# Machine Learning Refinements to Metallicity-Dependent Isotopic Abundances

Haoxuan Sun
Department of Physics & Astronomy
Macalester College
Saint Paul, MN 55105
<EMAIL_ADDRESS>

###### Abstract

The project aims to use machine learning algorithms to fit the free parameters
of an isotopic scaling model to elemental observations. The processes
considered are massive star nucleosynthesis, Type Ia SNe, the s-process, the
r-process, and p-isotope production. The analysis on the successful fits seeks
to minimize the reduced chi squared between the model and the data. Based upon
the successful refinement of the isotopic parameterized scaling model, a table
providing the 287 stable isotopic abundances as a function of metallicity,
separated into astrophysical processes, is useful for identifying the chemical
history of them. The table provides a complete averaged chemical history for
the Galaxy, subject to the underlying model constraints.

##  1 Introduction

The yields of stellar simulations are dependent on the star’s initial isotopic
composition. During hydrostatic burning phases, the initial composition is
crucial for neutron capture reactions on initial metals, which affects the
abundance of odd-z nuclei([10]). The detailed stellar abundances influence the
opacity of the star, which in turn influences the star’s structure as well as
the loss of mass and angular momentum, thereby altering the late stellar
evolution. Understanding $\gamma$-process abundances, which use s- and
r-process isotopes as seeds, requires information on the initial abundances of
heavy isotopes ([11] & [12]). The objective of galactic chemical evolution
(GCE) is to comprehend how the abundances of the elements and their isotopes
changed from the big bang to the present day, and it can be used to obtain the
isotopic abundances at any metallicity for use as inputs for stellar
simulations. Traditional GCE models typically require nucleosynthesis yields
from stellar simulations as inputs, but the problem with this approach is that
in order to provide self-consistent nucleosynthesis yields, the stellar
simulations require a full initial set of isotopic abundances
([13];[14];[15]). The construction of an astrophysical model of all stable
isotopes, based on physical principles for production sites and mechanisms, is
a complementary strategy to conventional GCE methods. The completed model then
provides the Galaxy’s average isotopic history, subject to the employed
approximations ([19]). Compared to full GCE calculations, this method is
comparatively simple and approximative, but it improves upon the standard of
scaling isotopic solar abundances by a constant factor.

The study of chemical abundance in physics is an important aspect of
understanding the composition of matter in the universe. Chemical abundance
refers to the relative amounts of different chemical elements present in a
given sample. This information is crucial for understanding the formation and
evolution of stars, galaxies, and the universe as a whole. All stellar
evolution models for nucleosynthesis necessitate a beginning point for
isotopic abundance ([16]). Except for the Sun, our understanding of the
isotopic abundances of stars is generally incomplete. We offer parameters for
models of a complete average isotopic decomposition as a function of
metallicity, which are fitted to observational data, as opposed to the
conventional forward galactic chemical evolution modeling that incorporates
star yields beginning with big bang nucleosynthesis. This method of machine
learning uses the grid search algorithm to identify the parameter values with
the lowest reduced chi-square value, resulting in a better fit of the model to
these real data. Our fittings of parameter finds the light elements with great
accuracy, within a $8.7\%$ uncertainty compared with the original results but
as well as a better performance in the heavier element corresponding to strong
$s-$process and main $s-$process.

##  2 Overview

This project primarily focuses on two main components aimed at improving the
astrophysical model completed by ([2]). The research employs the application
of machine learning algorithms to refine the model, using the grid search and
random forest algorithms, and analyzing the impact of these adjustments on
isotope abundances.

The background context of the problem lies in the need for an improved
astrophysical model that can accurately predict isotope abundances in stars.
This is crucial for understanding stellar nucleosynthesis processes and the
evolution of elements in the universe. The general solution provided by this
project involves the integration of machine learning techniques to optimize
parameters within the model, for the specific purpose of improving stellar
simulation inputs.

The grid search algorithm primarily optimizes most reaction parameters, with
the exception of the strong s-process. Conversely, the random forest algorithm
focuses on adjusting the scaling and shift factors of the s-process. We
evaluate our current progress by comparing it both numerically and visually to
previous work, utilizing Python-based code.

After successfully refining the model, we compute the abundances of all 287
isotopes for various reactions, considering different initial stellar
metallicities. The reactions are color-coded and stacked together to generate
a comprehensive abundance graph, which is further normalized using solar
isotope abundances.

Subsequently, we select specific isotopes for in-depth analysis and discuss
the overall trends observed. This holistic approach allows us to gain a deeper
understanding of the underlying astrophysical processes and evaluate the
effectiveness of the machine learning algorithms in improving the model.

##  3 Astrophysical Processes

The origin of the elements in the Universe is a fundamental question in
astrophysics. Understanding the processes responsible for the production of
isotopes is crucial for gaining insights into the galactic chemical evolution
models used in nucleosynthesis studies. Thus, various astrophysical processes
that contribute to the formation of isotopes, including hydrostatic burning,
supernovae, the $r$-process, and the $s$-process will be discussed.

Primary processes are mainly producing metals directly from H and He. Nuclear
fusion is the primary process by which atomic nuclei combine to form more
massive nuclei, releasing energy in the process ([1]). This process powers the
energy production in stars, including our sun, and is responsible for the
production of heavier elements [3]. While Type Ia supernova event is a
powerful and luminous explosion of a massive star at the end of its lifecycle.
It is one of the most energetic events in the universe and results in the
dispersion of elements and the formation of neutron stars or black holes
([4]).

Neutrino-induced nucleosynthesis, also known as neutrino nucleosynthesis, is a
process in which neutrinos interact with atomic nuclei, resulting in the
production of new elements. This process is particularly important in core-
collapse supernovae, where a large number of neutrinos are produced, and their
interactions contribute to the synthesis of some isotopes ([5]).

The r-process is a nucleosynthesis process that occurs in explosive
astrophysical events, such as core-collapse supernovae or neutron star
mergers. It involves the rapid capture of neutrons by seed nuclei, leading to
the production of heavy, neutron-rich isotopes, including many of the elements
heavier than iron. The R-process is responsible for the synthesis of about
half of the elements heavier than iron ([6]).

The $\nu p$-process, or neutrino-p process, is a nucleosynthesis process that
occurs in the neutrino-driven winds from protoneutron stars during a core-
collapse supernova. It is driven by the interaction of neutrinos with protons,
leading to the production of elements heavier than iron via rapid proton
captures on existing seed nuclei. The $\nu p$-process is thought to be
responsible for the synthesis of some neutron-deficient isotopes of heavy
elements ([7]).

Secondary processes are different from primary processes since they make
metals from pre-existing metals made earlier in a primary process and could be
divided into three major types. The weak s-process occurs in massive stars
(with masses greater than 10 solar masses) during core helium and shell carbon
burning. It synthesizes elements with atomic numbers between iron and
strontium (Z = 38), contributing to the production of elements in the lower
mass range of the s-process. The main s-process occurs in low-to-intermediate-
mass stars (with masses between 1 and 8 solar masses) during the asymptotic
giant branch (AGB) phase. It is responsible for producing elements with atomic
numbers between strontium (Z = 38) and bismuth (Z = 83). The main s-process is
responsible for synthesizing most of the s-process elements in the universe.
The strong s-process is a hypothetical nucleosynthesis process that would
occur in an environment with an extremely high neutron density. It is not
currently associated with any known stellar environment ([8]).

##  4 Model Description

In this project, we mainly utilize the existing model from [2] and briefly
discuss the models that we fit in the machine learning algorithm. Generally,
isotopes can be divided into light and heavy isotopes, with light isotopes
having masses lower than $56$, which is the threshold of ${{}^{56}Fe}$, and
heavy isotopes having masses greater than this threshold. Light isotopes are
primarily formed through BBN, Type Ia supernova, or massive star reactions. We
make assumptions on the Type Ia supernova model using the equation
$f\dot{X}_{56}^{\odot}/X_{56}^{Ia}$, where $X_{56}^{Ia}$ is the solar
abundance of ${{}^{56}Fe}$, and $X_{56}^{Ia}$ is a scaling factor. $f$ is a
free parameter in the model, determined by fitting the elemental scalings
against available data. We assume from [2] that massive star yields were
scaled to the remaining contribution to solar, resulting in the scaling model
for Type Ia supernova as $(1-f)\cdot X_{56}^{\odot}/X_{56}^{Ia}$. This model,
however, is only for the specific isotope ${{}^{56}Fe}$; we further assume
that for all other isotopes between $12\leq A\leq 56$, additional scaling
preserves the ratio of each isotopic contribution for massive star yields.

$\displaystyle X_{i,f}^{massive}$
$\displaystyle=(\frac{X_{i}^{\odot}}{X_{i,0}^{massive}+X_{i,0}^{Ia}})X_{i,0}^{massive}$
(1) $\displaystyle X_{i,f}^{Ia}$
$\displaystyle=(\frac{X_{i}^{\odot}}{X_{i,0}^{massive}+X_{i,0}^{Ia}})X_{i,0}^{Ia}$

In this equation, $X_{i,0}$ and $X_{i,\mathrm{f}}$ represent the original and
fitted abundances of isotope $i$ for massive or Type Ia contributions, denoted
by superscripts, and the solar abundance of isotope $i$ is given by
$X_{i}^{\odot}$.

After considering all model assumptions, we combine them into a more general
scaling model. To create a linear relationship for the scaling model, we apply
logarithmic methods to their respective abundances given by the normalized
Population III simulation. For each isotope, the abundances are scaled
according to:

$\displaystyle\log\left(X_{i}^{*}(\xi)\right)=m_{i}\left(\log(\xi)-\log\left(\xi_{\text{low
}}\right)\right)+\log\left(X_{i}^{\text{sim }}\right)$ (2)

where $X_{i}^{*}(\xi)$ is the massive abundance of isotope $i$ as a function
of the model parameter, and $\log\left(\xi_{\text{low }}\right)=-2.5$. The
slope is defined as,
$m_{i}\equiv\left(\log\left(X_{i,f}^{\text{massive
}}\right)-\log\left(X_{i}^{\text{sim
}}\right)\right)/\left(\log\left(\xi_{\odot}\right)-\log\left(\xi_{\text{low
}}\right)\right)$, where $X_{i,f}^{\text{massive }}$ is the massive star
contribution to the solar abundance, and obviously
$\log\left(\xi_{\odot}\right)=0$. Though, we finish the modeling of massive
star reaction for light isotope, type Ia supernova reaction occured in
creating light isotope is not negligible. One specific form which satisfies
all three constraints discussed is given, e.g., by a scaled and shifted
hyperbolic tangent base function,

$\displaystyle X_{i}^{\mathrm{Ia}}(\xi)=$ $\displaystyle
X_{i,\odot}^{\mathrm{Ia}}\cdot\xi\cdot[\tanh(a\cdot\xi-b)+\tanh(b)]/[\tanh(a-b)$
(3) $\displaystyle+\tanh(b)]$

Where $X_{i}^{Ia}(\xi)$ is the supernova abundance of isotope $i$ as a
function of the model parameter. The specific value for Type Ia onset is
determined by fitting the free parameters, $a$ and $b$, against available
data. Note the hyperbolic tangent function is tempered with a linear factor of
$\xi$ to ensure the behavior is linear near solar.After discussing the
assumption made for the light isotopes, heavy isotopes involve more
complicated models that need to be apply. Thus we are going to conclude the
existence for Neutron Capture and p-isotopes. This reaction will take place in
most of the heavier isotopes. And this Neutron Capture and p-isotopes include
the main $s$-process $(ls,hs$, and "strong" component), weak $s$-process,
$r$-process, $vp$-process, and $\gamma$-process contributions are
parameterized as power laws. In the equation, $X_{i}^{\odot}$ is the solar
abundance of the isotope i, where $X_{i}^{*}(\xi)$ with different subscripts
of $strong,ls,hs,ws,r,\mu p,\gamma$ is the respective processes’ abundance of
isotope i as a function of the model parameter.

$\begin{gathered}X_{i}^{\mathrm{strong}}(\xi)=c\left[1-\frac{\tanh(d\cdot\xi+g)}{\tanh(d+g)}\right]+X_{i,\odot}^{\mathrm{strong}}\\\
X_{i}^{\mathrm{ls}}(\xi)=X_{i,\odot}^{\mathrm{ls}}\cdot\xi^{l}\\\
X_{i}^{\mathrm{hs}}(\xi)=X_{i,\odot}^{\mathrm{hs}}\cdot\xi^{h}\\\
X_{i}^{\mathrm{ws}}(\xi)=X_{i,\odot}^{\mathrm{ws}}\cdot\xi^{w}\\\
X_{i}^{r}(\xi)=X_{i,\odot}^{r}\cdot\xi^{p}\\\
X_{i}^{vp}(\xi)=X_{i,\odot}^{vp}\cdot\xi^{p}\\\
X_{i}^{\gamma}(\xi)=X_{i,\odot}^{\gamma}\cdot\xi^{\frac{h+p}{2}+1}\end{gathered}$
(4)

While there are few isotopes undergoes the reaction Hydrogen Burning,
Classical Novae, v-process, and Galactic Cosmic Ray Spallation, but will be
scaled in an subtraction of mass fraction of the helium isotopes and mass
fraction of the helium isotopes from the solar value of total metallicity. The
remaining isotope of hydrogen, ${}^{1}\mathrm{H}$, was scaled according to,

$\displaystyle
X_{i}^{\mathrm{H}}(\xi)=X_{i,\odot}^{\mathrm{H}}\cdot\left[1.0-\xi\cdot
Z_{\odot}-Y(\xi)-D(\xi)\right]$ (5)

where $Y(\xi)$ is the mass fraction of the helium isotopes, $D(\xi)$ s the
mass fraction of deuterium, and $Z_{\odot}$ is the solar value of otal
metallicity.

Since all the scaling models are introduced for all these 287 different
isotopes, it is confident to say that there are complicity when with in the
same element, for example, Iron, could involve different reactions since $56$
isomass is the threshold for between heavy and light isotopes. For ${Fe}$, the
reaction will be

$\displaystyle\begin{aligned}
{}^{56}\mathrm{Fe}(\xi)=&{}^{56}\mathrm{Fe}_{\odot}^{\mathrm{Ia}}\cdot\xi\cdot[\tanh(a\cdot\xi-b)+\tanh(b)]/[\tanh(a-b)\\\
&+\tanh(b)]+\mathrm{Fe}_{\odot}^{*}\cdot
10^{m_{\mathrm{Fe}56}\cdot\left(\log(\xi)-\log\left(\xi_{\mathrm{low}}\right)\right)+\log\left(X_{\mathrm{Fe}56}^{\mathrm{sim}}\right)}\end{aligned}$
(6)
$\displaystyle{}^{57}\mathrm{Fe}(\xi)={}^{57}\mathrm{Fe}_{\odot}^{\mathrm{ws}}\cdot\xi^{w}+{}^{57}\mathrm{Fe}_{\odot}^{*}\cdot
10^{m_{\mathrm{Fe}57}\cdot\left(\log(\xi)-\log\left(\xi_{\mathrm{low}}\right)\right)+\log\left(X_{\mathrm{Fe}57}^{\mathrm{sim}}\right)}$

Thus a machine learning algorithm is thus to be used under these situation
since we are dealing with the models that could change gradually as the
parameters are changing which in this case are interconnected.

##  5 Methodology of Work

In the published study under discussion, the authors developed a novel
algorithm for constructing elemental abundance ratios $[X/\text{Fe}]$ using
isotopic scaling functions and fitting them to observational data. By
systematically analyzing isotopic contributions to the solar abundances of
various elements, they derived elemental scaling relations that account for
multiple nucleosynthetic processes. Their methodology not only provided a
robust way to evaluate elemental ratios $[X/\text{Fe}]$ and
$[\text{Fe}/\text{H}]$ as functions of the continuous technical parameter
$\xi$, but also enabled efficient comparison with observational data through
$\chi^{2}$ analysis.

Compared to the published work, the previous model has an over-reliance on Mg
data to fit free parameters. In this project, I employ machine learning
techniques to simultaneously fit all the elemental abundance data along with
all the free parameters, thereby addressing the limitations of the previous
model and offering a more comprehensive solution.

Similarly, the treatment of the strong $s$-process in the previous work
involved crudely fitting the strong $s$-process parameter by hand using Pb
data. In contrast, we consider more data points for this typical astrophysical
process and develop a better solution to find parameters. This project aims to
improve the understanding of nucleosynthesis and the chemical evolution of the
universe, making it a suitable topic for an honors thesis in astrophysics.

the observational data analysis was conducted through a rigorous and
systematic process. The data points, which represent various measurements of
chemical abundances of elements in stars, were analyzed in the context of
[X/Fe] versus [Fe/H] space. This space is used to study the chemical evolution
of galaxies and stellar populations, where [X/Fe] denotes the abundance ratio
of element X to iron (Fe), while [Fe/H] signifies the abundance of iron
relative to hydrogen (H).

To account for uncertainties in the measurements, a Gaussian spread was
assigned to each observational data point, based on a nominal uncertainty of
0.15 dex. The Gaussian spread is a probability density function, which
describes the distribution of the data points around their true values. It is
mathematically represented as:

$\displaystyle
f_{i}=\exp\left\\{-0.5\cdot\left(\left[\left(x_{i}-x_{0}\right)/\sigma_{x}\right]^{2}+\left[\left(y_{i}-y_{0}\right)/\sigma_{y}\right]^{2}\right)\right\\}$
(7)

Here, $f_{i}$ represents the Gaussian contribution of the $i^{th}$ data point,
$(x_{i},y_{i})$ are the coordinates of the data point in the [X/Fe] vs [Fe/H]
space, and $(x_{0},y_{0})$ are the true values of the data point. The
parameters $\sigma_{x}$ and $\sigma_{y}$ represent the standard deviations of
the Gaussian spreads in the $x$ and $y$ directions, respectively.

The [X/Fe] vs [Fe/H] space was then divided into a series of bins, which
facilitated the organization and analysis of the data. Within each bin, the
Gaussian contributions of all the data points were used to compute an average
and standard deviation. This process helped to mitigate the effects of
uncertainties in individual data points and provided a more reliable basis for
further analysis.

##  6 Machine Learning

Methods with machine learning involve supervised machine learning and
unsupervised machine learning, where the first one has variables given a
certain target and the other one operates upon only the input data without
outputs or target variables. Since previous work proves that there has already
been successful fitting on most of the models, we then choose the supervised
machine learning method to further improve the performance of the model. Here,
we introduced a method of grid search, where we input the range of the
variables and enable the machine to get the sufficient time to run through all
possible combinations within a line-space of 0.01. Though, this machine
learning algorithm is time intensive while also need strong computation
ability for computers, still this is the best method that could result in best
parameter fittings for our models. This grid search method, scans over
$7^{10}$ possible parameters given the existing data. By using this typical
machine learning method, we could cover all the processes that potentially
correlate between isotopes. The test success will be based on the least
reduced chi-square values. Previous work involves a method that assigns
Gaussians to data points based on uncertainties and divides the region into
grids. However, the exact correlation is unsupported by data, and the model’s
assumption of scaling may not be statistically accurate at very low
metallicities due to insufficient data and the influence of individual
astrophysical events. Thus the way that we come out to test whether model fits
or not is the reduced chi-square method. This method states the following
equation ([18]):

$\displaystyle
x_{r}^{2}=\frac{1}{N}\sum_{i=1}^{N}(\frac{x_{i}-\mu_{i}}{\sigma_{i}})^{2}$ (8)

Within reduce chi square equation, N represents the number of bins we choose
specifically, while $\mu_{i}$ is the mean of the data $\sigma_{i}$ be the
standard deviation and $x_{i}$ symbols the model use. In case we do not lose
the generality, we use the bin size of 200 for all isotopes fitting with
standard deviation value of 0.1. The bin size means that we cut the whole
graph with 200 sections but we ignore the initial sections of line because
there was some defects how the data is constructed. While we set each bin
size, we use the model data compared with the observed data and get one bin
size reduced chi-square and sum them up at last. As long as we got all the
chi-square, according to (paper), reduced chi-square will behave linearly in
the statistics such that we could sum all the isotopes reduced chi-square
values all together to get a final reduced chi-square value. In other words,
the least reduced chi-square corresponds a best parameter set for the result.

##  7 Grid Search

Grid search is a parameter optimization technique that searches over a
specified set of hyperparameter values to find the optimal combination for a
given model ([9]). In general, since the previous work fitted free parameters
well for most light isotopes, as evidenced by the frequent use of Magnesium
isotope data in his model, we set the range of parameters within $0.5$ of the
initial parameter values. For example, if the original parameter for a massive
star reaction is $2.5$, we instruct the machine to search within the range of
$2$ to $3$, outputting a reduced chi-square value for each parameter value
examined. The step size is set to $0.05$, meaning that for each iteration of
parameter search, there is a $0.05$ increase in parameter values and a
corresponding reduced chi-square value output. This logic is applied to all
reaction parameters in the model, generating a large dataset of reduced chi-
square values. We calculate the linear sum of the reduced chi-square values
for all parameter sets and identify the least value as the best fit set of
parameter values. Although grid search is a straightforward way to determine
the best fit parameter values, it is computationally intensive and time-
consuming. As a result, we explore a new algorithm for the second part of our
study, focusing on the strong $s-$process.

##  8 Random Forest

Random Forest is an ensemble learning method that builds multiple decision
trees and combines their predictions to improve accuracy and reduce
overfitting ([17]). As we have a numerical method to assess the effectiveness
of our machine learning approach, we conduct preliminary result estimations
before implementing the machine learning method. We focus on typical reactions
occurring in heavy elements, such as Lead, and believe that the previous
parameter values for these reactions are not representative of the true
values. Consequently, we closely examine the machine learning method for these
elements and introduce the Random Forest algorithm to better understand how
machine learning works for these specific reactions.

Using the Random Forest algorithm significantly reduces the time required to
compute ideal parameter values while maintaining the same logic as the reduced
chi-square method. We instruct the computer to follow the trend of reducing
reduced chi-square values when adjusting parameter values. If the computer
identifies an increasing trend in reduced chi-square values, it will change
the parameter values to reverse the trend and reduce the reduced chi-square
value. Once a minimum reduced chi-square value is reached, the computer will
continue searching for more parameter values and corresponding reduced chi-
square values unless a peak value has been found, at which point it will
output the results. This algorithm is specifically used for determining the
values of the strong $s-$process, as there are limited datasets to train on
and only a few isotopes undergo this process for reactions. The best training
results are achieved for the Lead isotope, which will be discussed in the
analysis section.

##  9 Analysis on Grid Search Algorithm

In this honor thesis, we aim to provide a more quantitative analysis of a
machine learning algorithm specifically designed for comparing astrophysical
models based on their parameter values. This innovative algorithm leverages a
reduced chi-square calculation to generate numerical values for each pair of
parameters, thereby enabling direct comparison of these values to determine
the best-fit model.

To assess the performance of our algorithm, we concentrate on astrophysical
processes, excluding strong $s$-processes, which are key to understanding the
nucleosynthesis of heavy elements in stars. We implement a grid search
algorithm and systematically evaluate the contributions of various isotopes to
the total reduced chi-square values, thus offering a more comprehensive
understanding of the model’s performance.

For instance, Magnesium isotopes demonstrate the lowest contribution to the
total reduced chi-square value, with a value of $0.78$. This finding supports
the methodology of previous work, which heavily relied on Magnesium data to
fit free parameters due to its reliable and abundant data. In stark contrast,
Germanium isotopes display the highest contribution to the reduced chi-square
value, at $46.9$. This significant discrepancy can be primarily attributed to
a lack of data points and a large inconsistency in the available data for
Germanium isotopes.

To further validate our findings, we conducted a visual comparison of graphs
generated using two distinct sets of parameters. Figure 1 presents the
published [Eu/Fe] results from West $\&$ Heger (2013), and Figure 2 showcases
the resulting model for [Eu/Fe] found by parameter fitting. The graphs employ
a green line for the model prediction, a blue line for the average data trend,
a black shaded area to convey uncertainty in individual data points, and blue
crosses to indicate specific data points, providing an in-depth visual
comparison of the astrophysical findings. As seen in Figures 1 and 2, the
salient difference is a better fit against the data averages using the machine
learning method. This better fit comes at the expense of having a slightly
non-zero slope to the model line at low metallicities, which makes less
physical sense – since at low metallicities, both Eu and Fe are made in
primary processes, hence their ratio should be a flat function of metallicity.
Nevertheless, we recognize that there is no available data to constrain the
model at such low metallicities, where the chemical evolution may have been
more stochastic and without a well defined averaged anyway.

The visual comparison reveals minimal changes in Magnesium isotopes, as
expected, considering the previous work relied on Magnesium isotopes to
determine several best-fit values due to the abundance of data. Figure 3
combines the published [Mg/Fe] results from West $\&$ Heger (2013), and Figure
4 displays the resulting model found by machine learning parameter fitting.
Both figures effectively illustrate the predicted model curve and the mean
data line, as well as the uncertainty in each data point represented by the
black shaded zone. Although the present model does not offer much improvement
over the previous work for this element, it does serve as a consistency check
demonstrating that the machine learning process reproduces what the previous
work did quite well.

Figure 1: Published Results for [Eu/Fe] from [2], the green line represents
the theoretical model curve, while the blue line signifies the observed
average data trend. The black shaded region encapsulates the uncertainty
associated with individual data points, which are further indicated by blue
cross symbols throughout the graph. Figure 2: The resulting model for [Eu/Fe]
found by parameter fitting. Displaying the green line as the model prediction,
the blue line as the average data trend, the black shaded area representing
uncertainty in individual data points, and blue crosses as particular data
points. Figure 3: Published Results for [Mg/Fe] from [2]. The green line
symbolizes the predicted model curve, whereas the blue line reflects the mean
data line. Uncertainty in each data point is showcased by the black shaded
zone, with blue crosses marking individual data points. Figure 4: The
resulting model for [Mg/Fe] found by parameter fitting.The green line denotes
the model’s projected outcomes, while the blue line corresponds to the average
data trajectory. Individual data point uncertainty is demonstrated by the
black shaded area, and blue crosses pinpoint specific data points.

##  10 Analysis on Random Forest Algorithm

In this section, we conducted a thorough analysis of the success of the random
forest algorithm in refining our astrophysical model. As mentioned in previous
sections, this algorithm is specifically employed in the region where the
$s$-process reactions occur. Due to this assumption, we have a limited data-
set available for training; however, our algorithm performed remarkably well
for isotopes with few data points provided.

For initial parameter values of $d=150$ and $g=-0.23$, the machine learning
algorithm successfully modified these values to $d=250$ and $g=-0.05$. This
adjustment effectively reduced the reduced chi-square value from $1.3$ to
$0.2$, which corresponds to an impressive $84\%$ decrease.

In addition to the numerical value comparisons for this particular algorithm,
we also employed visual comparisons to assess the model’s performance. Figures
5 and 6 compare the published results with strong tanh factors (West $\&$
Heger, 2013) and the enhanced model via the Random Forest algorithm. Both
figures use a green line for the model curve, a blue line for the average data
trend, a black shaded area for uncertainty, and blue crosses for specific data
points. These figures highlight the improved outcomes achieved through the
application of the Random Forest algorithm. Initially, we observed a
significantly low and inaccurate estimation of the curve for the Lead isotope,
as each data point had a substantial contribution to the model’s values. The
previous model appeared to place undue emphasis on the outlier data point
located on the left side of the graph. However, after implementing the machine
learning algorithm, we not only eliminated the outliers in the left part of
the graph but also identified the unrealistic line in our model, which is the
dotted green line on the left side where no data occurred, and the model is
also hypothetical in that region.

The striking difference between the initial model and the current refined
version highlights the algorithm’s success in improving the model. Our
findings demonstrate the potential of the random forest algorithm as a
valuable tool for refining astrophysical models and enhancing our
understanding of nucleosynthesis processes in stars.

Figure 5: Published Results of Model and Data Analysis with Strong Tanh
Factors ([2]): The green line portrays the model curve incorporating a strong
tanh scaling factor (150) and strong tanh shift factor (0.23). The blue line
exhibits the average data trend, while the black shaded region demonstrates
the uncertainty associated with individual data points. Blue crosses represent
specific data points, reflecting the refined outcomes presented in this study.
Figure 6: Enhanced Model and Data Comparison via Random Forest Algorithm: The
green line displays the model curve with strong tanh scaling factor (250) and
strong tanh shift factor (-0.05). The blue line represents the average data
trend, while the black shaded area illustrates individual data point
uncertainty. Blue crosses indicate specific data points within the graph,
demonstrating refined outcomes through the application of the Random Forest
algorithm.

##  11 Visualizing the Isotopic Table

After refining the astrophysical models, we incorporated the updated parameter
values into our Python code to calculate and output the abundance of different
reaction contributions for each of the 287 isotopes under investigation. To
effectively visualize the trends in isotope reaction abundance, we varied the
initial metallicity of the star from $[Z]$ ranging from $-4.5$ to $0.5$,
employing a step size of $0.5$. We generated individual graphs for all
isotopes and used a color-coding system to differentiate the various
reactions. To facilitate easier comparison, all calculated abundances were
normalized by dividing the value by the solar abundance.

The isotopic abundance table generated from our analysis will be further
examined in later sections, with a focus on the trends in metallicity and the
corresponding reactions that typically occur in these nucleosynthesis
processes.

By visually comparing the isotopic abundance table, we can observe the
relative contributions of various processes for different isotopes. In this
section, we examine three distinct isotopes representing a range from light to
heavy isotopes, each consisting of a variety of nuclear processes. The
abundance is depicted on the y-axis using a logarithmic scale after
normalization from the solar abundance, while the x-axis displays the
metallicity range, which spans from $-4.5$ to $0.5$.

This in-depth analysis of isotopic abundances and their relation to
metallicity provides valuable insights into the nucleosynthesis processes
occurring in stars. Understanding these trends and reactions can ultimately
contribute to our knowledge of stellar evolution and the formation of elements
in the universe.

###  11.1 ${}^{56}Fe$ Isotope

Iron is first formed in the late stages of massive star evolution through a
series of nuclear reactions during hydrostatic burning. These abundances,
however, are not returned to the interstellar medium. The iron abundance
returned proceeds from explosive nucleosynthesis during the star’s death.
According to our model, see Figure 7, the creation of low metallicity
${}^{56}\text{Fe}$ abundances is indeed due to the results of massive star
nucleosynthesis (which includes explosive nucleosynthesis and the yields from
stellar winds). At low metallicities, Type Ia SNe contribute negligible
amounts. When the metallicity increases, the massive star reactions contribute
about $30\%$ of the total solar abundance. At Type Ia onset, (see the change
in slope in Type Ia abundances, Figure 7), contributions increase to their
solar value. At higher metallicities, massive star contributions to the iron
abundance decreases, as less new iron is made relative to the initial
composition.

Figure 7: The abundance as a function of metallicity for ${}^{56}Fe$. Massive
star contributions (green) and Type Ia SNe contributions (pink) are shown.
This visualization permits an estimate of the relative contributions from each
process to the ${}^{56}Fe$ abundance at any desired metallicity.

###  11.2 ${}^{85}Ru$ isotope

The Rubidium-85 isotope (see Figure 8) is a trans-iron isotope with both
primary and secondary components to its abundance. According to our model,
this isotope is made from the weak $s$-process, $r$-process, and $ls$-process.
As seen in Figure 8, $r$-process contributions to this isotope dominates at
all metallicities. The contribution resulting from weak $s$-process and
$ls$-process obey visually distinct slopes from the $r$-process, due to the
different fit parameters found for their scaling functions.

Figure 8: The abundance as a function of metallicity for ${}^{85}Rb$.
$r$-process contributions (orange), weak $s$-process contributions (yellow)
and $ls$-process contributions (red) are shown. This visualization permits an
estimate of the relative contributions from each process to the ${}^{85}Rb$
abundance at any desired metallicity.

###  11.3 ${}^{102}Pd$ isotope

We also highlight the scaling behavior of a Palladium isotope 9, which, in
comparison to Rubidium and iron isotopes, has contributions from different
nucleosynthesis processes. Our findings indicate that the gamma process and
$\nu p$-process contribute to ${}^{102}Pd$. However, the $\nu p$-process
dominates the abundance at low metallicities due to its primary nature,
whereas at higher matellicities the gamma process begins to dominate. The
gamma process, or photo-disintegration events, can either have primary or
secondary seed nuclei as targets. Hence, this process can either be in-between
a secondary or tertiary process, although both in the present model and
previous work, the parameter for it is taken as the average of primary and
secondary parameters. The gamma process contributions show a tendency to
increase at higher metallicities, and these different low veresus high
metallicity behavior can be observed for other species, such as Strontium and
Barium (See the Appendix for a complete set of all isotopic abundance scalings
by process).

Figure 9: The abundance as a function of metallicity for ${}^{102}Pd$. Gamma
Process contributions (cyan) and $np$ process contributions (purple) are
shown. This visualization permits an estimate of the relative contributions
from each process to the ${}^{102}Pd$ abundance at any desired metallicity.

##  12 Comparison

In this subsection, we compare our findings with those reported in published
work. The results are shown in the following table 1. Our analysis
demonstrates a significant difference of $19.8\%$ between the published and
refined results, with reduced chi-square values of 278 and 223, respectively.
We attribute this improvement to the effectiveness of our machine learning
methods in refining the models and adjusting certain reactions’ parameters.
Furthermore, visual comparisons reveal agreement for lighter isotopes but
considerable differences and enhancements for heavier isotopes, corroborating
our quantitative findings.

Parameter | Best-fit Value | Description
---|---|---
$a$ | 4.93(5.024) | Type $Ia$ $tanh$ scaling factor
$b$ | 2.82(2.722) | Type $Ia$ $tanh$ shifting factor
$f$ | 0.69(0.693) | Fraction of solar ${}^{56}\text{Fe}$ from Type Ia
$p$ | 0.84(0.938) | Primary process exponent
$h$ | 1.41(1.509) |  $hs$-process exponent
$l$ | 1.13(1.227) |  $ls$-process exponent
$w$ | 1.13(1.230) | Weak $s$-process exponent
$c$ | $-2\times 10^{-11}(-2\times 10^{-11})$ | “Strong” $tanh$ coefficient
$d$ | 250(150) | “Strong” $tanh$ scaling factor
$g$ | -0.05(-0.23) | “Strong” $tanh$ shift factor
Table 1: Final results after machine learning algorithms with best-fit values
and comparisons with values from previous work (shown in brackets). The
corresponding description for each parameter is also provided.

##  13 Limitations

Like any other study, our research has limitations that may impact the
accuracy and generalizability of our findings. In this subsection, we address
these limitations, which encompass various aspects of our study such as data
sources, machine learning algorithms, and modeling choices.

###  13.1 Data Sources

For this project, we acquired solar isotopic abundances from [20]. The solar
abundances suffer from uncertainties in certain key isotopes, whose values are
unstable across competing solar abundance data-sets (e.g., ${}^{14}N$).
Nevertheless, the most significant uncertainties arise from spectroscopic
abundance data acquired from elemental observations from stars ([21]). These
uncertainties arise from factors such as modeling the stellar atmospheres, and
these uncertainties are data-set specific and can vary across individual stars
measured.

###  13.2 Machine Learning Algorithm

Our study acknowledges potential issues with the employed machine learning
algorithm, which fits according to available elemental data. Whereas all data
is fitted together, individual isotopic scalings may be poorly constrained if
a paucity of elemental data exists for them. In essence, elements with few
stellar abundance data do not receive a higher weighting considered to advance
the fit. All data is considered on equal footing, which results in possibly
poorly constrained fits for scalings with less data available.

The selection of the grid search algorithm for parameter tuning presents
additional limitations. Given that this algorithm relies on predefined
parameter values and does not extensively modify them, it may restrict the
optimization process and limit the potential improvements to the model.

By acknowledging and addressing these limitations, we aim to offer a more
comprehensive understanding of our study’s findings and their implications.
Recognizing these limitations also highlights areas where future research
could further refine our understanding of astrophysical processes and improve
the methodologies employed in our investigation.

##  14 Uncertainty Analysis

In this section, we will address the challenges associated with the data
points and the interpretation of our findings with related to uncertainty. Due
to the Gaussian assignment of the data points, the standard deviation is quite
large, which leads to a significant spread in the astrophysical data. Despite
this, our data model is able to track the data well and reinforce the model
line averages for isotopic abundance.

As previously mentioned in the magnesium fit section, astrophysical data often
suffers from a large spread, as demonstrated by the red line which represents
the data line with Gaussian assignments. This spread results in several fit
parameters yielding reduced chi-square values of less than 1. To mitigate this
issue, we have introduced a scaling function for the astrophysical processes
derived from nuclear physics. This approach ensures that our model selects the
best average data without overfitting.

However, it is important to note that due to the inherent uncertainties in
astrophysical data, our model provides only average chemical abundance
estimates and does not predict any particular isotopic abundance in the
interstellar medium. This limitation should be taken into account when
interpreting the results of this study and applying them to specific
astrophysical contexts.

##  15 Conclusion

In this project, our primary objective was to refine and enhance the existing
construction of Galactic isotopic decomposition for all stable isotopes, by
leveraging advanced machine learning algorithms. Understanding the
astrophysical processes responsible for isotope synthesis is crucial for
gaining insights into the origins of elements in the universe and improving
our knowledge of stellar nucleosynthesis processes. The scope of our research
encompassed the development of a comprehensive isotopic table, representing
the relationship between a star’s initial metallicity and isotopic abundance.
To achieve this, we examined various astrophysical processes responsible for
isotope synthesis using solar abundance data. We successfully modified the
model, achieving lower reduced chi-square values compared to the published
results by West et al. (2013). This work laid the foundation for isotopic
decomposition as a function of metallicity, based on elemental observational
data and underlying nucleosynthesis processes in complex environments like the
Galaxy. Our key scientific findings include: Verification and refinement of
light isotope parameter values: Through the application of machine learning
algorithms, we not only verified the previous work on parameter values but
also provided a more detailed analysis of fitting lines for specific groups of
light isotopes.Significant improvement in heavy isotope predictions: By
utilizing a larger dataset to train the model and replacing the simple
estimation methods from previous results, we achieved a substantial
improvement in the predictions for heavy isotopes.The project’s broader
implications involve providing isotopic abundances that can be used as initial
abundances for stellar models in future work or other nucleosynthesis studies.
Furthermore, our research contributes to the improvement of input isotopic
abundance patterns by fitting a complex model structure with interconnected
parameters, marking the first time this has been done in a systematic
manner.Looking ahead, we propose several future extensions of this project:
Integration with Stellar Evolution Software: Implementing the refined model
into a realistic simulation of stellar evolution software, such as MESA, will
allow us to analyze the impact of initial metallicity on each phase of a
star’s evolution and isotope output, providing invaluable insights into the
life cycle of stars.Expanding the Application of Machine Learning Algorithms:
We suggest broadening the application of the Random Forest Algorithm to
optimize more parameter fittings and exploring advanced neural network machine
learning techniques, such as deep learning, for further improvements in model
accuracy. Validation and Comparison of Isotopic Scaling Models: Validating the
success of the machine learning methods by applying different isotopic scaling
models will enable us to determine the most accurate model for predicting
isotopic abundances. Additionally, this will help enhance principle component
analysis based on physical nucleosynthesis processes, furthering our
understanding of stellar nucleosynthesis.By pursuing these future research
directions, we can continue to improve our understanding of isotopic
decomposition in relation to stellar metallicity and advance the development
of more accurate and comprehensive stellar models and nucleosynthesis studies.
This honors thesis serves as an essential stepping stone towards a deeper
understanding of the origins and evolution of elements in the universe.

## References

  * [1] E. Margaret Burbidge, G. R. Burbidge, William A. Fowler, F. Hoyle, "Synthesis of the Elements in Stars," Rev. Mod. Phys., vol. 29, no. 4, pp. 547–650, 1957.
  * [2] Christopher West, Alexander Heger, "Metallicity-dependent galactic isotopic decomposition for nucleosynthesis," The Astrophysical Journal, vol. 774, no. 1, pp. 75, 2013.
  * [3] Hans Albrecht Bethe, "Energy production in stars," Physical Review, vol. 55, no. 5, pp. 434, 1939.
  * [4] Stan Woosley, Thomas Janka, "The physics of core-collapse supernovae," Nature Physics, vol. 1, no. 3, pp. 147–154, 2005.
  * [5] K Takahashi, J Witti, H-T Janka, "Nucleosynthesis in neutrino-driven winds from protoneutron stars II. The r-process," Astronomy and Astrophysics, vol. 286, pp. 857–869, 1994.
  * [6] John J Cowan, Friedrich-Karl Thielemann, James W Truran, "The r-process and nucleochronology," Physics Reports, vol. 208, no. 4-5, pp. 267–394, 1991.
  * [7] Carla Fröhlich, Gabriel Martinez-Pinedo, M Liebendörfer, F-K Thielemann, E Bravo, WR Hix, K Langanke, Nikolaj Thomas Zinner, "Neutrino-induced nucleosynthesis of A> 64 nuclei: the $\nu$ p process," Physical Review Letters, vol. 96, no. 14, pp. 142502, 2006.
  * [8] Donald D Clayton, Principles of stellar evolution and nucleosynthesis, University of Chicago press, 1983.
  * [9] James Bergstra, Yoshua Bengio, "Random search for hyper-parameter optimization," Journal of machine learning research, vol. 13, no. 2, 2012.
  * [10] M Pignatari, R Gallino, M Heil, M Wiescher, F Käppeler, F Herwig, S Bisterzo, "The weak s-process in massive stars and its dependence on the neutron capture cross sections," The Astrophysical Journal, vol. 710, no. 2, pp. 1557, 2010.
  * [11] T Rauscher, A Heger, RD Hoffman, SE Woosley, "Nucleosynthesis in massive stars with improved nuclear and stellar physics," The Astrophysical Journal, vol. 576, no. 1, pp. 323, 2002.
  * [12] Marcel Arnould, Stéphane Goriely, "The p-process of stellar nucleosynthesis: astrophysics and nuclear physics status," Physics Reports, vol. 384, no. 1-2, pp. 1–84, 2003.
  * [13] FX Timmes, SE Woosley, Thomas A Weaver, "Galactic chemical evolution: Hydrogen through zinc," The Astrophysical Journal Supplement Series, vol. 98, pp. 617–658, 1995.
  * [14] C Chiappini, Gratton, R, "The chemical evolution of the galaxy: the two-infall model," The Astrophysical Journal, vol. 477, no. 2, pp. 765, 1997.
  * [15] RDD Costa, WJ Maciel, AV Escudero, "Chemical evolution of the Galactic bulge: single and double infall models," arXiv preprint arXiv:0901.4382, 2009.
  * [16] L Portinari, C Chiosi, A Bressan, "Galactic chemical enrichment with new metallicity dependent stellar yields," Astronomy and Astrophysics, vol. 334, pp. 505–539, 1998.
  * [17] Leo Breiman, "Random forests," Machine learning, vol. 45, pp. 5–32, 2001.
  * [18] Philip R. Bevington, D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed., McGraw-Hill, Boston, MA, 2003.
  * [19] Chiaki Kobayashi, Hideyuki Umeda, Ken’ichi Nomoto, Nozomu Tominaga, Takuya Ohkubo, "Galactic chemical evolution: carbon through zinc," The Astrophysical Journal, vol. 653, no. 2, pp. 1145, 2006.
  * [20] Katharina Lodders, "Solar elemental abundances," in Oxford Research Encyclopedia of Planetary Science, 2020.
  * [21] Anna Frebel, "Stellar archaeology: Exploring the Universe with metal-poor stars," Astronomische Nachrichten, vol. 331, no. 5, pp. 474–488, 2010.