# Exploring Supernova Gravitational Waves with Machine Learning

A. Mitra,1,2,3 B. Shukirgaliyev,4,5,6 Y. S. Abylkairov,4,7 and E.
Abdikamalov1,4
1Department of Physics, Nazarbayev University, 53 Kabanbay Batyr ave, 010000
Astana, Kazakhstan
2The Inter-University Centre for Astronomy and Astrophysics (IUCAA), Post Bag
4, Ganeshkhind, Pune 411007, India
3Kazakh-British Technical University, Almaty, Kazakhstan
4Energetic Cosmos Laboratory, Nazarbayev University, 53 Kabanbay Batyr ave,
010000 Astana, Kazakhstan
5Fesenkov Astrophysical Institute, 23 Observatory str, 050020 Almaty,
Kazakhstan
6Faculty of Physics and Technology, Al-Farabi Kazakh National University, 71
Al-Farabi ave, 050020 Almaty, Kazakhstan
7Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr ave,
010000 Astana, Kazakhstan E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Core-collapse supernovae (CCSNe) emit powerful gravitational waves (GWs).
Since GWs emitted by a source contain information about the source, observing
GWs from CCSNe may allow us to learn more about CCSNs. We study if it is
possible to infer the iron core mass from the bounce and early ring-down GW
signal. We generate GW signals for a range of stellar models using numerical
simulations and apply machine learning to train and classify the signals. We
consider an idealized favorable scenario. First, we use rapidly rotating
models, which produce stronger GWs than slowly rotating models. Second, we
limit ourselves to models with four different masses, which simplifies the
selection process. We show that the classification accuracy does not exceed
$\sim\\!70\%$, signifying that even in this optimistic scenario, the
information contained in the bounce and early ring-down GW signal is not
sufficient to precisely probe the iron core mass. This suggests that it may be
necessary to incorporate additional information such as the GWs from later
post-bounce evolution and neutrino observations to accurately measure the iron
core mass.

###### keywords:

Gravitational Waves – Supernovae: general

††pubyear: 2022††pagerange: Exploring Supernova Gravitational Waves with
Machine Learning–B.2

##  1 Introduction

The detection of gravitational waves (GWs) from black hole mergers gave birth
to GW astronomy (Abbott et al., 2016b), while the observation of neutron star
merger in gravitational and electromagnetic waves is the prime example of the
multi-messenger astronomy (Abbott et al., 2017). One of the most promising
multi-messenger sources that are yet to be detected in GWs are core-collapse
supernovae (CCSNe).

CCSNe are the powerful explosions of massive stars at the end of their life.
The collapse of the stellar core releases $\sim\\!10^{53}$ erg of its
gravitational binding energy. While most of this escapes in the form of
neutrinos, the rest powers the explosion (Bethe, 1990). Supernovae involve
powerful aspherical flows that generate GWs with energies up to
$\sim\\!10^{47}$ erg (e.g., Ott, 2009; Kotake, 2013; Kotake & Kuroda, 2017).
The explosion front breaks out of the stellar surface hours later (Waxman &
Katz, 2017), producing a blast of photons across the electromagnetic spectrum
(e.g, Nakamura et al., 2016).

The supernova dynamics, and thus the GW signal, strongly depends on rotation.
In non- or slowly rotating models, which represent the majority of CCSNe
(e.g., Heger et al., 2005), the explosion is governed by the _neutrino
mechanism_ (e.g., Kotake et al., 2012; Janka, 2012; Müller, 2020; Burrows &
Vartanyan, 2021; Mezzacappa, 2022, for recent reviews). A fraction of
neutrinos emitted by the protoneutron star (PNS) heats the post-shock
material, giving rise to neutrino-driven convection (Herant et al., 1992;
Burrows et al., 1995; Janka & Müller, 1995) and standing accretion shock
instability (SASI) (Blondin et al., 2003; Foglizzo et al., 2006; Müller et
al., 2012). These flows perturb the PNS and excite its oscillations,
generating powerful GWs (e.g., Murphy et al., 2009; Müller et al., 2013;
Cerdá-Durán et al., 2013; Yakunin et al., 2015; Andresen et al., 2017; Hayama
et al., 2018; Radice et al., 2019; Mezzacappa et al., 2020b; Raynaud et al.,
2022). Asymmetries in neutrino emission also contributes to the GW signal
(e.g., Kotake et al., 2007; Takiwaki & Kotake, 2018; Vartanyan & Burrows,
2020). The GWs are detectable for sources within our Galaxy with current
generation detectors (Gossan et al., 2016; Abbott et al., 2020b; López et al.,
2021; Szczepańczyk et al., 2021; Antelis et al., 2022), while the future
generation detectors will enable a more detailed observation (Srivastava et
al., 2019).

In rare rapidly rotating stars, the rotational kinetic energy powers the
explosion via the _magnetorotational mechanism_ (LeBlanc & Wilson, 1970;
Bisnovatyi-Kogan et al., 1976; Akiyama et al., 2003; Burrows et al., 2007;
Mösta et al., 2014; Kuroda et al., 2020; Obergaulinger & Aloy, 2020; Raynaud
et al., 2020). Due to the centrifugal force, the PNS forms with an oblate
perturbation, triggering PNS ring-down oscillations that last for $\sim\\!10$
ms (Ott et al., 2012; Fuller et al., 2015). In some cases, the PNS may be
subject to non-axisymmetric instabilities (e.g., Ott et al., 2007; Scheidegger
et al., 2010; Shibagaki et al., 2020; Takiwaki et al., 2021). When this
happens, the non-axisymetrically-deformed PNS emits GW for many rotation
periods, significantly enhancing the detectability (Abbott et al., 2020b). For
moderate rotation, both the rotational bounce and convection contribute to the
GW signal (Andresen et al., 2019; Pan et al., 2021; Jardine et al., 2022).

Once detected, it is possible to estimate the parameters of source using the
GWs emitted by the source (e.g., Christensen & Meyer, 2022, for a recent
review). Since slowly and rapidly rotating models have significantly different
dynamics, their GW signals can be confidently distinguished (Logue et al.,
2012; Powell et al., 2016; Chan et al., 2020; Szczepańczyk et al., 2021; Saiz-
Pérez et al., 2022). Both convection and SASI develop from stochastic
perturbations, so the GWs coming from slowly rotating models contain
stochastic components. In contrast, the bounce GW signal in rapidly rotating
stars can be determined precisely for a given (physical and computational)
model parameters (e.g., Zwerger & Mueller, 1997; Dimmelmeier et al., 2008).
Despite the presence of the stochastic contributions, the GW spectra contain
the frequencies of the physical processes happening in the central regions
(e.g., Kotake et al., 2011; Müller et al., 2013; Astone et al., 2018; Roma et
al., 2019; Srivastava et al., 2019; Powell & Müller, 2022). In particular, the
PNS oscillations is the dominant component of the signal (Murphy et al., 2009;
Cerdá-Durán et al., 2013; Morozova et al., 2018; Radice et al., 2019; Warren
et al., 2020), the frequencies of which can be related to the physical
parameters of the system, such as the mass and radius of the PNS (e.g.,
Dimmelmeier et al., 2006; Müller et al., 2013; Torres-Forné et al., 2019;
Sotani et al., 2021; Bizouard et al., 2021).

For rapidly rotating stars, Abdikamalov et al. (2014) explored the possibility
to infer the rotation and its distribution in the supernova core. They
considered $\sim\\!100$ different rotational configurations of a stellar model
with 5 different degrees of differential rotation. They found that, for a
source at $10$ kpc distance, it is possible to measure rotation with
$\sim\\!20\%$ accuracy for rapidly rotating models, in which the rotational
kinetic energy exceed $\sim\\!8\%$ of the potential binding energy. In slowly
rotating models, due to smaller GW amplitudes and higher stochastic
contribution, the error becomes larger. These estimates were further improved
by Engels et al. (2014); Edwards et al. (2014), and Afle & Brown (2021).
Hayama et al. (2016) showed that signs of rapid rotation can be found in the
circular polarization of the GW signal. Yokozawa et al. (2015) proposed to
combine GW and neutrino observations to infer the rotation from the time
delays between the bursts of these two signals. Pajkos et al. (2019, 2021)
proposed to combine the the core-bounce signal with the dominant frequency
mode of the PNS in the pre-explosion to constrain the structure of the
progenitor star.

Richers et al. (2017) studied the dependence of the GW signal on the
parameters of the equation of state (EOS) of high-density matter in
protoneutron stars and treatment the electron capture rate during collapse.
They find a modest impact of these parameters to the bounce and the early
$\sim\\!20$ ms post-bounce GW signal. This emphasizes the importance of
accurate modeling (e.g., Lentz et al., 2012; Kotake et al., 2018; Pan et al.,
2018, 2019; Mezzacappa et al., 2020a; Andresen et al., 2021). Using deep
convolutional neural networks, Edwards (2021) classified these EOSs $72\%$
correctly, while their most probable five EOSs were found with $97\%$ accuracy
(see also Chao et al., 2022).

In this work, we study if it is possible to probe the iron core mass of CCSN
progenitor from the bounce and early ring-down GW signal alone. Ott et al.
(2012) showed that progenitors with different masses will produce similar GWs
if they have a similar angular momentum distributions at a given mass
coordinate in the stellar core. This suggest that the progenitor mass at most
has a subtle effect on the bounce dynamics. We extend this work further by
using machine learning (ML) for signal classification and studying a wider
ranges of progenitors and rotational configurations. We look at idealized
optimistic scenario for measuring mass from the bounce and early ring-down GW
signal alone. First, we consider rapidly rotating models, which produce strong
GWs. Second, we use the deleptonization method that is known to artificially
amplify the differences between progenitors. Third, we limit ourselves to
model progenitors with four different iron core masses only, which simplifies
the selection process. Despite these ideally favorable conditions, we show
that the iron core mass cannot be accurately measured from the GW bounce
signal alone. This suggests that, to measure the mass, one has to incorporate
additional information such as longer post-bounce signal or neutrino
observations, or both.

##  2 Methodology

###  2.1 Gravitational waveforms

We consider four progenitor models with zero-age main sequence masses ranging
from $12$ to $40$ $M_{\odot}$. At the pre-collapse stage, these models develop
iron cores with masses ranging from $1.3M_{\odot}$ to $1.8M_{\odot}$, as shown
in Table 1, which also provides the central densities and entropies. The
$12M_{\odot}$ and $40M_{\odot}$ models are produced by Woosley & Heger (2007),
while the $15M_{\odot}$ model is evolved by Heger et al. (2005) with magnetic
field prescription by Spruit (2002). The $27M_{\odot}$ model is produced by
Woosley et al. (2002). All models have solar metallicity.

To obtain the GW signals, we perform simulations using the CoCoNuT code
(Dimmelmeier et al., 2002a, b; Dimmelmeier et al., 2005). The simulations are
continued until 25 ms after bounce. This time interval contains full bounce
and ring-down signal from a CCSN event (Abdikamalov et al., 2022). We use the
same computational setup as in Richers et al. (2017) with the SFHo nuclear
equation of state (Steiner et al., 2013). During collapse, our code uses the
$Y_{\mathrm{e}}(\rho)$ deleptonization method of Liebendörfer et al. (2005).
Following Richers et al. (2017), we use $Y_{\mathrm{e}}(\rho)$ profile
obtained from spherically symmetric radiation hydrodynamics simulations using
the GR1D code (O’Connor, 2015). This method is known to amplify the contrast
between progenitors with different masses (Müller, 2009; Pajkos et al., 2021):
it produces variations in the inner core mass at bounce of $\simeq 10\%$
between progenitors with different masses, whereas full neutrino-transport
simulations yields inner core masses that are practically independent of the
progenitor (Müller, 2009). However, this weakness of the method represents the
strength of our work: as we show below, even with the artificial
amplification, the difference in the GW signals is too small to be
distinguishable. In more realistic models, the distinction should be even
smaller.

Following Abdikamalov et al. (2014), for each progenitor we consider about
$100$ different rotational profiles ranging from slow to rapid rotation, where
rotation has little to large effect on the core dynamics, respectively. At
distance $\varpi$ from the rotation axis, the angular velocity of pre-collapse
models is given by

$\Omega(\varpi)=\Omega_{0}\left[1+\left(\frac{\varpi}{A}\right)^{2}\right]^{-1},$
(1)

where $A$ is a measure of degree of differential rotation, $\Omega_{0}$ is the
central angular velocity (Komatsu et al., 1989). As in Abdikamalov et al.
(2014), we consider five different values of $A$, ranging from $300$ to
$10,\\!000$ km, which correspond to the limits of, respectively, extreme
differential and uniform rotations in the stellar core. For a given
$\Omega_{0}$ and $A$, we impose the rotation law (1) to the $12M_{\odot}$
model, as described in Abdikamalov et al. (2014). We then map the specific
angular momentum distribution homologously to the other progenitors with
different masses. This ensures that the specific angular momentum at a given
enclosed mass coordinate is similar in all four progenitors. As summarized in
Table 2, the total number of rotational configurations we consider is 97, 99,
102, and 104 for s12, s15, s27, and s40 models, respectively. The difference
in this number is caused by the fact that at the extreme rapid rotation limit,
due to a fine balance between gravity and centrifugal force in the pre-
collapse stage, some of the models do not collapse for some progenitors and
collapse for other.

We extract the GW waveforms from the simulations using the Newtonian
quadrupole formula in the first moment of momentum density formulation
(Dimmelmeier et al., 2002a, b; Dimmelmeier et al., 2005), which yields
accurate waveforms for stellar core collapse simulations (Reisswig et al.,
2011).

ZAMS mass | $M_{\mathrm{IC}}$ | $s_{\mathrm{c}}$ | $\rho_{\mathrm{c}}$
---|---|---|---
[$M_{\odot}$] | [$M_{\odot}$] | [$k_{\mathrm{B}}/\mathrm{baryon}$] | [$10^{9}\,\mathrm{g/cm}^{3}$]
12 | 1.3 | $0.60$ | $12.0$
15 | 1.4 | $0.62$ | $8.74$
27 | 1.5 | $0.77$ | $4.96$
40 | 1.8 | $1.06$ | $2.17$
Table 1: Progenitor model parameters used in our work. The ZAMS mass is the initial (the so-called zero-age) mass of the star. Parameter $M_{\mathrm{IC}}$ is the inner core mass, while $s_{\mathrm{c}}$ and $\rho_{\mathrm{c}}$ are the central entropy and density before collapse. Progenitor Mass | No. of Events
---|---
[$M_{\odot}$] |
12 | 97
15 | 99
27 | 102
40 | 104
Total | $402$
Table 2: Number of the generated GW events grouped by their progenitor mass
types.

###  2.2 Machine learning

Figure 1: Flowchart of the analysis pipeline of our study.

Our analysis pipeline can be divided in the following steps:

  * •

Preprocessing: Customize the waveforms for ML input.

  * •

Noise: Add noise to the signals to model the detector noise.

  * •

ML: Optimize hyperparameters111Hyperparameters are parameters of the
algorithm, which are tuned to optimize the learning. Their values are chosen
before initialising learning. They should not be confused with parameters of
the data, which are obtained during the learning. The choice of hyperparameter
has a direct impact on the performance of the ML model and hence they are
optimized to produce the best performance. and apply ML algorithm.

  * •

Analysis: Explore confusion matrix and other metrics.

A flowchart depiction of the pipeline is illustrated in Fig. 1.

To make the waveforms suitable to ML analysis, we condense all candidate
waveforms into a single data cube with the progenitor mass of each candidate
treated as target class. We map each candidates to the same cadence along the
time axis with uniform starting and ending points of $-5$ and $+15$ ms around
the bounce time. The time of bounce corresponds to time equal to zero. We
generate $10,000$ interpolated points along the time axis per candidate,
resulting in a data cube of dimensions $($402$\times 10000\times 2)$.

####  2.2.1 Random Forest

hyperparameters | Range
---|---
No. of estimators | $[100,400]$
Max. features | [auto, sqrt, log2]
Max. depth | $[10,250]$
Min. samples split | $[2,100]$
min. samples leaf | $[2,100]$
Bootstrap | [True, False]
OOB score | [True, False]
Warm start | [True, False]
Table 3: Hyperparameters and their ranges (or states) used during random grid
search. Figure 2: Mean test score of grid search results for
$\mathrm{SNR}=100$ plotted against individual hyperparameters listed in Tab.
3.

Random forest (RF) is a learning method that operates by constructing an
ensemble of decision trees (Morgan & Sonquist, 1963). It can be applied to
classification and regression problems. For classification problems, the RF
output is the class selected by the largest number trees, while for regression
problems, the result is the average prediction of all trees. RF classifies
samples using a forking path of decision points. We move from one decision
point to another and at each point we apply a rule that decides which branch
to follow (Ao et al., 2019). At the end, we arrive at a leaf, which has a
corresponding class label. We conclude the selection process by assigning the
data to the class (Denisko & Hoffman, 2018).

Tree-based approaches such as the RF have generated significant interest
recently (Kennedy et al., 2021; Hernandez Vivanco et al., 2020; Bonjean et
al., 2019) because they are robust, difficult to overfit, requires minimal
computational resources, and has availability of methods to assist in
interpreting the results (Hastie et al., 2010; Breiman, 2001), while being
quick to implement. It is also efficient in learning highly non-linear
relations between the input and the labels, making them suitable for large
training data set and large number of input variables.

We use the accuracy score as the performance metric for optimizing our RF
classifier network and to evaluate their performance. To arrive at the final
classifier settings for running the pipeline, we split our train-test data
into $90-10\%$ ratio and tune several hyperparameters using a randomised grid
search algorithm (Buitinck et al., 2013). Table 3 lists all the
hyperparameters used in the pipeline and their values (or ranges) considered
during the grid search. We perform a $100$ fold randomized cross validation
for tuning the hyperparameters. Since cross validation can produce an unbiased
estimator of a model on unseen data (Kuhn & Johnson, 2013), we do not use a
separate validation dataset. For comparison and consistency of our results, we
test additional algorithms, as discussed in Appendix B).

####  2.2.2 Detector Noise

The GW detectors can be impacted by a wide variety of noises, stemming from,
e.g., quantum sensing, seismic disturbances, suspension and mirror coating
thermal vibrations, and transient perturbations such as anthropogenic
interventions, weather, equipment malfunctions as well as occasional transient
noise of unknown origin (Aasi et al., 2015; Abbott et al., 2016a; Covas et
al., 2018; Driggers et al., 2019; Cabero et al., 2019; Ormiston et al., 2020).
In this analysis, we model the combination of all these possible sources of
noise as additive white Gaussian noise (AWGN) (Bergmans, 1974; McClaning &
Vito, 2002; Ding et al., 2010). While AWGN cannot model instrumental noise
transients and loud GW bursts contributing to non-Gaussian and non-stationary
features (Abbott et al., 2020a), as we show below, this approximation is
adequate to establish the upper limit for the classification accuracy, which
is the main goal of this work.

For a discrete time sequence signal, $x(t_{i})\equiv x_{i}$, the continuum
signal $G(t)$ can be defined with the help of a linear correlator as

$G\equiv\int q(t)x(t)dt,$ (2)

where $q(t)$ encapsulates the information from the inverse noise covariance
matrix. This signal can be matched to a known waveform from a linear matched
filter. For zero signal, the expectation value $E$ of $G(t)$ becomes zero. One
can define the signal to noise ratio (SNR) as

$\frac{S}{N}=\frac{E\\{G\\}}{\sqrt{{\left(\sigma\left\\{G\right\\}\right)}^{2}}},$
(3)

where the numerator is the expectation222It is derived from the power spectral
density of the signal noise (Flanagan & Hughes, 1998). and denominator is the
variance of the signal. In other words, it can be defined as the ratio of
signal power to variance. The SNR is a measure of the strength of a signal
observed by a detector with a given level of noise. It is directly
proportional to the amplitude of the signal buried in the noise.

In our analysis, we add AWGN noise $h_{\mathrm{noise}}$ to GW strain $h$ in
the time domain before the application of ML analysis, for every single chosen
value of SNR. The mean value of noise is assumed to be 0. The standard
deviation $\sigma_{\mathrm{noise}}$ of $h_{\mathrm{noise}}$ is governed by the
value of SNR. A consistency check on the stationarity and Gaussianity of the
waveform distribution can be done by transforming them in Fourier space and
studying their real and imaginary parts individually (Abbott et al., 2020a).
If noise distributions for both these components yield mean value of $0$, then
white Gaussianity can be assumed. In this analysis, we have performed this
check on our noise data and found them to be consistent with this requirement.

As the final step before applying ML, we standardize the dataset using the
Z-score normalization (Cheadle et al., 2003; Milligan & Cooper, 1988). Roughly
speaking, this normalization moves the mean of the signal to $0$ and adjusts
the average strength to $1$.

##  3 Results

Figure 3: GW strain as a function of time for different progenitors. The top,
center, and bottom panels show slowly, moderately rapidly, and rapidly
rotating models, respectively. The time value of $0$ ms corresponds to the
time of bounce.

The top, center, and bottom panels of Fig. 3 show the GW strain as a function
of time for three sets of models with slow, moderate and rapid rotations,
respectively. The models in each set have similar distributions of angular
momentum in their cores. The plot shows that they produce GW signals that are
similar to each other. This supports the observation that different
progenitors produce similar bounce GW signal for the same distribution of
rotation across mass coordinate (Ott et al., 2012). Slowly rotating models
exhibit contribution from prompt convection $\gtrsim 6\,\mathrm{ms}$ after
bounce. Since convection is a stochastic process, there is no obvious
correlation with the progenitor mass within the time scale considered in this
work.

The minor difference in the GW signal between different progenitor masses is
caused by the differences in the specific entropy of the iron core. The latter
tend to increase with the progenitor mass, but the dependence can be non-
monotonic (e.g., Woosley et al., 2002). The variation of the inner core mass
is amplified by our treatment of deleptonization (Müller, 2009). As discussed
above, our deleptonization scheme produces inner core masses bounce that are
$\simeq 10\%$ larger in progenitors with large masses than those in with small
masses. In full neutrino transport simulations, the inner core mass is
practically independent of the progenitor mass (Müller, 2009). For this
reason, we can only put the upper limit to the accuracy of the classification
accuracy, which, as we show below, is too low for any reliable identification
of the progenitor mass from GW bounce signal along.

Figure 4: The best accuracy score returned by the RF classifier pipeline for
classifying the progenitor mass for different input time ranges using the
rolling window of fixed width of $8$ ms. The time value of $0$ ms corresponds
to the time of bounce.

Figure 4 shows the best accuracy score for classifying the progenitor mass for
different ranges of time windows computed at maximum SNR of $100$. This score
measures how many labels the model gets right out of the total number of
predictions. The highest accuracy is achieved for range $[-2,\ +6]$ ms, which
is also the range used in the analysis of Richers et al. (2017). This is
expected since most of the bounce and ring-down GW signal is emitted in this
time interval (e.g., Abdikamalov et al., 2022). Moreover, the prompt
convection develops $\gtrsim\\!6$ ms after bounce, which, due to its
stochastic nature, makes it harder to identify the source parameters from the
signal. Consequently, we carried out the rest of the analysis with the wave
spectra in the range $[-2,\ +6]$ ms.

We run the above pipeline for six different SNR values $1,10,25,50,70$, and
$100$. We then compute the best accuracy score for each run. This score
measures how many labels the model gets right out of the total number of
predictions. Fig. 5 shows the accuracy rate of the classifier as a function of
the SNR. The accuracy curve is obtained using cubic interpolation between the
discrete SNR values. At the highest SNR of $100$, we see that there is $\sim
70\%$ probability of correct classification of the progenitor mass.

This finding is mirrored in the behavior of the confusion matrix used to asses
the classifier performance shown in Fig. 6. Along each rows, the fraction of
correct prediction is shown in percentage. The sum of these fractions is $1$.
The correct predictions are along the diagonal. All classes have more than
$50\%$ correct predictions. The s40 model is the most successfully predicted
class with $10$ out of $11$ cases accurately predicted.

Apart from studying the accuracy metric and the confusion matrix, we also
check the classification performance using the recall (also known as
completeness) and precision (also known as correctness) of each progenitor
mass class. The recall is defined as

$\mathrm{recall=\frac{TP}{TP+FN}},$ (4)

while precision is defined as

$\mathrm{precision=\frac{TP}{TP+FP}},$ (5)

where TP, FN and FP stand for the number of true positives, false negatives,
and false positives for classification of target labels, respectively.
Precision measures the quality of positive predictions made by the machine,
i.e., the fraction of results which are relevant. Recall reflects the
percentage of total true positive results correctly classified by the
algorithm (hence also called as true positive rate or sensitivity). However,
there is always a trade-off between recall and precision, and one can not
maximize both these metrics at the same time (Davis & Goadrich, 2006).

The recall and precision for our data are shown as a function of SNR in Fig. 7
and 8, respectively. We see from Fig. 7 that the s40 model achieves the
highest recall. For $\mathrm{SNR}\gtrsim 15$, we see that recall value for s40
peaks off around $1$, implying zero or low false negatives for
$\mathrm{SNR}\gtrsim 15$ (see Eq. 4), making it the most successfully
classified progenitor. This is also consistent with the values of the
confusion matrix shown in Fig. 6, which has an accuracy of $\sim 91\%$ for
s40. The s15 model is seen to be the worst performing model. s12 and s27 show
similar trends in their recall and precision curves. Except in model s27,
where recall improves in high SNR range and correspondingly the precision
falls, thus negating any additional gain in classification results, achieved
by the improved recall at high SNR values. In conclusion, we find that the
recall and precision plots in combination, show overall poor classification
performance and also beyond SNR $=40$ the improvement in performance levels
off and minor gain is achieved.

To understand the classification mechanism and which signal features the
classifier labelled as most dominating, we use the Gini importance score or
the Gini index (Menze et al., 2009; Nembrini et al., 2018). The Gini index is
commonly used as the splitting criterion in classification trees, the
corresponding impurity importance is often called Gini importance or mean
decrease in impurity. While it is debated that correlation between data
features can hinder importance interpretation, it is still a robust metric for
analyzing feature importance. Fig. 9 shows the Gini score mapped to the time
axis of the input signals. It is seen that the GW signal features in the
neighbourhood of $\sim\\!2.5$ ms are regarded as most relevant by the
classifier for splitting nodes across the whole tree. This signifies that in
this region the maximum purity in node splitting occurs compared to the rest
of the region in the waveforms. In this region, the GW signal is dominated by
the ring-down oscillations of the PNS, suggesting that the difference in the
initial mass influences the properties of PNS oscillations. However, Gini
importance does have sensitivity to correlation of features which influences
the quality of node splits along decision tree structures. Therefore higher
score is obtained with uncorrelated signals or pure node splits. In our plot,
the low Gini score except around $2.5$ ms implies that there must be
correlation of features in GW signals, present in these regions. This might
have reduced the purity during node splits and thus blurred identification
between features.

To verify the robustness of these finding, we have explored sensitivity of the
ML algorithm by performing our analysis with multiple different classifiers,
including the XGBoost and Neural Network. As we show in Appendix B, the main
finding of our paper remains the same: in all cases, the performance of the
classifier is not satisfactory to recover target labels.

Figure 5: Classifier accuracy score as a function of the signal SNR. It is
seen that for low SNR values ($\leq 30$), accuracy rises sharply with SNR,
beyond which accuracy increases marginally with maximum $\sim 0.70$ at SNR
$=100$. Figure 6: Confusion matrix for $\mathrm{SNR}=100$ presented for test
data of $41$ candidates which is $10\%$ of the full strength catalogue used.
The fraction of correct prediction per class can be seen along the diagonal.
Figure 7: Recall as a function of the SNR for each progenitor mass. Recall
gives insight into the number of positive cases that are misclassified by the
model as negatives (i.e. number of false negatives). Higher recall value
signifies higher amount of correctly classified positives. In this plot we can
see that except for s40 model, recall values are below $0.7$. They only
improve up to $\mathrm{SNR}\sim 40$, except for s27 which shows a gain in
recall at high SNR values. However, its corresponding precision value falls in
the same (high) SNR region (Fig. 8) and thus nullifying any additional gain
achieved in classification, by the improved recall performance in the high SNR
range.  Figure 8: Precision as a function of the SNR for each progenitor mass.
Precision gives insight into the quality of positive predictions made by the
model. In this case it defines the number of labels the model correctly
predicted divided by the total number of labels the model predicted. In this
plot we see that beyond SNR$=40$, precision does not change significantly
until it reaches maximum $\mathrm{SNR}=100$, where some improvement is
observed apart from s27 model, which counteracts this behaviour by showing
improvement in its recall values in the same SNR region (Fig. 7).

Figure 9: GW strain for SNR $=15$ (top panel) and Gini importance score
(bottom panel) plotted as a function of time. Highest importance score
(scaled) is seen around $\sim 2.5$ ms after bounce. The Gini importance score
indicates the features where maximum purity in Random Forest node splitting
(selection process) occurred.

##  4 Conclusion

We have presented a detailed analysis of whether bounce and early ring-down GW
signals from future observations of rotating CCSNe can be used for identifying
iron core masses. We used GW data obtained from numerical simulations and
injected white Gaussian noise to mimic the detector noise. We developed a ML
classifier with the aim to identify the target classes of these input signals.
We considered an idealized optimistic scenario for identifying the mass: we
use rapidly rotating models that emit strong GWs, include only progenitor with
only four different masses, which simplifies the selection process. In
addition, to generate GW signals, we used numerical simulations that use a
deleptonization scheme that artificially amplify differences in the collapse
and post-bounce dynamics of progenitors with distinct masses.

Despite this idealized favorable scenario, we were unable to identify the
mass, purely based on bounce and early ring-down GW signal alone. More
realistic treatment of noise modelling like non-Gaussian signals can only
reduce the chances of any possible identification of the progenitor mass
labels. The fact that even for SNR of 100, the classifier failed to give a
high accuracy score, does point to the fact that this is an intrinsic
phenomenon and independent of detector quality. Our results therefore show
that the information about the iron core mass is not contained in the bounce
and early ring-down GW signal. We have also performed similar analysis in
Fourier space and classification performance was within $2\%$ of the accuracy
score in time domain space. We thus conclude that the weak relation between GW
waveforms and the iron core mass is an intrinsic property of the system and
the classification via ML is not possible.

Additionally, we have explored sensitivity of the ML algorithm by performing
our analysis with multiple different classifiers (e.g., XGBoost and Neural
Network, as discussed in Appendix B). In all cases, our results showed that
the performance of the classifier was not satisfactory to recover target
labels. We do however note that, as with any ML analysis, numerous other
hyperparameter combinations are possible, as do multiple other algorithms. We
anticipate to further perform our analysis in future with more complex ML
model and representative data. However, we do not expect significant deviation
from the results presented in this paper. We expect that to constrain the iron
core mass one has to incorporate more information such us longer-term post-
bonce GW signal and/or neutrino emission (e.g., O’Connor & Ott, 2013; Yokozawa
et al., 2015; Kuroda et al., 2017; Nagakura & Vartanyan, 2022).

## Acknowledgements

We thank Sherwood Richers for carefully reading the manuscript and for helpful
comments. This research has been funded by the Science Committee of the
Ministry of Education and Science of the Republic of Kazakhstan (Grant No.
AP13067834 and AP08856149) and the Nazarbayev University Faculty Development
Competitive Research Grant Program No 11022021FD2912 (ssh2022007). High-
performance workstations of ECL/NU have been used to perform all simulations,
data analysis, and ML calculations (http://ecl.nu.edu.kz/computational-
facilities/).

## Data Availability

The data used in this work is available from authors upon request. The
gravitational waveforms are publicly available at
https://zenodo.org/record/7090935.

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##  Appendix A Principal Component Analysis

Principal component analysis (PCA) is a well-known method for dimension
reduction generally applied before a full scale ML analysis (Abdi & Williams,
2010). It was used in several works in the context of CCSN GWs (e.g., Rover et
al., 2009; Logue et al., 2012; Engels et al., 2014; Edwards et al., 2014; Afle
& Brown, 2021; Suvorova et al., 2019). In our work, we have the advantage of
having a relatively small data size by ML standards and the availability of
high computational resources without any time constraint. This means we can
run a full scale ML analysis without the requirement to apply a dimension
reduction technique, before classification stage. We still performed a PCA
analysis of our data for further insight.

We illustrate our findings in Fig. 10, which shows the eigenvalues ($y$-axis)
as a function of the number of components along the $x$-axis. It helps us to
estimate the reduced dimension of the data set with minimal loss in accuracy.
The principal components are constructed from the linear combination of
eigenvectors obtained from the covariance matrix of the data. The principal
components are uncorrelated. The maximum variance (information) is carried by
the first component, which decreases along the successive components. In other
words, the first few principal components help us capturing most of the
information of the data. We show that about $75\%$ of the data, which
corresponds to the number of principal components of $\simeq 300$, is adequate
to capture the entire variance, as seen along the $y$-axis of the plot with
cumulative variance $\simeq\\!1$. Furthermore, only $25\%$ of the data, which
is represented by $\simeq\\!100$ by principal components, is sufficient to
capture $90\%$ of the total variance in the input data. This can speed up the
calculations by a few orders magnitude compared to applying RF on the full
size data set. For future analysis with potentially more complex algorithms
(e.g., long short term memory (Hochreiter & Schmidhuber, 1997)) and for
quicker reproduction of results in a bigger survey pipeline, applying a PCA
can prove to be necessary.

Figure 10: Amount of variance captured as a function of the number of
principal components. The red dashed horizontal lines represent the $90$ and
$100\%$ cumulative variance mark while the black dashed vertical line
corresponds to $N=300$ which is the mark where cumulative variance reaches
$\sim 1$.

##  Appendix B Alternative Algorithms

The analysis presented in this paper is based on a RF classifier, which
performed best compared to the other choices that we considered. In this
section, we will summarize the two other algorithms that we tried: the XGBoost
and Neural Network based classifiers.

###  B.1 XGBoost Classifier

The Extreme gradient Boosting classifier, in short XGBoost classifier, is a
tree optimisation technique based on gradient boosting (Chen & Guestrin, 2016;
Friedman, 2001). If the output of an ensemble of $K$ trees is given as

$y_{i}=\sum_{k=i}^{K}f_{k}(\chi_{i}),$ (6)

such that each $f(\chi)=w_{q}(\chi_{i})$, where $x$ is the input vector and
$w_{q}$ is the score of the corresponding leaf $q$, then the objective
function $J$ that the XGBoost algorithm tries to minimise at each step $t$ is

$j(t)=\sum_{i=1}^{n}\left(y_{i},\hat{y}^{t-1}+f_{t}(x_{i})\right)+\sum_{i=1}^{t}\Omega(f_{i}).$
(7)

The first term $L$ contains the train loss function of mean squared error
between the true class $y$ and the predicted class $\hat{y}$ for the $n$
samples. The last (second) term on the right is for regularization term
controlling the model complexity and preventing an overfit. The second term
$\Omega(f)$ also defines the model complexity and can be expressed as

$\Omega(f)=\gamma T+\frac{1}{2\lambda\sum_{j=i}^{T}\left(w_{j}^{2}\right)},$
(8)

where $T$ is the number of leaves, $\gamma$ is the pseudo-regularization
hyperparameter, which is dependent on each dataset. Parameter $\lambda$ is the
L2 norm for leaf weight.

With the XGBoost classifier, we follow the exact same pipeline as with the RF
classifier except tuning for the XGBoost specific hyperparameters in the grid
search stage. The hyperparamters used in this algorithm is listed in Table 4.

Hyperparameters
---
No. of estimators
Learning rate
Max. depth
Booster
reg$(\alpha)$
reg$(\lambda)$
Base_score
Table 4: Hyperparameters that are tuned in the grid search step in the XGBoost
classification analysis.

Fig. 11 illustrates the accuracy values as a function of the SNR, similar to
Fig. 5. We can see that for $\mathrm{SNR}=100$, i.e. the highest value used in
this analysis, the accuracy is $\sim 50\%$. This value is significantly
smaller compared to RF classifier’s accuracy score of $\sim 70\%$, shown in
Fig. 5. Since both XGBoost and Random Forest classifiers are some of the most
robust algorithms, this finding reinforces our finding that classifying the
core mass seems impossible from the bounce and early ring-down GW signal
alone.

Figure 11: Accuracy as a function of SNR with XGBoost classifier, similar to
Fig. 5. It is seen that beyond SNR$=50$, the accuracy score does not improve.

###  B.2 Deep Learning: Neural Network Classifier

We run a separate convolutional neural network (CNN) based time series
classifier algorithm on our data, based on the architecture presented in Wang
et al. (2016). They provided an end-to-end fully connected time series
classification architecture without any need for feature engineering and data
processing. This architecture was shown to be successful with such time series
classification, outperforming all other neural network based architectures.
This motivates a classification analysis of our data with this pipeline. A
flowchart cartoon of the architecture is illustrated in Fig. 12.

We assess the model performance with sparse categorical crossentropy metric.
As seen from the training and validation curve in Fig. 13, the model heavily
over-fits with the given architecture. The overfit can be seen from the noisy
distribution of points in the curves. For the best case scenario of
$\mathrm{SNR}=100$, we achieve a classification accuracy of $\sim 55\%$ (Fig.
13). From this, we conclude that the neural network pipeline performance is
consistent with our RF classification results, but the network still needs
significant improvement in its design. In our case, we expect a simpler
architecture could reduce the over-fit problem.

Figure 12: Architecture of the fully connected neural network used in this
analysis based on Wang et al. (2016). Figure 13: Accuracy score for the neural
network for $\mathrm{SNR}=100$. The highly noisy pattern of the validation
curve suggests that the model heavily overfits.