# Abundance Patterns of $\alpha$ and Neutron-capture Elements in the Helmi
Stream

Guilherme Limberg Universidade de São Paulo, Instituto de Astronomia,
Geofísica e Ciências Atmosféricas, Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil  Rafael M. Santucci Universidade Federal de
Goiás, Instituto de Estudos Socioambientais, Planetário, Goiânia, GO
74055-140, Brazil  Universidade Federal de Goiás, Campus Samambaia, Instituto
de Física, Goiânia, GO 74001-970, Brazil  Silvia Rossi Universidade de São
Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas,
Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil  Anna B. A. Queiroz  Leibniz-Institut
f$\ddot{u}$r Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam,
Germany  Cristina Chiappini  Leibniz-Institut f$\ddot{u}$r Astrophysik Potsdam
(AIP), An der Sternwarte 16, D-14482 Potsdam, Germany  Laboratório
Interinstitucional de e-Astronomia - LIneA, RJ 20921-400, Rio de Janeiro,
Brazil  Stefano O. Souza Universidade de São Paulo, Instituto de Astronomia,
Geofísica e Ciências Atmosféricas, Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil  Hélio D. Perottoni Universidade de São Paulo,
Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Departamento de
Astronomia,
SP 05508-090, São Paulo, Brazil  Angeles Pérez-Villegas Instituto de
Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 106, C.
P. 22800, Ensenada, B. C., México  Fabrícia O. Barbosa Universidade de São
Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas,
Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil

###### Abstract

We identified 8 additional stars as members of the Helmi stream (HStr) in the
combined GALAH+ DR3 and Gaia EDR3 catalog. By consistently reevaluating
claimed members from the literature, we consolidate a sample of 22 HStr stars
with parameters determined from high-resolution spectroscopy and spanning a
considerably wider (by $\sim$0.5 dex) metallicity interval
($-2.5\lesssim\rm[Fe/H]<-1.0$) than previously reported. Our study focuses on
$\alpha$ (Mg and Ca) and neutron-capture (Ba and Eu) elements. We find that
the chemistry of HStr is typical of dwarf spheroidal (dSph) galaxies, in good
agreement with previous $N$-body simulations of this merging event. Stars of
HStr constitute a clear declining sequence in [$\alpha$/Fe] for increasing
metallicity up to $\rm[Fe/H]\sim-1.0$. Moreover, stars of HStr show a median
value of $+0.5$ dex for [Eu/Fe] with a small dispersion ($\pm$0.1 dex). Every
star analyzed with $\rm[Fe/H]<-1.2$ belong to the $r$-process enhanced
($\rm[Eu/Fe]>+0.3$ and $\rm[Ba/Eu]<0.0$) metal-poor category, providing
remarkable evidence that, at such low-metallicity regime, stars of HStr
experienced enrichment in neutron-capture elements predominantly via
$r$-process nucleosynthesis. Finally, the extended metallicity range also
suggests an increase in [Ba/Eu] for higher [Fe/H], in conformity with other
surviving dwarf satellite galaxies of the Milky Way.

Galaxy: stellar halo – Galaxy: formation – Galaxy: evolution – Galaxy:
kinematics and dynamics – stars: abundances – stars: Population II – galaxies:
dwarf

##  1 Introduction

Within the hierarchical assembly paradigm, the Galactic stellar halo (or
simply “halo”) is expected to retain the chemodynamical signatures of merging
events between the Milky Way and dwarf galaxies of various masses in the past
(Helmi, 2008, 2020). Therefore, studies of surviving satellite galaxies
provide insights about the formation of the Milky Way itself. Unfortunately,
measurements of elemental abundances in the atmospheres of individual stars
from these distant systems are extremely difficult (e.g., Tolstoy et al.
2009), hindering our ability to directly investigate their star-forming
environments and chemical-evolution histories.

An alternative approach is to pinpoint which stars in the halo were accreted
and, out of these, which ones share a common origin. A widely utilized
strategy to find the stellar remnants of these ancient building blocks is to
search for their clumping in phase space. The first identification of a
kinematically-cohesive group of stars with this method was presented in a
seminal work by Helmi et al. (1999). Throughout the years, this substructure
has been known as the Helmi stream (hereafter “HStr”; see Helmi 2020 for a
recent review).

In order to investigate the chemical profile of HStr, Roederer et al. (2010)
observed 12 likely members of this substructure. More recently, Aguado et al.
(2021a) acquired spectra for 7 more candidates. Interestingly, both efforts
reported that stars of HStr were enriched in neutron-capture elements
predominantly via the “rapid” process ($r$-process) in comparison to the
“slow” one ($s$-process; see Sneden et al. 2008 and Frebel 2018 for reviews).
In recent studies, Limberg et al. (2021a) and Gudin et al. (2021) demonstrated
that, indeed, several $r$-process-enhanced (RPE; [Eu/Fe] $>+0.3$ and [Ba/Eu]
$<0.0$) metal-poor ($\rm[Fe/H]<-1.0$; Beers & Christlieb 2005) stars are
dynamically associated with HStr. It appears that chemical-abundance
information for larger samples of genuine members of HStr might allow us to
investigate the site(s) for the occurrence of the $r$-process in its (now)
destroyed progenitor system. In addition, stars of HStr have the enormous
advantage of being much closer (and hence brighter) than any surviving dwarf
spheroidal (dSph) or ultra-faint dwarf (UFD) satellite galaxy.

In this Letter, we identified additional stars of HStr in the third data
release (DR3) of the Galactic Archaeology with HERMES (GALAH+ DR3; Buder et
al. 2020) survey combined with Gaia Early Data Release 3 (EDR3; Gaia
Collaboration et al. 2020). We also searched the literature to consolidate a
sample covering an [Fe/H] interval considerably wider than previously
reported. The extended metallicity range reveals a declining trend in
[$\alpha$/Fe] within $-2.0\lesssim\rm[Fe/H]<-1.0$, providing clear evidence
that HStr is the debris of a long-vanished dwarf galaxy. This sample further
allowed us to investigate neutron-capture elements. We demonstrate that the
[Ba/Eu] pattern is consistent with the dwarf-galaxy progenitor hypothesis. We
also confirm that every star of HStr with $\rm[Fe/H]<-1.2$ belongs to the
aforementioned RPE class. The employed data is described in Section 2. Section
3 contains our dynamical and chemical analyses. The summary of our results and
a brief discussion are presented in Section 4.

##  2 Data

###  2.1 Stream Candidates

We considered the complete GALAH+ DR3 high-resolution ($R\sim$ 28,000)
spectroscopic catalog. First, we removed stars with flagged555Whenever a
quality flag is set to “0” in the GALAH+ DR3 catalog, it represents a reliable
and/or real estimate of the given parameter. For details on the quality
assessment, we refer the reader to Buder et al. (2020). stellar parameters
($\texttt{flag\\_sp}\neq 0$) and metallicities ($\texttt{flag\\_fe\\_h}\neq
0$). Since we are interested in finding stars associated with a substructure
of low-metallicity, we further constrain our sample to $\rm[Fe/H]\leq-0.7$
(keeping $\sim$18,000 stars). Moreover, all abundances analyzed (Sections 3.2
and 3.3) are true measurements ($\texttt{flag\\_X\\_fe}=0$, where “X” is any
given chemical species). We put particular attention to the estimated [Eu/Fe],
since it relies on a single line in GALAH, Eu II at 6645 Å, which is difficult
to detect in warm metal-poor stars (Sneden et al., 2008). Hence, all stars
with reliable [Eu/Fe] in the sample are in the upper-giant-branch phase
($T_{\rm eff}<4800$ K and $\log g<2.0$). At this wavelength, the typical
signal-to-noise ratio is $60\leq\texttt{snr\\_c3\\_iraf}\leq 80$ for these
objects. We also visually inspected the spectra of stars found to be
associated with HStr (Section 3.1) to ensure the presence of this absorption
feature (see Figure 1).

Figure 1: Examples of GALAH+ DR3 spectra in the wavelength region of the Eu II
line (6645 Å; orange-shaded area). For the convenience of the reader, we also
identify the strong absorption feature as Ni I (blue stripe). Both panels show
stars selected as confident members of HStr (see Section 3.1). The black solid
lines are the normalized spectra, while the gray-shaded regions represent
their respective $1\sigma$ uncertainties. The red dashed line shows the
position of the pseudo-continuum to guide the eye.  Figure 2: Left: velocity
distribution defined by $(v_{\phi},\sqrt{v_{R}^{2}+v_{z}^{2}})$. Dashed line
marks the $||\boldsymbol{v}-\boldsymbol{v}_{\rm circ}||\geq 220$ km s-1
boundary between disk and halo (see text). Middle: $(L_{z},L_{\perp})$. Dashed
lines delineate the selection boxes (A and B; see Section 3.1) from Koppelman
et al. (2019). Right: inclination vs. eccentricity. The dashed ellipse marks
the selection from Aguado et al. (2021a). Green ellipses delimit 3-$\sigma$
ranges for the distributions of HStr stars presented by Limberg et al.
(2021a). In all panels, the regions dominated by disk-like orbits are also
indicated. Stream members are shown as red symbols with black (confident) or
red (likely) edges. Open symbols are discarded candidates. Star symbols are
members selected from the GALAH sample and circles are from the literature
compilation. Gray dots represent low-metallicity ($\rm[Fe/H]\leq-0.7$) stars
from GALAH.

To achieve a more complete view of the chemical patterns of HStr, over a large
metallicity range, we also compiled stars that had been previously suggested
to be associated with the substructure. We require that these stars have
atmospheric parameters and abundances estimated from analyses of high-
resolution spectra, in conformity with GALAH. The fundamental source is Aguado
et al. (2021a), because these authors had already performed a search in the
literature for HStr candidates. This sample includes stars observed mostly by
the authors themselves and by Roederer et al. (2010). It also contains stars,
attributed to HStr by Yuan et al. (2020), observed over the course of the
APOGEE (Majewski et al., 2017) survey. We have further included stars
indicated by Limberg et al. (2021a) and Gudin et al. (2021) in our literature
compilation. Stellar parameters and abundances for these references were
obtained during the main effort of the $R$-Process Alliance (Hansen et al.,
2018; Ezzeddine et al., 2020).

###  2.2 Dynamical Properties

We cross-matched (1.5′′ search radius) all samples with Gaia EDR3 (Gaia
Collaboration et al., 2020) to acquire accurate parallaxes (re-calibrated
following Lindegren et al. 2020a) and absolute proper motions. These
astrometric information, the high-resolution spectroscopic data (Section 2.1),
as well as mid- and near-infrared photometry from WISE (Wright et al., 2010)
and 2MASS (Skrutskie et al., 2006), respectively, were used to estimate
heliocentric distances via isochrone fitting in a Bayesian framework with the
StarHorse code (Queiroz et al., 2018). For this exercise, we did not consider
targets classified as spurious astrometric solutions
($\texttt{fidelity\\_v1}<0.5$; Rybizki et al. 2021). The medians of the
resulting posterior probability distribution functions were taken as nominal
values (see Appendix A). We refer to Queiroz et al. (2020) for a complete
description of the assumptions regarding stellar-evolution models and priors.
We also discarded stars with re-normalized unit weight errors of the reduced
astrometric $\chi^{2}$ outside the recommended interval ($\texttt{RUWE}>1.4$;
Lindegren et al. 2020b) and those with relative uncertainties $\geq$15% in
their derived distances.

Given the high quality of the spectra at hand, we considered radial velocities
(RVs) from the various sources mentioned in Section 2.1. For the targets
observed by Aguado et al. (2021a), neither improved RVs were determined by the
authors nor these are provided by Gaia’s DRs. Hence, we adopted RVs from the
low-resolution spectroscopy of SDSS/SEGUE (York et al., 2000; Yanny et al.,
2009) and LAMOST (Cui et al., 2012), with errors no worse than 5 km s-1.

We calculated the orbits of all stars for 5 Gyr forward, with the publicly
available library AGAMA (Vasiliev, 2019), under the axisymmetric Galactic
potential model of McMillan (2017). The Galactic parameters are, for
consistency, from McMillan (2017) as well. Specifically, the distance from the
Sun to the Galactic center is $R_{\odot}=8.2$ kpc, the circular velocity at
this position is $v_{\rm circ}=232.8$ km s-1, and the peculiar motion of the
Sun is $(U,V,W)_{\odot}=(11.10,12.24,7.25)$ km s-1 (Schönrich et al., 2010).
We accounted for the uncertainties in distances, proper motions, and RVs,
assuming Gaussian profiles for those, by performing 1000 realizations of each
star’s orbit with a Monte Carlo procedure. The medians of the resulting
distributions were taken as our nominal values in the derived dynamical
quantities.

##  3 Analysis

###  3.1 Stream Membership

In order to accomplish a consistent analysis of abundance information
(Sections 3.2 and 3.3), it is necessary to delineate a clear stream membership
criteria. Then, we need to implement such selection consistently for all
samples described in Section 2. Conveniently, previous studies carried out
similar exercises for HStr.

First, we removed stars with disk-like kinematics utilizing the plane defined
by ($v_{\phi},\sqrt{v_{R}^{2}+v_{z}^{2}}$), where ($v_{R},v_{\phi},v_{z}$) is
the velocity vector in the cylindrical coordinate system. Stars with
$v_{\phi}>0$ are in prograde motion, rotating in the same orientation as the
disk. For this purpose, we applied the cut
$||\boldsymbol{v}-\boldsymbol{v}_{\rm circ}||\geq 220$ km s-1 (dashed line in
the left panel of Figure 2), where $\boldsymbol{v}$ is the complete velocity
vector of a given star.

Figure 3: Left: $(L_{z},E)$. Middle: action-space map. In both panels, the
regions dominated by disk-like orbits are also indicated. Right: HStr stars in
Cartesian Galactic positions $(X,Z)_{\rm Gal}$. The arrows point in the
direction of motion of the stars, scaled by their amplitudes. Green ellipses
delimit 3-$\sigma$ ranges for the distributions of HStr stars presented by
Limberg et al. (2021a). In all panels, stream members are shown as red symbols
with black (confident) or red (likely) edges. Open symbols are discarded
candidates. Star symbols are members selected from the GALAH sample and
circles are from the literature compilation (Section 2.1). Gray dots represent
low-metallicity ($\rm[Fe/H]\leq-0.7$) stars from GALAH.

Second, we employed the criteria suggested by Koppelman et al. (2019), who
carried out an in-depth chemodynamical characterization of HStr. These
authors’ selection boxes are drawn in the middle panel of Figure 2 within the
space defined by the in-plane ($L_{\perp}$) and vertical ($L_{z}$) components
of the total angular momentum ($L=\sqrt{L_{\perp}^{2}+L_{z}^{2}}$). For
prograde motion, $L_{z}>0$. Although $L_{\perp}$ is not fully conserved in an
axisymmetric potential, it is commonly used for identification of
substructures since it preserves a reasonable amount of clumping over time
(e.g., Helmi 2008, 2020). Stars that respect the more restrictive box (A) are
considered confident members of the stream (symbols with black edges), while
objects occupying the more permissive one (B) are taken to be likely
associated (red edges).

The third criterion used in this work is equivalent to the one described by
Aguado et al. (2021a), characterized by orbital inclination
($i=\arccos{(L_{z}/L)}$; from this definition, $i<90^{\circ}$ for prograde
motion) and eccentricity ($e$). These authors vetted their stream candidates
with a 4-$\sigma$ range around the center of the distribution of HStr members
originally found by Myeong et al. (2018) in this parameter space. In the right
panel of Figure 2, the dashed ellipse reproduces these authors’ selection.

A comparison between these criteria and the 3-$\sigma$ distributions of HStr
stars from Limberg et al. (2021a) is also provided in all panels of Figure 2
(green ellipses). The application of this three-step selection to the low-
metallicity sample from GALAH yields a total of 8 HStr candidates (red star
symbols). Out of these, 3 stars are classified as confident and 5 as probable
members. We also evaluated the literature compilation with the same approach,
resulting in a total of 14 additional stars to be considered for analysis of
chemical-abundance data (red circles). Amongst these stars, 4 are originally
from Roederer et al. (2010), 3 are from Yuan et al. (2020), and 5 were
observed by Aguado et al. (2021a). The single star selected from Limberg et
al. (2021a) has been confirmed to be associated with the stream, while only 1
from Gudin et al. (2021) can be considered a member. The final list of 22
vetted HStr stars, alongside relevant information about them, is provided in
Appendix A.

As a sanity check, we have examined the newly-identified stream members in
orbital energy ($E$) and actions ($J_{R}$, $J_{\phi}=L_{z}$, and $J_{z}$). For
consistency, these are also compared to the distributions independently found
by Limberg et al. (2021a). In Figure 3, we show the 3-$\sigma$ ranges for HStr
found by these authors in $(L_{z},E)$ and in the action-space map (left and
middle panels, respectively). All confident members overlap with the
delineated regions, while a couple of the probable ones fall outside of them,
but are located near the boundaries. This qualitative inspection helps
consolidate our set of criteria as truly representative of HStr.

Despite carrying out a selection exclusively with a kinematic/dynamical
approach, stars of HStr are cohesively distributed in configuration space
(right panel of Figure 3). These objects are piercing through the Galactic
plane, streaming downwards in the Cartesian Galactic position plane
$(X,Z)_{\rm Gal}$. Such behavior is commonly interpreted as the partial phase
mixing of the debris of a shredded dwarf galaxy (Helmi, 2008; Myeong et al.,
2018; Koppelman et al., 2019).

###  3.2 $\alpha$ Elements

With our collection of 22 stream members (15 confident and 7 likely), we
investigate abundance trends of $\alpha$ elements (Mg and Ca) derived
exclusively from high-resolution spectroscopy. In this context, we can compare
our [$\alpha$/Fe]–[Fe/H] distributions with those presented by Aguado et al.
(2021a), who analyzed a mixture between low- and high-resolution data. We can
also compare these chemical-abundance patterns with those from surviving dwarf
satellite galaxies of the Milky Way and speculate about the nature of HStr’s
parent system.

We present the $\alpha$-element profiles in the top row of Figure 4. The most
prominent, immediately perceptible feature is the clear decrease in
[$\alpha$/Fe] (both Mg and Ca) with increasing metallicity, but plateauing at
$\sim$0.35 for $\rm[Fe/H]\lesssim-2.0$, characterizing a “knee” (e.g.,
Matteucci & Greggio 1986). This point delimits the transition between core-
collapse and type Ia supernovae dominated epochs in chemical evolution and was
first conjectured for external galaxies by Matteucci & Brocato (1990), later
observed by Shetrone et al. (2003). A similar $\alpha$-pattern signature was
presented by Aguado et al. (2021a). However, their results were limited to
lower metallicities ($\rm[Fe/H]\lesssim-1.5$). Hence, the inclusion of the
GALAH sample confirms the continuity of this $\alpha$-element trend of HStr
stars up to $\rm[Fe/H]\sim-1.0$ with an excellent agreement. We stress that
considering exclusively the newly found members from GALAH, this pattern is
still noticeable. Consequently, even if small biases between sources might
exist, these qualitative conclusions are robust against them.

The comparison to dwarf satellite galaxies of the Milky Way is presented in
Figure 4 as well. Abundances have been compiled from the SAGA database (Suda
et al., 2008, 2017), favouring references with neutron-capture elements
available. The Sculptor dSph (Hill et al., 2019) follows a similar sequence in
[$\alpha$/Fe] for the same metallicity range
($-2.5\lesssim\rm[Fe/H]\lesssim-1.0$) to that of HStr stars, but with a
steeper decline. With the aid of chemical-evolution models, such feature has
been interpreted by Aguado et al. (2021a) as a dSph-galaxy progenitor with a
slower star-formation rate, but accompanied by a smaller wind efficiency, when
compared to Sculptor. This apparent dSph-like origin is in keeping with
previous $N$-body simulations of this merging event by Koppelman et al.
(2019).

###  3.3 Neutron-capture Elements

In this work, we are also interested in evaluating the hypothesis that the
progenitor of HStr was enriched in neutron-capture elements (Ba and Eu)
predominantly via the $r$-process, as suggested by Roederer et al. (2010, and
followed-up by ) and reinforced with a dynamical counterpart to the argument
by Limberg et al. (2021a, also ). Here, we consider the abundances of Ba and
Eu, since these chemical species serve as diagnostic for $r$-process
enhancement in metal-poor stars as first noted by Spite (1992), although the
existence of this kind of objects was established much earlier by Spite &
Spite (1978). Roederer et al. (2010) were capable of determining abundances of
these elements for some of their stream candidates. Out of 4 stars observed by
these authors and vetted as confident members (Section 3.1), 3 of them have
[Ba/Fe] and [Eu/Fe] measured. All stream stars from the GALAH sample have non-
flagged ($\texttt{flag\\_Ba\\_fe}=0$) [Ba/Fe], but only 3 of them have usable
($\texttt{flag\\_Eu\\_fe}=0$) Eu abundances. In total, 8 HStr stars can be
employed for our investigation of neutron-capture nucleosynthesis in this
substructure.

Figure 4: Top row: $\alpha$ elements, [Mg/Fe] (left) and [Ca/Fe] (right)
versus [Fe/H]. Upper dashed lines approximately follow the evolution of the
Milky Way, with a “knee” at $\rm[Fe/H]=-1.0$ (e.g., Matteucci & Greggio 1986).
The solar level is also indicated in both panels. Bottom row: neutron-capture
elements. Left: [Eu/Fe]–[Fe/H]. The dashed lines at $\rm[Eu/Fe]=+0.3$ and
$+1.0$ delineate the characteristic $r$-process-enhancement of the $r$-I and
$r$-II, respectively, classes of RPE metal-poor stars (Beers & Christlieb,
2005). Right: [Ba/Eu]–[Fe/H]. The solar level is marked as well. In all
panels, stream members are shown as red symbols with black (confident) or red
(likely) edges. Star symbols are members selected from the GALAH sample and
circles are from the literature compilation (Section 2.1). Colored dots
illustrate stars from dSph or UFD galaxies compiled in the SAGA database (Suda
et al., 2008, 2017). Purple: Fornax (dSph; Letarte et al. 2010; Lemasle et al.
2014). Green: Sculptor (dSph; Hill et al. 2019). Black: Reticulum II (UFD; Ji
et al. 2016). Dark-blue and dark-red diamonds are stars from Sausage/Enceladus
and Sequoia, respectively, observed by Aguado et al. (2021b). Gray dots in all
panels are Milky Way stars, also from SAGA database. We complement these with
data from the $R$-Process Alliance (Gudin et al., 2021).

The [Eu/Fe] as a function of metallicity is displayed in the bottom left panel
of Figure 4. Every analyzed member of HStr have $\rm[Eu/Fe]>+0.3$, with a
median value of $+0.5$ and median absolute deviation of $\pm 0.1$. This is,
indeed, consistent with the other dSph galaxies in SAGA database within the
same metallicity range. However, field metal-poor stars are known for having
$\langle\rm[Eu/Fe]\rangle\sim+0.4$ (Shetrone, 1996; Fulbright, 2000; Sneden et
al., 2008), which is unsurprising given the understanding that the Galactic
halo was formed from the accretion of many small dwarf galaxies. Therefore,
the elevated Eu abundance alone is not necessarily indicative of the same
progenitor, unlike the peculiar $\rm[\alpha/Fe]$–[Fe/H] tendency discussed in
Section 3.2. The most metal-poor star in the sample, despite showing higher
levels of Eu enrichment ($\rm[Eu/Fe]=+1.1$), has [Eu/H] similar to other low-
metallicity ($\rm[Fe/H]<-1.2$) stars of HStr. Nevertheless, more observations
of very metal-poor ($\rm[Fe/H]<-2.0$) stars from this substructure and, of
course, detection of Eu will certainly help constrain the conditions that
existed in the early chemical-enrichment site(s) of its parent system.

We can investigate the predominance between the $s$\- and the $r$-processes in
the neutron-capture nucleosynthesis through [Ba/Eu]. Stars with
$\rm[Ba/Eu]>0.0$ were enriched mostly via the $s$-process, while the birth
environments of those with $\rm[Ba/Eu]<0.0$ were polluted primarily by the
$r$-process (Spite 1992; see Sneden et al. 2008 and Frebel 2018 for reviews).
In the bottom right panel of Figure 4, we present the [Ba/Eu]–[Fe/H] plane.
All stars of HStr with $\rm[Fe/H]<-1.2$ occupy the sub-solar region of this
parameter space. Therefore, these are moderately RPE
($+0.3<\rm[Eu/Fe]\leq+1.0$) metal-poor stars, attributed to the so-called
$r$-I regime (Beers & Christlieb, 2005). The only exception is the lowest-
metallicity one that belongs to the highly RPE category ($r$-II;
$\rm[Eu/Fe]>+1.0$).

Another important discovery is that stars of HStr follow a similar sequence in
[Ba/Eu] as a function of [Fe/H] to that of Sculptor (also perceptible from
Figure 4), increasing towards higher metallicities. The single star with
$\rm[Ba/Eu]>0.0$ is the most metal-rich in the sample, but is also consistent
with Sculptor for this [Fe/H].Accounting for possible systematic effects could
shift the [Ba/Eu] down by up to $\sim$0.4 dex by, for instance, considering
non-local thermodynamic equilibrium in abundance calculations (Mashonkina &
Christlieb, 2014). Despite that, the result that most stars from HStr
experienced strong chemical enrichment via the $r$-process would not be
altered. Furthermore, the [Ba/Eu] of HStr members would still be in good
agreement with stars from Sculptor. This finding corroborates the hypothesis
that this surviving satellite of the Milky Way experienced an evolution
similar to the progenitor of the stream. If Sculptor is a “textbook” dSph
galaxy (Hill et al., 2019), HStr might be the remnant of an ancient system of
similar kind. Hence, spectroscopic studies of its member stars provide a way
to refine our understanding about dSph galaxies in general. Most importantly,
it represents a marvelous opportunity to study the emergence of $r$-process
elements in these environments, especially at the lowest metallicities, but
with stars bright enough to have high-resolution spectra readily acquired from
ground-based facilities. In the context of the recent observations of
gravitational waves (Abbott et al., 2017a), alongside an electromagnetic
counterpart (Abbott et al., 2017b), of a neutron star merger (GW170817),
comparison to $r$-process nucleosynthesis frequencies and yields should help
elucidate whether or not this source is responsible, and to what degree, for
the production of heavy atomic nuclei.

Likewise, we can compare the Ba- and Eu-to-iron ratios in the stream with
those from other prominent kinematic/dynamical halo substructures, also
suggested to be of accreted origin. In the bottom row of Figure 4, we display
the [Eu/Fe] and [Ba/Eu] for metal-poor stars from Gaia-Sausage/Enceladus
(Belokurov et al., 2018; Helmi et al., 2018) and Sequoia (Myeong et al., 2019)
with neutron-capture abundances recently published by Aguado et al. (2021b).
These targets cover a narrow metallicity range, within
$-2.2<\rm[Fe/H]\lesssim-1.5$, but comparable to the interval for most HStr
stars with such elements available. Overall, these stars from both
Sausage/Enceladus and Sequoia are, apparently, more $r$-process rich (medians
of $+0.6$ and $-0.7$ for [Eu/Fe] and [Ba/Eu], respectively) than those of HStr
($+0.5$ and $-0.4$) for similar [Fe/H] values. However, given the small
numbers considered, it is difficult to achieve meaningful conclusions at this
time.

During the preparation of this manuscript, Gull et al. (2021) made available
Eu abundances for stars claimed to be associated with HStr. However, their
target-selection function was based on pre-Gaia data and they did not employ a
dynamical selection approach, relying solely on kinematics. After reevaluating
their stars, only 2 out of their 12 candidates would be classified as members
according to our criteria, both of which were already being considered in our
study since they were previously recognized by either Roederer et al. (2010,
CD–36 1052) or Limberg et al. (2021a, HE 0324–0122). As a consequence, the
authors were unable to identify an $\alpha$ “knee” for a similar metallicity
range. Also, many of their stars clearly deviate in both [Eu/Fe] and [Ba/Eu],
showing no cohesive sequence as functions of [Fe/H] and a large spread.

Other recent works have utilized orbital and/or phase-space criteria to select
members of halo substructures and/or stellar streams for high-resolution
spectroscopy (e.g., Monty et al. 2020). However, such efforts are still
incipient. This work sheds light on the possibility of taking advantage of
dynamical information to, for instance, accelerate the discovery of RPE stars
(see the discussion by Limberg et al. (2021b). Moreover, since these objects
likely share a common origin, this approach should be more useful than
randomly drawing from extensive lists of cool low-metallicity stars (Limberg
et al., 2021b).

##  4 Summary

In this Letter, we employed astrometric and spectroscopic data from Gaia EDR3
and GALAH+ DR3, respectively, to identify 8 new members of HStr, which allows
us to cover a considerably wider (by $\sim$0.5 dex) metallicity interval
($-2.5\lesssim\rm[Fe/H]<-1.0$) than previously reported by Aguado et al.
(2021a). We also reevaluated candidates from the literature to consolidate a
sample of 22 stars of this substructure. Thanks to the now-extended
metallicity range, our study clearly shows a declining trend in [$\alpha$/Fe]
(Mg and Ca) with increasing [Fe/H]. Considering exclusively the newly found
members from GALAH, this pattern is still noticeable. Consequently, our
qualitative conclusions are valid even in the presence of small biases between
abundances extracted from different sources. We were also able to confirm that
stars of HStr constitute an $\alpha$-element pattern similar to the Sculptor
dSph galaxy up to $\rm[Fe/H]\sim-1.0$. This apparent dSph origin is in good
agreement with $N$-body simulations (Koppelman et al., 2019) of this merging
event.

We confirm that, at low metallicities ($\rm[Fe/H]\lesssim-1.2$), the
progenitor system of HStr experienced enrichment in neutron-capture elements
predominantly via the $r$-process, as first conjectured by Roederer et al.
(2010). All analyzed stars in this metallicity regime are RPE ones, with
median values for [Eu/Fe] and [Ba/Eu] of $+0.5$ and $-0.4$, respectively. The
behavior of HStr in [Ba/Eu]–[Fe/H] is also coherent with stars from Sculptor.
In particular, the extended metallicity range suggests an increase in [Ba/Eu]
for higher [Fe/H], reinforcing the hypothesis that the stream originated from
the disruption of a dwarf galaxy of similar kind. Finally, stars from
Sausage/Enceladus and Sequoia are, apparently, more $r$-process rich than HStr
ones. In the upcoming years, stars from these accreted halo substructures will
serve as important laboratories to understand the early nucleosynthesis of
heavy elements in their long-vanished dwarf-galaxy progenitor systems.

We thank the anonymous referee for a careful review of our work and
constructive suggestions that helped improve the manuscript. The authors also
thank João A. Amarante, Sean Ryan, and Vinicius Placco for their important
suggestions to this work. G.L. acknowledges CAPES (PROEX; Proc.
88887.481172/2020-00) for the funding of his Ph.D. R.M.S. acknowledges CNPq
(Proc. 436696/2018-5 and 306667/2020-7). S.R. acknowledges support from FAPESP
(Proc. 2015/50374-0 and 2014/18100-4), CAPES, and CNPq. C.C. acknowledges
partial support from the ChETEC COST Action (CA16117), supported by COST
(European Cooperation in Science and Technology). S.O.S. acknowledges the
FAPESP Ph.D. fellowship 2018/22044-3. A.P-V. and S.O.S acknowledge the DGAPA-
PAPIIT grant IG100319. H.D.P. thanks FAPESP Proc. 2018/21250-9. F.O.B.
acknowledges CAPES (PROEX; Proc. 88887.604787/2021-00). This research has been
conducted despite the ongoing dismantling of the Brazilian scientific system.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium. Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has also made
use of the SIMBAD database and the cross-match service provided by CDS,
Strasbourg, France. This work made use of the Third Data Release of the GALAH
Survey (Buder et al., 2020), which is based on data acquired through the
Australian Astronomical Observatory (AAT). We acknowledge the traditional
owners of the land on which the AAT stands, the Gamilaraay people, and pay our
respects to elders past and present. This paper includes data that has been
provided by AAO Data Central (datacentral.aao.gov.au). This publication also
makes use of data products from the Two Micron All Sky Survey (Skrutskie et
al., 2006) and the Wide-field Infrared Survey Explorer (Wright et al., 2010).

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##  Appendix A Final list of members

Table LABEL:tab:members contains relevant information about our compilation of
stars of HStr from both GALAH and the literature, including uncertainties. The
universal identifier for these objects is their Gaia EDR3 IDs. In the second
column, we provide the classification of the stars either as confident (1) or
likely (2) members (Section 3.1). Positions ($\alpha,\delta$), proper motions
($\mu_{\alpha}\cos{\delta},\mu_{\delta}$), and parallaxes ($\varpi$; re-
calibrated following Lindegren et al. 2020a) are included as well. The adopted
RV values are also listed. Heliocentric distances ($d_{\rm SH}$) estimated
with StarHorse are in the ninth column. Lower and upper limits for these
quantities represent the 16th and 84th percentiles of their distributions. The
stellar atmospheric parameters and elemental abundances determined from the
analyses of the high-resolution spectra (Section 2.1) of the studied stars are
provided by the end of the table.

Notes.

† Newly-recognized members in this work.

Table 1: Consolidated list of Helmi stream members Star name | Class | $\alpha$ | $\delta$ | $\mu_{\alpha}\cos{\delta}$ | $\mu_{\delta}$ | ${\varpi}$ | RV | $d_{\rm SH}$ | $T_{\rm eff}$ | $\log g$ | [Fe/H] | [Mg/Fe] | [Ca/Fe] | [Ba/Fe] | [Eu/Fe]
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(Gaia EDR3) |  | (deg) | (deg) |  (mas yr-1) |  (mas yr-1) | (mas) |  (km s-1) | (kpc) | (K) | (cgs) |  |  |  |  |
2336548524982631808 | 2 | 0.5695 |  $-$24.8971 |  $-$3.635$\pm$0.013 |  $-$8.153$\pm$0.012 |  0.178$\pm$0.024 |  78.8$\pm$2.0 |  5.84${}^{+0.74}_{-0.54}$ |  5020$\pm$100 |  1.73$\pm$0.30 |  $-$1.81$\pm$0.19 | $\dots$ | $\dots$ |  $+$0.10$\pm$0.10 |  $+$0.52$\pm$0.20
2543922018619120000 | 1 | 6.5826 | 0.6263 |  7.200$\pm$0.045 |  $-$22.060$\pm$0.035 |  0.574$\pm$0.059 |  168.0$\pm$3.0 |  1.73${}^{+0.13}_{-0.17}$ |  6305$\pm$102 |  4.04$\pm$0.52 |  $-$2.02$\pm$0.10 |  $+$0.38$\pm$0.05 |  $+$0.36$\pm$0.12 |  $-$0.40$\pm$0.10 | $\dots$
2781151719714308352 | 1 | 12.4166 | 15.5549 |  7.534$\pm$0.055 |  $-$29.597$\pm$0.034 |  0.644$\pm$0.053 |  114.0$\pm$3.0 |  1.55${}^{+0.11}_{-0.13}$ |  6315$\pm$102 |  4.06$\pm$0.52 |  $-$1.94$\pm$0.11 |  $+$0.38$\pm$0.07 |  $+$0.36$\pm$0.09 |  $-$0.39$\pm$0.11 | $\dots$
2537418854016682368 |  1† | 15.0695 | 1.2378 |  1.247$\pm$0.038 |  $-$20.811$\pm$0.031 |  0.469$\pm$0.032 |  187.3$\pm$1.0 |  2.16${}^{+0.17}_{-0.21}$ |  5299$\pm$189 |  3.52$\pm$0.26 |  $-$1.14$\pm$0.22 |  $+$0.06$\pm$0.24 |  $+$0.02$\pm$0.14 |  $-$0.34$\pm$0.20 | $\dots$
2518385517465704960 | 1 | 31.7239 | 4.5957 |  $-$6.045$\pm$0.043 |  $-$37.618$\pm$0.032 |  0.828$\pm$0.053 |  181.8$\pm$5.0 |  1.20${}^{+0.06}_{-0.06}$ |  6143$\pm$105 |  4.63$\pm$0.51 |  $-$1.99$\pm$0.11 |  $+$0.31$\pm$0.08 |  $+$0.16$\pm$0.09 |  $-$0.29$\pm$0.10 | $\dots$
4696863298741726976 |  2† | 40.8047 |  $-$65.1743 |  $-$0.895$\pm$0.015 |  3.278$\pm$0.013 |  0.157$\pm$0.013 |  255.2$\pm$0.7 |  6.18${}^{+0.61}_{-0.74}$ |  4898$\pm$128 |  2.09$\pm$0.24 |  $-$1.14$\pm$0.13 |  $-$0.09$\pm$0.14 |  $+$0.02$\pm$0.11 |  $+$0.40$\pm$0.14 | $\dots$
2497639347957300096 | 2 | 41.3709 |  $-$1.0151 |  3.173$\pm$0.019 |  $-$9.668$\pm$0.015 |  0.287$\pm$0.027 |  205.6$\pm$0.1 |  3.84${}^{+0.12}_{-0.12}$ |  4168$\pm$15 |  0.77$\pm$0.10 |  $-$1.69$\pm$0.10 |  $+$0.14$\pm$0.02 |  $-$0.03$\pm$0.05 | $\dots$ | $\dots$
4696089036397251200 |  1† | 41.4447 |  $-$65.5653 |  $-$2.994$\pm$0.014 |  1.705$\pm$0.012 |  0.150$\pm$0.013 |  143.0$\pm$0.5 |  6.76${}^{+0.54}_{-0.74}$ |  4463$\pm$95 |  1.19$\pm$0.25 |  $-$2.04$\pm$0.08 |  $+$0.13$\pm$0.06 |  $+$0.24$\pm$0.07 |  $+$0.51$\pm$0.08 |  $+$0.67$\pm$0.09
5049085217270417152 | 1 | 41.9060 |  $-$36.1075 |  3.187$\pm$0.010 |  12.591$\pm$0.014 |  1.380$\pm$0.023 |  304.7$\pm$0.8 |  0.73${}^{+0.01}_{-0.01}$ |  6070$\pm$200 |  2.30$\pm$0.30 |  $-$1.65$\pm$0.10 |  $+$0.36$\pm$0.10 |  $+$0.33$\pm$0.10 |  $+$0.13$\pm$0.15 |  $+$0.48$\pm$0.10
3267948604442696448 | 1 | 51.7594 | 1.5422 |  $-$19.554$\pm$0.015 |  $-$41.520$\pm$0.014 |  0.968$\pm$0.025 |  186.2$\pm$2.0 |  1.04${}^{+0.02}_{-0.03}$ |  5170$\pm$100 |  2.50$\pm$0.30 |  $-$2.39$\pm$0.13 |  $+$0.29$\pm$0.15 |  $+$0.28$\pm$0.15 |  $+$0.50$\pm$0.20 |  $+$1.07$\pm$0.20
4668621380510613248 |  2† | 64.7304 |  $-$67.3146 |  $-$1.803$\pm$0.017 |  $-$0.399$\pm$0.015 |  0.136$\pm$0.016 |  148.1$\pm$0.4 |  6.19${}^{+0.54}_{-0.63}$ |  4738$\pm$91 |  1.79$\pm$0.26 |  $-$1.15$\pm$0.07 |  $-$0.11$\pm$0.05 |  $+$0.03$\pm$0.06 |  $+$0.58$\pm$0.06 |  $+$0.33$\pm$0.08
2959451922593403904 |  2† | 77.0546 |  $-$25.5669 |  $-$9.151$\pm$0.012 |  $-$13.330$\pm$0.014 |  0.315$\pm$0.017 |  100.4$\pm$0.6 |  3.36${}^{+0.25}_{-0.29}$ |  4859$\pm$111 |  2.00$\pm$0.23 |  $-$2.27$\pm$0.12 |  $+$0.23$\pm$0.10 |  $+$0.30$\pm$0.07 |  $-$0.07$\pm$0.08 | $\dots$
680735037764135808 | 1 | 122.1393 | 24.3130 |  $-$49.704$\pm$0.026 |  $-$37.350$\pm$0.019 |  1.130$\pm$0.037 |  $-$88.1$\pm$5.0 |  0.88${}^{+0.02}_{-0.03}$ |  6070$\pm$102 |  4.50$\pm$0.51 |  $-$2.38$\pm$0.10 |  $+$0.38$\pm$0.07 |  $+$0.34$\pm$0.14 |  $+$0.50$\pm$0.13 | $\dots$
814087862430876160 | 1 | 142.4193 | 41.0978 |  $-$39.395$\pm$0.027 |  $-$20.315$\pm$0.021 |  1.186$\pm$0.041 |  $-$222.0$\pm$3.0 |  0.88${}^{+0.02}_{-0.03}$ |  5973$\pm$105 |  4.64$\pm$0.51 |  $-$2.19$\pm$0.11 |  $+$0.34$\pm$0.06 |  $+$0.26$\pm$0.12 |  $+$0.40$\pm$0.14 | $\dots$
5399644708578669696 |  1† | 170.7452 |  $-$34.1081 |  $-$11.629$\pm$0.014 |  $-$5.05$\pm$0.011 |  0.183$\pm$0.016 |  70.8$\pm$0.4 |  5.23${}^{+0.44}_{-0.45}$ |  4523$\pm$87 |  1.45$\pm$0.31 |  $-$1.24$\pm$0.07 |  $+$0.18$\pm$0.05 |  $+$0.07$\pm$0.06 |  $+$0.11$\pm$0.07 |  $+$0.32$\pm$0.08
840616123070439552 | 1 | 177.6522 | 54.1241 |  $-$3.861$\pm$0.011 |  $-$1.818$\pm$0.012 |  0.306$\pm$0.022 |  $-$277.7$\pm$0.1 |  2.85${}^{+0.16}_{-0.27}$ |  4907$\pm$25 |  2.16$\pm$0.10 |  $-$1.14$\pm$0.10 |  $+$0.08$\pm$0.02 |  $+$0.16$\pm$0.03 | $\dots$ | $\dots$
6153268944831195648 |  2† | 190.2765 |  $-$38.6669 |  $-$9.576$\pm$0.015 |  $-$11.931$\pm$0.012 |  0.256$\pm$0.018 |  $-$43.7$\pm$0.6 |  4.13${}^{+0.36}_{-0.43}$ |  4897$\pm$131 |  2.26$\pm$0.10 |  $-$1.38$\pm$0.12 |  $+$0.14$\pm$0.11 |  $+$0.15$\pm$0.10 |  $+$0.04$\pm$0.12 | $\dots$
3630429997250686592 |  2† | 202.5277 |  $-$7.6950 |  $-$6.231$\pm$0.025 |  $-$6.512$\pm$0.014 |  0.194$\pm$0.023 |  $-$228.9$\pm$0.9 |  5.43${}^{+0.76}_{-0.59}$ |  4962$\pm$193 |  2.43$\pm$0.33 |  $-$1.26$\pm$0.20 |  $+$0.29$\pm$0.23 |  $+$0.28$\pm$0.14 |  $+$0.05$\pm$0.20 | $\dots$
3742101345970116224 | 1 | 205.8611 | 15.5752 |  $-$49.997$\pm$0.017 |  $-$14.872$\pm$0.011 |  1.898$\pm$0.029 |  $-$285.9$\pm$0.8 |  0.54${}^{+0.01}_{-0.00}$ |  5290$\pm$200 |  1.10$\pm$0.30 |  $-$2.15$\pm$0.10 |  $+$0.34$\pm$0.10 |  $+$0.33$\pm$0.10 |  $-$0.02$\pm$0.24 |  $+$0.34$\pm$0.10
1275876252107941888 | 1 | 226.7244 | 30.0102 |  13.941$\pm$0.009 |  $-$7.761$\pm$0.013 |  0.683$\pm$0.027 |  $-$280.3$\pm$0.8 |  1.47${}^{+0.06}_{-0.06}$ |  4330$\pm$200 |  0.60$\pm$0.30 |  $-$1.45$\pm$0.10 |  $+$0.31$\pm$0.16 |  $+$0.29$\pm$0.10 |  $+$0.06$\pm$0.25 |  $+$0.65$\pm$0.15
1770226575557939840 | 1 | 322.8380 | 13.1278 |  4.082$\pm$0.012 |  $-$16.745$\pm$0.010 |  0.258$\pm$0.022 |  71.4$\pm$0.1 |  4.06${}^{+0.34}_{-0.28}$ |  4975$\pm$16 |  1.82$\pm$0.10 |  $-$1.85$\pm$0.10 |  $+$0.18$\pm$0.03 |  $+$0.01$\pm$0.07 | $\dots$ | $\dots$
6536758912468029184 | 1 | 351.2973 |  $-$39.9912 |  4.400$\pm$0.023 |  $-$7.736$\pm$0.028 |  0.561$\pm$0.022 |  286.8$\pm$0.8 |  1.80${}^{+0.10}_{-0.12}$ |  6650$\pm$200 |  3.70$\pm$0.30 |  $-$2.55$\pm$0.10 |  $+$0.32$\pm$0.10 |  $+$0.41$\pm$0.12 |  $-$0.31$\pm$0.30 | $\dots$
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |


:
LEAVE UNSET 2021 LEAVE UNSET LEAVE UNSET Machine Learning for Health (ML4H)
2021

# Machine Learning for Mechanical Ventilation Control

Daniel Suo Google LLCPrinceton University Naman Agarwal11footnotemark: 1
Wenhan Xia11footnotemark: 122footnotemark: 2 Xinyi Chen11footnotemark:
122footnotemark: 2 Udaya Ghai11footnotemark: 122footnotemark: 2 Alexander
Yu11footnotemark: 1 Paula Gradu11footnotemark: 1 Karan Singh11footnotemark:
122footnotemark: 2 Cyril Zhang11footnotemark: 122footnotemark: 2 Edgar
Minasyan11footnotemark: 122footnotemark: 2 Julienne LaChance22footnotemark: 2
Tom Zajdel22footnotemark: 2 Manuel Schottdorf22footnotemark: 2 Daniel
Cohen22footnotemark: 2 Elad Hazan11footnotemark: 122footnotemark: 2

###### Abstract

Mechanical ventilation is one of the most widely used therapies in the ICU.
However, despite broad application from anaesthesia to COVID-related life
support, many injurious challenges remain.

We frame these as a control problem: ventilators must let air in and out of
the patient’s lungs according to a prescribed trajectory of airway pressure.
Industry-standard controllers, based on the PID method, are neither optimal
nor robust.

Our data-driven approach learns to control an invasive ventilator by training
on a simulator itself trained on data collected from the ventilator. This
method outperforms popular reinforcement learning algorithms and even controls
the physical ventilator more accurately and robustly than PID.

These results underscore how effective data-driven methodologies can be for
invasive ventilation and suggest that more general forms of ventilation (e.g.,
non-invasive, adaptive) may also be amenable.

##  1 Introduction

Mechanical ventilation is a widely used treatment with applications spanning
anaesthesia (Coppola et al., 2014), neonatal intensive care (van Kaam et al.,
2019), and life support during the current COVID-19 pandemic (Meng et al.,
2020; Wunsch, 2020; Möhlenkamp and Thiele, 2020). Despite its use in ICUs for
decades, mechanical ventilation can still lead to ventilator-induced lung
injury (VILI) for patients (Cruz et al., 2018).

We focus on pressure-controlled invasive ventilation (PCV) (Rittayamai et al.,
2015). In this setting, an algorithm controls two valves (see Figure 1(a))
that let air in and out of a fully-sedated patient’s lung according to a
target waveform of lung pressure (see Figure 1(b)). We consider the control
task only on ISO-standard (ISO, 2018) artificial lungs.

(a) A simplified respiratory circuit showing the airflow through the
inspiratory pathway, into and out of the lung, and out the expiratory pathway.
We shade the components that our algorithms can control in green.

(b) An example run of three breaths where PID (dark gray line) controls lung
pressure (blue line) according to a prescribed target waveform (orange line).

Despite its importance, ventilator control has remained largely unchanged for
years, relying on PID (Bennett, 1993) controllers and similar variants to
track patient state according to a prescribed target waveform. A ventilator
controller must adapt quickly and reliably across the spectrum of clinical
conditions, which are only indirectly observable given a single measurement of
pressure.

A model that is highly expressive may learn the dynamics of the underlying
systems more precisely and thus adapt faster to the patient’s condition.
However, such models usually require a large amount of data to train, which
can take prohibitively long to collect by purely running the ventilator. We
opt instead to learn a simulator to generate artificial data, though learning
such a simulator for a partially observed non-linear system is itself a
difficult problem. As part of our contributions, we present better-performing,
more robust results and present resources for future researchers.

##  2 Scientific background

### Control of dynamical systems

We begin with some formalisms of the control problem. A partially-observable
discrete-time dynamical system is given by the following equation:
$x_{t+1}=f(x_{t},u_{t}),o_{t+1}=g(x_{t+1})$, where $x_{t}$ is the underlying
state of the dynamical system, $o_{t}$ is the observation of the state
available to the controller, $u_{t}$ is the control input and $f,g$ are the
transition function and observation functions respectively. Given a dynamical
system, the control problem is to minimize the sum of cost functions over a
long-term horizon:
$\min_{u_{1:T}}\sum_{t=1}^{T}c_{t}(x_{t},u_{t})\quad\text{s.t.}\;\;x_{t+1}=f_{t}(x_{t},u_{t})$.

A ubiquitous technique for the control of dynamical systems is the use of
linear error-feedback controllers, i.e. policies that choose a control based
on a linear function of the current and past errors vs. a target state. That
is, $u_{t+1}=\sum_{i=0}^{k}\alpha_{i}\varepsilon_{t-i}$, where
$\varepsilon_{t}=x_{t}-{x}^{\star}_{t}$ (or
$\varepsilon_{t}=o_{t}-{o}^{\star}_{t}$ if the system is partially-observable)
is the deviation from the target state at time $t$, and $k$ represents the
history length of the controller. PID applies a linear control with
_proportional_ , _integral_ , and _differential_ coefficients,
$u_{t}=\alpha\varepsilon_{t}+\beta\sum_{i=0}^{k}\varepsilon_{t-i}+\gamma(\varepsilon_{t}-\varepsilon_{t-1})$.
This special class of linear error-feedback controllers, motivated by physical
laws, is a simple, efficient and widely used technique (Åström and Hägglund,
1995). It is currently the industry standard for (open-source) ventilator
control.

### The physics of ventilation

The goal of ventilator control is to regulate the pressure sensor measurements
to follow a target waveform $p_{t}^{\star}$ via controlling the air-flow into
the system which forms the control input $u_{t}$. As a dynamical system, we
can denote the underlying state of the ventilator-patient system as $x_{t}$
evolving as $x_{t+1}=f(x_{t},u_{t}),$ for an unknown $f$ and the pressure
sensor measurement $p_{t}$ is the observation available to us. The cost
function can be defined to be a measure of the deviation from the target; e.g.
the absolute deviation $c_{t}(p_{t},u_{t})=|p_{t}-p_{t}^{\star}|$. The
objective is to design a controller that minimizes the total cost over $T$
time steps.

### Challenges and benefits of a model-based approach

Physics-based dynamics models ((Nadeem, 2021)) rely on idealized physical
behaviors, lagged and partial observations, and underspecification of how the
true system may vary. As a result, we adopt a learned model-based approach due
to its sample-efficiency and reusability. A reliable simulator enables much
cheaper and faster data collection for training a controller, and allows us to
incorporate multitask objectives and domain randomization (e.g. different
waveforms, or even different patients). An additional goal is to make the
simulator _differentiable_ , enabling direct gradient-based policy
optimization through the system’s dynamics (rather than stochastic estimates
thereof).

We show that in this partially-observed (but single-input single-output)
system, we can query a reasonable amount of training data in real time from
the test lung, and use it offline to learn a differentiable simulator of its
dynamics (_“real2sim”_). Then, we complete the pipeline by leveraging
interactive access to this simulator to train a controller (_“sim2real”_). We
demonstrate that this pipeline is sufficiently robust that the learned
controllers can outperform PID controllers tuned directly on the test lung.

##  3 Experimental Setup

### Test lung and ventilator

To develop simulators and control algorithms, we run mechanical ventilation
tasks on a physical test lung (IngMar, 2020) using the open-source ventilator
designed by Princeton University’s People’s Ventilator Project (PVP) (LaChance
et al., 2020).

### Abstraction of the simulation task

We treat the mechanical ventilation task as episodic by separating each
inspiratory phase from the breath timeseries and treating those as individual
episodes. This approach reflects both physical and medical realities.
Mechanically ventilated breaths are by their nature highly regular and feature
long expiratory phases that end with the ventilator-lung system close to its
initial state, thereby justifying the episodic nature. Further, the
inspiratory phase is indeed the most relevant to clinical treatment and the
harder regime to control with prevalent problems of under- or over-shooting
the target pressure and ringing.

(a) The ventilator cluster we constructed to run our experiments, featuring 10
ventilators, 4 air compressors, and 2 control servers. Each ventilator is re-
calibrated after each experimental run for consistency across ventilators and
over time.

(b) As an example, we compare our method (learned controller on learned
simulator) to the best P-only, I-only, and PID controllers relative to a
target waveform (dotted line). Whereas our controller rises quickly and stays
very near the target waveform, the other controllers take significantly longer
to rise, overshoot, and, in the case of P-only and PID, ring the entire
inspiratory phase.

##  4 Learning a data-driven simulator

The evaluation for any simulator can only be performed using a black-box
metric, since we do not have explicit access to the system dynamics, and
existing physics models are poor approximations to the empirical behavior.

Let $\mathcal{D}$ be a distribution over sequences of controls denoted
$\mathbf{u}=\\{u_{1},u_{2},...,u_{T}\\}$.

We define the open-loop distance w.r.t. horizon $T$ and control sequence
distribution $\mathcal{D}$ as
$\|f_{1}-f_{2}\|_{ol}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}_{\mathbf{u}\sim\mathcal{D}}\left[\sum_{t=1}^{T}\|f_{1}(x_{t,1},u_{t})-f_{2}(x_{t,2},u_{t})\|\right]$.

### Data collection

balance physical safety against the need to explore the space of controls and
resulting pressures by choosing a known good PID for a given lung setting and
introducing random exploratory perturbations according to the following two
policies:

  1. 1.

Boundary exploration: To the very beginning of the inhalation, add an
additional control sampled uniformly from $(c^{a}_{\min},c^{a}_{\max})$ and
decrease this additive control linearly to zero over a time frame sampled
randomly from $(t^{a}_{\min},t^{a}_{\max})$. This policy capitalizes on the
fact that the beginning of a breath is safer and exploration here is more
valuable since it is the driving phase of inspiration.

  2. 2.

Triangular exploration: sample a maximal additional control from a range
$(c^{b}_{\min},c^{b}_{\max})$ and an interval $(t^{b}_{\min},t^{b}_{\max})$,
within the inhalation. Start from $0$ additional control at time
$t^{b}_{\min}$, increase the additional control linearly until
$(t^{b}_{\min}+t^{b}_{\max}))/2$, and then decrease to $0$ linearly until
$t^{b}_{\min}$. This policy helps explore the intrinsic delay in the system
between a control and a resulting change in system state.

For each breath during data collection, we choose policy $(a)$ with
probability $p_{a}$ and policy $(b)$ with probability $(1-p_{a})$. The ranges
in $(a)$ and $(b)$ are lung-specific.

### Training Task(s).

The simulator aims to learn the unknown dynamics of the inhalation phase. We
approximate the state of the system (which is not observable to us) by the
sequence of the past pressures and controls upto a history length of $H_{c}$
and $H_{p}$ respectively. The task of the simulator can now be distilled down
to that of predicting the next pressure $p_{t+1}$, based on the past $H_{c}$
controls $u_{t},\ldots,u_{t-H_{c}}$ and $H_{p}$ pressures
$p_{t},\ldots,p_{t-H_{p}}$. We define the training task by constructing a
regression data set whose inputs come from contiguous overlapping sections of
$H_{p},H_{c}$ within the collected trajectories and the task is to predict the
following pressure.

##  5 Learning controllers from learned physics

We focus on the following two tasks: (1) Performance: improve performance for
tracking desired waveform in ISO-specified benchmarks. Specifically, we
minimize the combined $L_{1}$ deviation from the target inhalation behavior
across all target pressures on the simulator corresponding to a single lung
setting of interest. We also compare performance against several well-studied
RL algorithms (Figure 4(a)). (2) Robustness: improve performance using a
single trained controller. Specifically, we minimize the combined $L_{1}$
deviation from the target inhalation behavior across all target pressures and
across the simulators corresponding to several lung settings of interest.

### Controller architecture

Our controller is comprised of a PID baseline upon which we learn a deep
network correction controlled with a regularization parameter $\lambda$. This
residual setup can be seen as a regularization against the gap between the
simulator and the real dynamics. In particular this prevents the controller
training from over-fitting on the simulator. We found this approach to be
significantly better than the directly using the best (and perhaps over-
fitted) controller on the simulator.

### Experiments

To make comparisons between PID and our controllers on the physical system, we
compute a score for each controller on a given test lung setting (e.g.,
$R=5,C=50$) by averaging the $L_{1}$ deviation from a target pressure waveform
for all inspiratory phases, and then averaging these average $L_{1}$ errors
over six waveforms specified in ISO (2018). We choose $L_{1}$ as an error
metric so as not to over-penalize breaths that fall short of their target
pressures and to avoid engineering a new metric. We determine the best
performing PID controller for a given setting by running exhaustive grid
searches over $P,I,D$ coefficients for each lung setting (details for both our
score and the grid searches can be found in the Appendix).

(a) We show that for each lung setting, the controller we trained on the
simulator for that setting outperforms the best-performing PID controller
found on the physical test lung.

(b) The controller we trained on all six simulators outperforms the best PID
found over the the same six settings on the physical test lung. Of note, our
wins are proportionally greater when trained on all six settings whereas
individual lung settings are more achievable by PID alone.

(a) Performance comparison of our controller with PPO/DQN. The score is
calculated by average per-step L1 distance between target and achieved
pressure.

(b) Convergence behavior demonstration. Our methods converge using orders of
magnitudes fewer samples, even when considering training data for our
simulator.

##  6 Conclusions and future work

We have presented a machine learning approach to ventilator control,
demonstrating the potential of end-to-end learned controllers by obtaining
improvements over industry-standard baselines.

There remain a number of areas to explore, mostly motivated by medical need.
The lung settings we examined are by no means representative of all lung
characteristics (e.g., neonatal, child, non-sedated) and lung characteristics
are not static over time; a patient may improve or worsen, or begin coughing.
Ventilator costs also drive further research. As an example, inexpensive
valves have less consistent behavior and longer reaction times, which
exacerbate bad PID behavior (e.g., overshooting, ringing), yet are crucial to
bringing down costs and expanding access. Learned controllers that adapt to
these deficiencies may obviate the need for such trade-offs.

## References

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