be
easilyobtained for rich stellar samples in compact clusters.
Inves-tigations of stars in clusters normally rely only on
’static’,rather than time-dependent imaging, which cannot be
ad-dressed by classic ML methods. Moreover, it is now well
un-derstood that parameter degeneracies encoded in the
evolu-tionary models make the problem of inferring stellar
massesand ages from photometric measurements a non-linear prob-lem.
The solution of such problems calls for the employmentof artificial
NNs.
There have been several recent studies that employ neu-ral
network approaches to solve prediction tasks in astron-
omy similar to the problem that we analyze in this paper.Sharma
et al. (2020) train a convolutional neural network ona suite of
spectral libraries in order to classify stellar spec-tra according
to the Harvard scheme and successfully applytheir approach to data
from the Sloan Digital Sky Survey(SDSS) database. Kounkel et al.
(2020) leverage Gaia DR2photometry and parallaxes to construct a
neural networkthat predicts age, extinction and distance of stellar
clustersin the Milky Way, allowing them to study the star
formationactivity in the spiral arms. Cantat-Gaudin et al. (2020)
usea similar neural network approach, also predicting
physicalparameters of stellar clusters from Gaia data, but use 2D
his-tograms of the observed CMDs as inputs. Olney et al. (2020)use
a deep convolutional neural network to predict surfacetemperature,
metallicity and surface gravity of young stel-lar objects (YSOs)
based on spectra from APOGEE. Withintheir training set construction
they employ another convolu-tional neural network to infer physical
parameters of YSOs,i.e. ages, masses, extinction, surface
temperature/gravity,from photometry in 9 bands of the Gaia system,
as well asdistance, stellar radius and luminosity. This auxiliary
net-work is trained on synthetic isochrone data and
successfullyrecovers surface temperatures for YSOs on real Gaia
obser-vations.
For many applications in natural sciences, the forwardprocess of
determining measurements from a set of underly-ing physical
parameters is well-defined, whereas the inverseproblem is ambiguous
because multiple parameter sets canresult in the same observation
(i.e., degeneracies). Classi-cal neural networks attempt to address
this ambiguity bysolving the inverse problem directly. However, to
fully char-acterise degeneracies, the full posterior parameter
distribu-tion, conditioned on an observed measurement, must be
de-termined. A particular class of neural networks, so-called
in-vertible neural networks (INNs), is well suited for this
task(e.g. Ardizzone et al. 2019a). Unlike classical neural
net-works, INNs learn the forward process, using additional la-tent
output variables to capture the information otherwiselost. This
invertibility allows a model of the correspondinginverse process to
be learned implicitly, providing the fullparameter posterior
distribution for a given observation andcorresponding distribution
of the latent variables. INNs aretherefore a powerful tool in
identifying multi-modalities, pa-rameter correlations, and
unrecoverable parameters.
In this paper we present the application of invertibleneural
networks to the regression problem of predictingphysical parameters
of individual stars based on observedphotometry. Note that we do
not perform an exhaustiveanalysis of the approach, but rather aim
to provide an intro-duction to the method, highlighting our first
successes. Thispaper is the first in a series, in which we adapt
and developthe approach, as well as explore its limitations.
As mentioned above, in general this regression task isprone to
errors due to the many sources of degeneracy in themapping from
physical to observable space, such as metallic-ity, extinction,
variability, binarity and the intrinsic overlapof certain phases in
stellar evolution in the observable space,e.g. the red giant branch
and the pre-main-sequence. Sinceour primary goal is to test the
viability of the method, inthis paper we neglect some of these
factors, adopting thefollowing simplifying assumptions: 1) We only
deal with sin-gle metallicity populations, 2) we obtain an estimate
of the
MNRAS 000, 1–38 (2020)
