††thanks: Corresponding author

# Deep-Learning Classification and Parameter Inference of Rotational Core-
Collapse Supernovae

Solange Nunes<EMAIL_ADDRESS>Centro de Física das Universidades do
Minho e do Porto (CF-UM-UP), Universidade do Minho, 4710-057 Braga, Portugal
Gabriel Escrig<EMAIL_ADDRESS>Departamento de Física Teórica, Universidad
Complutense de Madrid,
Pl. de las Ciencias 1, 28040 Madrid, Spain  Osvaldo G. Freitas
<EMAIL_ADDRESS>Centro de Física das Universidades do Minho e do Porto (CF-
UM-UP), Universidade do Minho, 4710-057 Braga, Portugal  Departamento de
Astronomía y Astrofísica, Universitat de València, Dr. Moliner 50, 46100,
Burjassot (València), Spain  José A. Font<EMAIL_ADDRESS>Departamento
de Astronomía y Astrofísica, Universitat de València, Dr. Moliner 50, 46100,
Burjassot (València), Spain  Observatori Astronòmic, Universitat de València,
Catedrático José Beltrán 2, 46980, Paterna (València), Spain  Tiago Fernandes
<EMAIL_ADDRESS>Centro de Física das Universidades do Minho e do Porto
(CF-UM-UP), Universidade do Minho, 4710-057 Braga, Portugal  Antonio Onofre
<EMAIL_ADDRESS>Centro de Física das Universidades do Minho e do Porto
(CF-UM-UP), Universidade do Minho, 4710-057 Braga, Portugal  Alejandro Torres-
Forné<EMAIL_ADDRESS>Departamento de Astronomía y Astrofísica,
Universitat de València, Dr. Moliner 50, 46100, Burjassot (València), Spain

###### Abstract

We test Deep-Learning (DL) techniques for the analysis of rotational core-
collapse supernovae (CCSN) gravitational-wave (GW) signals by performing
classification and parameter inference of the GW strain amplitude
($D\cdot\Delta h$) and the maximum (peak) frequency $(f_{\textrm{peak}})$,
attained at core bounce. Our datasets are built from a catalog of numerically
generated CCSN waveforms assembled by Richers et al. 2017\. Those waveforms
are injected into noise from the Advanced LIGO and Advanced Virgo detectors
corresponding to the O2 and O3a observing runs. For a signal-to-noise ratio
(SNR) above 5, our classification network using time series detects Galactic
CCSN GW signals buried in detector noise with a false positive rate (FPR) of
0.10% and a 98% accuracy, being able to detect all signals with SNR > 10\. The
inference of $f_{\textrm{peak}}$ is more accurate than for $D\cdot\Delta h$,
particularly for our datasets with the shortest time window (0.25 s) and for a
minimum SNR = 15. From the calibration plots of predicted versus true values
of the two parameters, the standard deviation ($\sigma$) and the slope
deviation with respect to the ideal value are computed. We find
$\sigma_{D\cdot\Delta h}=52.6$ cm and $\sigma_{f_{\textrm{peak}}}=18.3$ Hz,
with respective slope deviations of 11.6% and 8.3%. Our best model is also
tested on waveforms from a recent CCSN catalog built by Mitra et al. 2023,
different from the one used for the training. For these new waveforms the true
values of the two parameters are mostly within the $1\sigma$ band around the
network’s predicted values. Our results show that DL techniques hold promise
to infer physical parameters of Galactic rotational CCSN even in the presence
of real (non-Gaussian) noise conditions from current GW detectors.

##  I Introduction

The gravitational collapse of the core of massive stars and the subsequent
explosion as a supernova (a core-collapse supernova event, or CCSN hereafter)
is one of the most interesting sources of gravitational waves (GW) to be
detected in the coming years (see [1] for a recent review). CCSN are thought
to be the end result of stars with a zero age main sequence (ZAMS) mass larger
than about $8M_{\odot}$. The core of these stars, consisting of iron-group
nuclei, can no longer sustain nuclear burning and collapses. As the core
reaches densities above nuclear saturation density and the equation of state
(EoS) stiffens, the infalling material bounces back launching an outward-
moving shock wave. Numerical simulations have long helped understand the
processes by which the shock wave can gain sufficient energy to power a
supernova explosion. Two main mechanisms have been proposed, neutrino-driven
explosions and magneto-rotational explosions (see [2, 1] and references
therein). Rotation strongly affects the dynamics of CCSN and, in turn, its GW
emission. Most CCSN are non-rotating or slowly rotating [3]. In these models,
the GW emission is largely stochastic as it is triggered by convection and by
the standing accretion shock instability. In rapidly rotating stars, however,
the early part of the GW signal is well determined. In these models the
explosion is powered by the rotational kinetic energy (see [4] and references
therein), leading to a bounce GW signal that is well understood. The GW strain
of rapidly-rotating cores reaches its maximum right before bounce and its
value depends on the corresponding degree of oblateness of the core. Numerical
simulations have shown that for about 10 ms after core bounce the newly formed
proto-neutron star (PNS) undergoes a series of oscillations driven by the
excitation of axisymmetric fluid modes. Shortly after the GW signal becomes
stochastic as a result of the growth of non-axisymmteric hydrodynamical
instabilities. One to three CCSN events are expected to occur per galaxy per
century [5, 6] and less than 10% are likely to be from fast-rotating
progenitors [7].

To date, all confident GW signals observed by the Advanced LIGO and Advanced
Virgo detectors correspond to coalescing compact binaries (CBCs) [8, 9, 10,
11]. The most recent optically-targeted searches for GW from CCSN, using data
from the first and second observing runs of LIGO and Virgo, found no evidence
of significant candidates [5, 12]. As the sensitivity of the detectors
increases the probability of detecting GW signals from CCSN will also
increase, especially within our Galaxy and nearby satellite galaxies [5, 13,
14, 12, 15, 16, 17]. The successful detection of GW signals from CCSN may help
clarify the underlying physical processes occurring in the cores of massive
stars during their gravitational collapse along with providing a brand new
channel to infer properties of the source, allowing to probe the properties of
the PNS, the nuclear EoS, the rotation of the core, and the explosion
mechanism [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 17, 29, 30, 31, 32].

The detection of GW signals from CBCs is achieved through matched filtering
[33]. This requires template waveform models to be pre-computed with a
faithful representation of true GW. On the contrary, by their stochastic
nature, CCSN signals are unmodelled, requiring a completely different approach
for their detection. In this case, a coherent time-frequency analysis of the
data in a network of detectors is used [34, 12]. Correspondingly, the
estimation of the source parameters in the case of CBC signals is done through
Bayesian inference. This method uses a large number of waveforms (or
“approximants”) that cover a wide parameter space and applies stochastic
sampling techniques to evaluate their likelihood functions. Besides being
computationally expensive, it may become unsustainable as the number of
detections increases (but see [35, 36] for likelihood-free alternatives, and
[37, 38] for novel computational proposals). However, there are various
reasons why in general a Bayesian approach is not the optimal one for
inference of CCSN GW. On the one hand, post-bounce signals rapidly become
stochastic a few milliseconds after core bounce. On the other hand, the amount
of CCSN waveforms available is severely limited by the computational cost of
the simulations and by the impossibility to build the waveforms using the same
standard approximations employed for compact binaries. For the fairly brief,
“deterministic” part of the early bounce signal, matched-filtering analysis or
Bayesian model selection can be used to infer source properties (see e.g. [18,
29]). We also note that the dominant features of CCSN waveforms can be
extracted with Principal Component Analysis (PCA), where a mapping between
their measured eigenvectors and the physical parameters of the progenitor star
can be created [24]. Moreover, for close enough sources, it is also possible
to distinguish neutrino-driven CCSN from magneto-rotational CCSN using PCA
[39, 40, 19, 41, 30].

An alternative to the methods mentioned before are Machine Learning (ML) and
Deep Learning (DL) techniques which have been shown to be significantly less
time-consuming than Bayesian approaches in performing parameter estimation of
CBC signals [42, 35, 43, 44, 36, 45, 46]. The potential of ML for GW data
analysis is becoming increasingly relevant and the number and scope of
applications is already important (see [47, 48, 49, 50] for recent reviews).
Those include, e.g., detection methods for CBC signals [51, 52, 53], signal
quality improvements [54, 55], waveform generation [56, 57, 58], and noise
transients simulations in detectors [59, 60]. In the context of CCSN GW
signals the application of ML techniques is also an active field of research
[61, 62, 63, 28, 64, 32, 30, 65]. The potential of Convolutional Neural
Networks (CNNs) to detect CCSN GW signals was first shown by [61] using
phenomenological waveforms injected in Gaussian noise. CNNs were also employed
by [62] with CCSN GW signals injected in detector noise. At a false alarm
probability of 10%, neutrino-driven explosions at 10 kpc yield an expected
true alarm probability of 55% (76%) for current (future) LIGO-Virgo-KAGRA
(LVK) detectors, while for magneto-rotational explosions at 50 kpc the
corresponding values increase to 84% (92%). Approaches based on learned
dictionaries were first applied in [28] for CCSN signal classification,
obtaining 85% true classifications on signals with a SNR from 15 to 20 (see
also [30]). In [63] a Mini-Inception Residual Network (ResNet) for time-
frequency images (also implementing convolutional layers) was used to improve
the detectability of CCSN GW signals with the LVK pipeline coherent Wave Burst
[34]. For a dataset comprising phenomenological CCSN waveforms in background
noise from O2, an efficiency of around 80% was obtained for ${\rm SNR}\sim
16$. Recently [32] have used ML to investigate if it is possible to infer the
iron core mass from the bounce and early ring-down GW signal of rapidly
rotating CCSN models and [64] have compared the performance of different CNNs
and long short-term memory networks for multi-label classification of CCSN
simulated signals and noise transients using real data. CNNs have also been
used by [65] to classify GW from CCSN using spectrograms from numerical
simulations injected onto real noise data from O3. We also note the recent
work by [30] which presents a comprehensive comparison of different methods
(Bayesian model selection, dictionary learning, and CNNs) to determine the
explosion mechanism from a GW CCSN detection using up-to-date waveforms from
simulations (including, in particular, new 3D long-duration magneto-rotational
CCSN waveforms that cover the full explosion phase).

In this work, we present a DL approach for parameter estimation using GW from
rapidly rotating CCSN. We train Residual CNNs using datasets built from the
numerically generated CCSN waveforms computed by [20] which are injected into
noise from the O2 and O3a runs of the Advanced LIGO and Advanced Virgo
detectors. In addition, our best model is also tested on a different set of
waveforms for rapidly rotating CCSN built by [32] which allows us to assess
the generality of our approach. We follow closely the recent work of [29] and
perform our inference on two key parameters, the GW strain amplitude,
$D\cdot\Delta h$, and the maximum (peak) frequency, $f_{\textrm{peak}}$,
attained at core bounce. As discussed in [29], the results from [20] for a 12
$M_{\odot}$ progenitor and a large group of EoS show that when the ratio of
rotational-kinetic energy to gravitational potential energy, $T/|W|$, is below
0.06 (i.e. for slowly rotating cores), the first parameter is proportional to
$T/|W|$ and the second one is proportional to the square root of the central
density, $\sqrt{\rho_{c}}$. These relationships allow, in principle, the
inference of PNS properties from the GW detection of a rotating CCSN. In [29]
inference on these two parameters was conducted using Bayesian model selection
employing a master waveform template built from the waveform catalog of [20].
Our approach, based on DL, can thus be regarded as complementary to the
Bayesian approach of [29]. As we show in this work, for a rotating Galactic
CCSN, the inference of these two parameters using neural networks can be
successfully achieved, yielding results within the $1\sigma$ band from the
expected (true) values.

This paper is organized as follows: In Section II we describe the CCSN
waveforms we select to train our models. In Section III we discuss a first
approach in which the analysis of the signals is performed using spectrograms,
focusing on signal classification only. Section IV presents a second, more
complete approach. Here, DL methods are applied to time series and we discuss
results for both signal classification and parameter inference. Finally, our
conclusions are outlined in Section V.

##  II Waveform selection

Our models are trained with waveforms from the Richers et al. catalog [20].
This catalog is composed of 1824 numerical waveforms obtained from the
collapse of a 12$M_{\odot}$ progenitor, using 18 different EoS and covering a
parameter space of 98 rotation profiles. The simulations from [20] focus on
the bounce signal, including the collapse phase and the post-bounce evolution
up to $\sim 50$ ms after bounce. The bounce signal has been extensively
studied in previous works [66, 67, 68, 69, 18, 20, 70, 71] and it is
considered the key signature for the presence of rotation in CCSN. As
mentioned before, our study will focus only on the part of the signal
corresponding to the bounce and the following 10 ms.

The waveforms were obtained through a combination of 1D simulations of the
collapse-phase deleptonization with the code GR1D [72] and 2D core-collapse
simulations with the CoCoNuT Code [73]. To produce the collapse-phase
deleptonization, Richers et al. used an approximation proposed by Liebendörfer
[74] where the electron fraction ($Y_{e}$) is parameterized as a function of
density ($\rho$) only from spherically symmetric (1D) non-rotating general
relativity hydrodynamic simulations. This approximation was also used for the
rotating case, as electron captures and neutrino interactions with matter are
local and depend on the density in the collapse phase. In this case, the
rotational flattening of the collapsing core can be considered relatively
small. The axisymmetric (2D) CCSN simulations were performed assuming the
Conformal Flat Condition (CFC) approximation for the spacetime metric and an
initial rotational profile ($\Omega$) of the pre-collapse star given by

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

where $\Omega_{0}$ is the maximum initial (pre-collapse) rotation rate,
$\varpi$ is the distance from the axis of rotation and $A$ is a measure of the
degree of differential rotation.

Following [29], from the Richers et al. catalog [20] we only consider
waveforms with $\Omega_{0}\geq 3.0~{}{\textrm{rad}}{\textrm{s}}^{-1}$. This is
done to avoid the numerical noise present in models with slow rotation rates,
which gets amplified when performing the normalization of the signals.
Moreover, we do not use any of the simulations that do not collapse within the
first second (see Table III of [20]). Considering the above constraints only
999 waveforms remain from the catalog. While the injections on the datasets
used for the analyses with spectrograms employ the full catalog of 1824
waveforms, datasets created for the analyses with time series only apply the
selected 999 waveforms.

The waveforms assume an optimally oriented source. Therefore, for an
axisymmetric system there is no cross-polarization ($h_{\times}$) and only the
plus-polarization component ($h_{+}$) is different from zero, i.e.
$h_{\times}=0$ and $\quad h_{+}=h_{+}^{\rm opt}\sin^{2}\theta$. Here, $\theta$
is the inclination angle between the rotational axis of the core and the line
of sight of the observer. We fix this angle to $\pi/2$ rad, simplifying the
expression above to $h_{+}=h_{+}^{\textrm{opt}}$. Each waveform on the catalog
provides several physical parameters, from which we used the strain in units
of distance ($D\cdot h_{+}^{\textrm{opt}}$), the timestamps
$(t-t_{\textrm{b}})$, the peak frequency ($f_{\textrm{peak}}$), the maximum
rotational velocity $\Omega_{0}$, and the strain amplitude ($D\cdot\Delta h$).

##  III Analysis with Spectrograms

###  III.1 Dataset generation and neural network details

Our first analysis of the waveforms is based on time-frequency representations
of the signals, or spectrograms. These diagrams represent the signal amplitude
in pixel intensity, with time on the $x$-axis and frequency on the $y$-axis.
The use of spectrograms is not arbitrary. Since the bounce and post-bounce
phases of CCSN involve many frequencies, more significant patterns may be
found in the time-frequency representation (produced by splitting the input
time series into shorter sections and taking Fourier/wavelet transforms of
these sections) than in the time-strain representation. This pattern, being
contained in a short time interval and having many frequencies, is seen in a
spectrogram as an intense narrow vertical band, as shown in Figure 1 (right).

In order to train and validate a neural network (NN) it is necessary to have a
large enough data bank. For this purpose, we use a dataset generated with all
the 1824 waveforms from the Richers et al. catalog [20], as described in
Section II. Each of these waveforms is injected at a random time position into
sections of actual detector noise from O2 (corresponding to August 2017), with
a duration of 4096 ms. We assume a constant sky position, fixing the distance
to the source to 20 kpc and the right ascension, declination, and polarization
angle to zero. As the detectors do not have the same sensitivity in the whole
frequency range, a whitening procedure is applied to normalize the intensity
of each frequency. The frequency range we consider is between 20 and 1000 Hz,
where the LIGO and Virgo detectors are most sensitive. We also notch out fixed
frequencies affected by known artifacts, namely the frequencies at 60, 120,
and 240 Hz for the LIGO detectors and at 50, 100, and 200 Hz for the Virgo
detector, corresponding to the US and European power grid frequency and their
first harmonics, respectively. The sampling frequency is set to 16384 Hz.
High-resolution time-frequency maps of the injected waveforms are constructed
using the Q-transform [75].

The resulting images have dimensions of 256$\times$256 pixels. Since we use
three detectors, the images are coded with RGB colors where red, green, and
blue correspond to the spectrograms of LIGO-Hanford, LIGO-Livingston, and
Virgo detectors, respectively. In addition, the intensity of each color varies
between 0 and 255, so that when injecting a very intense signal, the
normalization makes the noise almost imperceptible. In the left panel of
Figure 1 an RGB image of only real background noise is shown, while the right
panel displays an RGB image with an injected CCSN GW signal. In the latter,
the signal stands out over the LIGO-Hanford and LIGO-Livingston noise. It
shows two narrow stripes with a rapid variation in frequency, associated with
the burst, and two subsequent features appearing shortly after, connected with
the oscillations of the PNS. The noise is barely noticeable in the green and
red channels due to the normalization while in the blue channel the intensity
of the injections is much lower than the noise (hence, the normalization makes
the noise stand out). Moreover, a delay can be observed between the detection
of the two interferometers.

Figure 1:  RGB images of only real noise (left) and with an injected GW into
real noise conditions (right) where the Hanford, Livingston, and Virgo
interferometer spectrograms are represented in red, green, and blue,
respectively.

A bank of $10^{4}$ images was generated, where half of them are only noise
(labeled as “background”) and the other half have CCSN GW signals injected
into noise (labeled as “signal”). In addition, the SNR of each signal was
evaluated, serving as a filter to eliminate weak signals (SNR < 5) that would
otherwise only confuse our NN. Subsequently, and for a better understanding of
our models, two other sets of images were generated for testing. One dataset
consists of 1500 images arranged in groups differentiated by five SNR units,
and the other dataset consists of 4000 images with distances ranging between 5
kpc and 40 kpc.

The SNR of a signal is a combination of the individual SNR for each GW
interferometer in the network, LIGO-Hanford, LIGO-Livingston and Virgo,
defined as

$\textrm{network SNR}=\sqrt{\sum_{i=1}^{3}{\textrm{SNR}_{i}}^{2}}.$ (2)

The individual $\textrm{SNR}_{i}$ is found by calculating the cross-
correlation matrix between the post-injected strain and the injected template
after projecting into the corresponding detector. Both strains are treated
with whitening, band-pass and notch filtering, as explained before. The
maximum value of this cross-correlation matrix is considered as the
$\textrm{SNR}_{i}$.

As our study is based on images we work with CNNs. These must be capable of
differentiating between background and signal. We use a pre-trained ResNet101
model which implements an architecture with a residual CNN 101 layers deep
[76]. We load the ResNet101 model pre-trained with over one million images
from the ImageNet database [77]. The pre-trained network can classify images
into $1000$ object categories. As a result, the network has learned feature-
rich representations for a wide range of images, although we will only need it
to distinguish between two classes, background and signal. To accomplish this,
we need to adapt the network by changing its last layer of 1000 neurons to one
with only 2 of them. For the generation of the datasets, functions from the
python [78] libraries SciPy [79], PyCBC [80], and GWpy [75], were used. We
also used the libraries fast.ai [81] for the classification tests.

###  III.2 Classification results with spectrograms

Our classification test aims at distinguishing the elements of the dataset
where a CCSN GW signal has been injected from those where only detector
background noise is present. For a training of 25 epochs, the network found
the minimum error in the 13${}^{\textrm{th}}$ epoch. The model obtained had an
accuracy 111The accuracy is defined as $\frac{TP+TN}{TP+TN+FP+FN}$, where TP
(TN) corresponds to the correctly classified signals (background) and FP (FN)
corresponds to the backgrounds (signals) that are misclassified. of 82%,
providing the confusion matrix displayed in Figure 2. This figure shows that
92% of the background images are correctly classified, with a FPR of 8%. While
the model only classifies 69% of the signals correctly, a true positive rate
(TPR) of 90% is obtained. These results show that our model detects signals
with high confidence, even if it only detects a modest amount of GW signals.

Figure 2:  Confusion matrix for the classification test with the
13${}^{\textrm{th}}$ epoch weights showing the fractions of strains predicted
as background noise or signal versus their actual labels.

With the previous training we further tested our model by performing
additional studies with different datasets sorted by SNR and by distance to
the source. Figure 3 (a) shows the relationship between the percentage of
success of the network in detecting signals and the SNR of the injected GW. It
is interesting to highlight that an accuracy of 95% and a precision of 93% is
reached for SNR > 30\. Although the network was trained to detect a GW
generated at a distance of 20 kpc, it can also be used to infer GW signals at
different distances. In Figure 3 (b), we observe that at shorter distances,
the percentage of success is higher, as expected, decreasing as the source
gets farther away. We observe a discrepancy for a distance of 20 kpc with
respect to the value obtained previously during validation. This is due to the
fact that when generating images for training and validation, those with SNR <
5 were eliminated, as mentioned above, a condition that we did not apply here.

(a)

(b)

Figure 3: Percentage of success in the validation of our network to detect a
CCSN GW signal as a function of SNR (a) and distance (b).

##  IV Analysis with Time Series

###  IV.1 Dataset generation

In our second approach, we apply DL techniques in the time domain. In this
case we only use the 999 selected waveforms from the Richers et al. catalog,
as described in Section II. For each of the selected waveforms we scale the
strain $D\cdot h_{+}^{\textrm{opt}}$ to a distance of the source between 5 and
20 kpc (i.e. a CCSN that occurs in our galaxy). For the background noise we
use public available data from both LIGO (Hanford and Livingston) and Virgo,
with an initial GPS time $t_{\textrm{GPS}}=1253326755~{}\textrm{s}$, which
corresponds to the end of September 2019, i.e. around the middle of the O3
run. We define a random time $t_{0}\in[6,1750]~{}\textrm{s}$ and select a
segment of the noise with a window $[t_{0}-5,t_{0}+10]~{}\textrm{s}$ for the
background sample. The waveforms were down-sampled to match the noise sampling
frequency. In our analysis we used a sampling frequency of 4096 Hz, allowing
faster performance without losing information. Each waveform was projected
into each detector with random values for the angles defining the sky position
(declination, right ascension, and polarization angle). Then, the projected
waveforms are injected into noise samples for each detector, having the time
of bounce of the signal ($t_{\textrm{b}}$) randomly set in the interval
$[t_{0},t_{0}+0.80]~{}\textrm{s}$, for the specific case of datasets with one-
second time window. For completeness, we also consider datasets of 0.50 s and
0.25 s with $t_{\textrm{b}}$ in the range $[t_{0},t_{0}+0.40]~{}\textrm{s}$
and $[t_{0},t_{0}+0.20]~{}\textrm{s}$, respectively.

We perform whitening on the resulting strains (background and signal) through
inverse spectrum truncation using the Amplitude Spectral Density (ASD) of the
three detectors’ noise. We then apply a band-pass filter from 20 to 1000 Hz
and notch filters at the individual frequencies of 60, 120, and 240 Hz, for
the two LIGO strains, and 50, 100, and 200 Hz for the Virgo strains. We
finally apply a crop on the strains to get the expected time window (1.00,
0.50, and 0.25 s).

For the classification and parameter estimation tests in the time domain we
use datasets of $10^{4}$ elements, where each element is composed of three
time series, one for each detector. Examples of these time series are given in
Figures 4 and 5, corresponding to one element of the signal and background
samples, respectively. Table 1 characterizes the datasets used for
classification and parameter estimation with time series. For the generation
of the datasets, functions from the python [78] libraries SciPy [79], PyCBC
[80], and GWpy [75], were used. We also used the libraries fast.ai [81] and
tsai [82] for the classification and parameter estimation tests, described in
the next sections.

(a)  (b)

Figure 4: Example of a dataset’s element labeled as signal, with a waveform
injected into noise from O3a, with a time window of 1 second (a) and 0.04
seconds (b).

(a)  (b)

Figure 5: Example of a dataset’s element labeled as background, with only
noise from O3a, with a time window of 1 second (a) and 0.04 seconds (b).

###  IV.2 Deep Neural Network: architecture and methodology

Time-domain GW signals of CCSN are tested using a Residual Convolutional
Neural Network (ResCNN), an implementation of CNN with residual learning
blocks. This network, proposed in [83], has 6 convolutional layers and uses
batch normalization techniques and different activation functions. The most
used activation function is the Rectified Linear Unit (ReLU). Other versions
of this activation function are also used in the architecture, i.e. Leaky ReLU
(LReLU), Parametric ReLU (PReLU), and Exponential Linear Unit (ELU). It is
desirable to use different activation functions in different layers to achieve
better performance [83].

As the analysis of the time series needs the precise location of the signal
features, the pooling operation was removed. Another reason for this choice is
the fact that pooling reduces the number of inputs given to the next layer,
limiting the information available. For this architecture, the usual Fully
Connected Layer at the end of the network was replaced by a global average
pooling and a final Softmax layer, so that the output of the network is a
probability distribution of its own predictions. The architecture of this
network is represented in Figure 6.

Parameters | Classification |  Parameter Estimation
---|---|---
$\textrm{SNR}>5$ $\textrm{progenitor's mass}=12\ M_{\odot}$ $\textrm{distance}\in[5,20]$ kpc $\textrm{inclination}=\frac{\pi}{2}$ $\textrm{declination}\in[-\pi,\pi]$ $\textrm{right ascension}\in[0,2\pi]$ $\textrm{polarization}\in[0,2\pi]$ |  Training set: 80% Validation set: 20% |  Training set: 70% Validation set: 30%
Table 1: Description of the datasets used to perform classification using time
series and parameter estimation. Figure 6:  Representation of the ResCNN’s
architecture as presented in [83].

We use the function fit$\\_$one$\\_$cycle to perform the training of the
network, defining the maximum Learning Rate (LR) and the weight decay. This
function of fastai internally uses the usual fit function but applies the
1cycle policy for better results [84]. The maximum LR and weight decay used
should be adapted to each dataset and architecture [85]. In our case, to find
the best values for our datasets we used another fastai function called
lr$\\_$find based on the LR range test [86]. We verified that for our datasets
the best results obtained were for weight decays and maximum LR of the order
of $10^{-3}$.

For our parameter estimation computations, before giving any dataset to the
network we perform a standard normalization of the signals’ parameters. To
find the predictions of the network in both classification and parameter
estimation, we use MC Dropout [87], passing each signal through the network
100 times. The prediction of the network is the mean value of all the 100
evaluations. Finally, we use PCA (from scikit-learn [88]) to evaluate the
prediction distributions for the parameter estimation tests. PCA can be used
for dimensionality reduction, being able to identify the most relevant
information (features) in datasets. In this work, we only use the PCA function
to perform Singular Value Decomposition (SVD) on the calibration plots, with
predicted values versus true values of $D\cdot\Delta h$ and
$f_{\textrm{peak}}$. We extract the two orthogonal eigenvectors to calculate
the slope and with the singular values we obtain the standard deviation of the
distributions, $\sigma$.

###  IV.3 Classification results with time series

A classification test aims at obtaining a model that best allows to
distinguish elements of a dataset where a CCSN GW signal has been injected
from those of only background noise. Our best results are achieved for a
12-epoch training and considering a 0.50 threshold, which led to a 0.09
minimum validation loss. This model has an accuracy of 98%, providing the
confusion matrix shown in Figure 7. Only 45 actual signals were predicted as
background, which corresponds to 4.5% of all GW signals present in the
validation set. In addition, only one element labeled as background was
misclassified as a signal, giving a FPR of 0.10%. Figure 8 displays the
receiver operating characteristic (ROC) curve of this classification model,
with a 0.99 area under the curve (AUC).

Figure 7:  Confusion Matrix for the classification test with 12 epochs giving
the number of strains predicted as background or signal versus their assigned
labels. Figure 8:  ROC Curve of the classification test with 12 epochs.

To better understand the results of the misclassified 45 signals (false
negatives), Figure 9 shows the distribution of the model’s scores for each of
those signals. In this plot we show our two key CCSN parameters, $D\cdot\Delta
h$ (top panel) and $f_{\textrm{peak}}$ (bottom panel), as a function of the
SNR. The different colors of the points represent the prediction scores given
by the model. All misclassified signals have SNR $\leq$ 10, which is expected
since in those cases it is difficult to distinguish the signal from the random
peaks of the noise.

Figure 9:  Values of $D\cdot\Delta h$ (top panel) and $f_{\textrm{peak}}$
(bottom panel) as a function of the SNR for the 45 misclassified signals on
the validation set. The color bar represents the score given by the model to
each element.

###  IV.4 Parameter inference results with time series

A parameter inference test was performed to predict the values of the peak
amplitude $D\cdot\Delta h$ (the difference between the highest and lowest
points in the bounce signal normalized to the distance) and the peak frequency
$f_{\textrm{peak}}$ (the highest frequency measured in the first 6 ms after
bounce). As discussed above (see also [29]) these two parameters are the most
relevant ones to characterize the waveform at bounce when $T/|W|<0.06$, as
$D\cdot\Delta h\propto T/|W|$ and $f_{\textrm{peak}}\propto\sqrt{\rho_{c}}$
for a large group of EoS [20].

Three datasets of $10^{4}$ elements each were analyzed, for different choices
of time windows, i.e. 1.00, 0.50, and 0.25 s, as mentioned previously. For all
the parameter estimation tests, we trained the network for 30 epochs setting
the LR to $2\times 10^{-3}$ and the weight decay to $10^{-3}$. As expected,
there is a relation between the size of the time window used and the training
time needed for each epoch. For the 1.00 s dataset, each epoch took around 90
s to complete. By decreasing the window to 0.50 s, the time needed for each
epoch also decreased to around 50 s. And lastly, for the 0.25 s dataset, each
epoch took around 27 s to complete. As we decrease the time window of the
dataset, the training time also decreases, almost in a linear way.

Figure 10: Calibration plots of predicted vs true $D\cdot\Delta h$ (left) and
predicted vs true $f_{\textrm{peak}}$ (right) with SNR $\geq 5$ for the
different datasets (corresponding to different time windows). The white dashed
line represents the ideal case, where the prediction value is equal to the
true value (slope is 1), and the white dotted line represents the slope of the
distribution calculated with the PCA function. The specific values of the
slopes are indicated in the legends. Figure 11: Calibration plots of predicted
vs true $D\cdot\Delta h$ (left) and predicted vs true $f_{\textrm{peak}}$
(right) with SNR $\geq 15$ for the different datasets. As in Fig. 10 the white
dashed line represents the ideal case (slope 1) and the white dotted line
represents the slope of the distribution calculated with the PCA function.

Figure 10 shows the calibration plots for the three datasets, considering SNR
$\geq$ 5\. These plots represent predicted values versus true values for
$D\cdot\Delta h$ (left) and $f_{\textrm{peak}}$ (right). Two white lines are
present in each plot. The dotted lines represent the slope of each
distribution, which should be close to the ideal slope, 1.00, represented by
the white dashed line. The best results are obtained for the 0.50 s dataset
(middle panel in the figure). Comparing the calibration plots for the two
parameters, our models yield more precise predictions for $f_{\textrm{peak}}$
than for $D\cdot\Delta h$. Moreover, low values of $D\cdot\Delta h$ and
$f_{\textrm{peak}}$ generally tend to be overestimated.

It is worth noticing that horizontal scattered lines are visible in Figure 10
for all the distributions of the calibration plots. In order to understand
this feature we generated new datasets with a minimum SNR of 15, consistent
with other studies which place constraints on the minimum SNR estimated to
detect GWs from CCSN [14]. The new results for the parameter estimation of
$D\cdot\Delta h$ and $f_{\textrm{peak}}$ on the three different datasets are
shown in Figure 11. With this choice of minimum SNR the horizontal scattered
lines disappear. We also see an improvement in the values of the slopes, with
the best results corresponding to the 0.25 seconds dataset. In this case, for
the $D\cdot\Delta h$ distribution, the slope is 0.884 and the standard
deviation is $\sigma_{D\cdot\Delta h}=52.6$ cm. For the $f_{\textrm{peak}}$
distribution, the results yield a slope of 0.917 and a standard deviation of
$\sigma_{f_{\textrm{peak}}}=18.3$ Hz.

Figure 12:  Probability distribution of the deviations of $D\cdot\Delta h$
(top panel) and $f_{\textrm{peak}}$ (bottom panel), i.e., the difference
between the labeled value and the predicted value, for the 0.25 seconds
dataset and a minimum SNR of 15.

The probability distributions of the difference between true and predicted
values of $D\cdot\Delta h$ and $f_{\textrm{peak}}$ are shown in the two plots
of Figure 12, for the 0.25 seconds dataset with minimum SNR of 15. For
$D\cdot\Delta h$, a mean value of $\mu_{[D\cdot\Delta
h_{\textrm{true}}-D\cdot\Delta h_{\textrm{pred.}}]}=0.5~{}\pm~{}76.6$ cm is
obtained. As for the $f_{\textrm{peak}}$ case, the mean value obtained is
$\mu_{[f_{\textrm{peak, true}}-f_{\textrm{peak,
pred.}}]}=~{}-0.2~{}\pm~{}26.4$ (Hz). We can clearly see that the inference of
$f_{\textrm{peak}}$ is much more accurate than $D\cdot\Delta h$. Taking into
account that in our datasets $D\cdot\Delta h\in[100,850]$ cm and
$f_{\textrm{peak}}\in[650,950]$ Hz, we can consider the mean values and
standard deviations obtained quite reasonable.

###  IV.5 Testing the network with a different catalog

We now discuss the performance of our network with a different set of CCSN
signals. In the absence of an actual discovery of a CCSN GW signal to test our
best model, we resort to employing waveforms from a different catalog within
the same parameter space of the datasets used for training. The new waveforms
were selected from the catalog created recently by Mitra et al. [32, 89]. This
catalog contains 404 waveforms of progenitors with masses ranging from 12 to
40$M_{\odot}$, albeit only for a single EoS, SFHo [90]. While training our
network with the Richers at al. catalog [20] we only considered 12$M_{\odot}$
progenitors. Therefore, we do the same for this additional test. The CCSN
simulations of [32, 89] also employ the general relativistic hydrodynamics
code CoCoNuT [73], as in Richers’ simulations. While the simulations extend up
to 25 ms after bounce, we only consider the part of the signals in the first
10 seconds after bounce in order to keep consistency between training and
testing.

Model A1 |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
Waveform | s12A1005.5 | s12A1006 | s12A1006.5 | s12A1007 | s12A1007.5 | s12A1008 | s12A1008.5 | s12A1009 |
$D\cdot\Delta h$ (cm) | 422 $\pm$ 50 | 446 $\pm$ 54 | 501 $\pm$ 59 | 591 $\pm$ 76 | 676 $\pm$ 57 | 730 $\pm$ 49 | 770 $\pm$ 45 | 757 $\pm$ 45 |
| [374] | [449] | [533] | [611] | [678] | [747] | [799] | [832] | –
$f_{\textrm{peak}}$ (Hz) | 755 $\pm$ 12 | 752 $\pm$ 12 | 762 $\pm$ 12 | 790 $\pm$ 25 | 827 $\pm$ 26 | 848 $\pm$ 16 | 883 $\pm$ 17 | 892 $\pm$ 30 |
| [770] | [764] | [783] | [826] | [843] | [863] | [986] | [923] |
Model A3 |  |  |  |  |  |  |  |  |
Waveform | s12A3003.5 | s12A3004 | s12A3004.5 | s12A3005 | s12A3005.5 | s12A3006 | s12A3006.5 | s12A3007 | s12A3007.5
$D\cdot\Delta h$ (cm) | 449 $\pm$ 58 | 493 $\pm$ 63 | 543 $\pm$ 61 | 647 $\pm$ 56 | 648 $\pm$ 39 | 637 $\pm$ 36 | 596 $\pm$ 31 | 561 $\pm$ 29 | 512 $\pm$ 36
| [361] | [464] | [564] | [643] | [666] | [629] | [583] | [560] | [520]
$f_{\textrm{peak}}$ (Hz) | 765 $\pm$ 11 | 764 $\pm$ 13 | 763 $\pm$ 18 | 787 $\pm$ 19 | 808 $\pm$ 17 | 831 $\pm$ 15 | 846 $\pm$ 13 | 851 $\pm$ 14 | 831 $\pm$ 19
| [772] | [773] | [780] | [798] | [816] | [841] | [858] | [862] | [859]
Model A4 |  |  |  |  |  |  |  |  |
Waveform | s12A4003 | s12A4003.5 | s12A4004 | s12A4004.5 | s12A4005 | s12A4005.5 | s12A4006 | s12A4006.5 |
$D\cdot\Delta h$ (cm) | 445 $\pm$ 59 | 484 $\pm$ 58 | 538 $\pm$ 61 | 597 $\pm$ 52 | 570 $\pm$ 33 | 520 $\pm$ 35 | 477 $\pm$ 28 | 442 $\pm$ 22 |
| [354] | [462] | [551] | [589] | [559] | [506] | [468] | [440] | –
$f_{\textrm{peak}}$ (Hz) | 761 $\pm$ 12 | 759 $\pm$ 12 | 759 $\pm$ 15 | 775 $\pm$ 14 | 790 $\pm$ 16 | 811 $\pm$ 14 | 808 $\pm$ 12 | 798 $\pm$ 15 |
| [768] | [765] | [770] | [786] | [802] | [816] | [820] | [817] |
Model A5 |  |  |  |  |  |  |  |  |
Waveform | s12A5003 | s12A5003.5 | s12A5004 | s12A5004.5 | s12A5005 |  |  |  |
$D\cdot\Delta h$ (cm) | 463 $\pm$ 62 | 506 $\pm$ 68 | 550 $\pm$ 61 | 560 $\pm$ 42 | 532 $\pm$ 38 |  |  |  |
| [387] | [494] | [552] | [544] | [506] | – | – | – | –
$f_{\textrm{peak}}$ (Hz) | 759 $\pm$ 12 | 760 $\pm$ 16 | 766 $\pm$ 14 | 780 $\pm$ 14 | 801 $\pm$ 15 |  |  |  |
| [764] | [764] | [775] | [797] | [813] |  |  |  |
Table 2: Parameter estimation results for the 30 selected waveforms from the
Mitra et al. catalog [32, 89]. The models are grouped by the degree of
differential rotation $A$, corresponding, from top to bottom to 300 km (A1),
634 km (A3), 1268 km (A4), and 10000 km (A5). For each waveform we report the
prediction, for both $D\cdot\Delta h$ and $f_{\textrm{peak}}$, in the form
$\mu\pm\sigma$, and the true value in brackets. When the true values are
within $1\sigma$ of the predictions they are written in boldface.

We selected 30 waveforms of this new catalog and, for each one of them,
generated a dataset with 100 elements as we did before. We fixed the time
window to 0.25 s and the minimum SNR to 15, which corresponds to the
conditions that led to our best model. Moreover, we allowed the model to vary
its response by using MCDropout. By performing signal injections in different
noise conditions as well as changes in the model response, we obtain a more
realistic implementation of a true GW detection. For a given dataset,
prediction values were determined for each element and, with the collection of
the 100 predicted values, the mean value, $\mu$, and the standard deviation,
$\sigma$, were calculated for both parameters, $D\cdot\Delta h$ and
$f_{\textrm{peak}}$.

The results of the parameter estimation test with this new catalog are
reported in Table 2. The name of the waveforms in this table encodes the
characteristics of the signal: s12 signifies that the mass of the progenitor
star is 12$M_{\odot}$, A1, A3, A4 and A5 correspond to the degree of
differential rotation of the pre-collapse star (see Eq. (1)), of 300, 634,
1268 and 10000 km, respectively, and the remainder of the string name gives
the central angular velocity in $\textrm{rad}\,\textrm{s}^{-1}$. Most of our
predictions are within 1$\sigma$ from the true values, confirming that our
model would be able to infer physical parameters of a Galactic CCSN GW signal
in more realistic conditions. For $D\cdot\Delta h$ only four waveforms are
outside of this interval but all are covered by the 2$\sigma$ band. In the
case of the $f_{\textrm{peak}}$ predictions, only seven waveforms have true
values outside of the 1$\sigma$ interval, but all are within 2$\sigma$.

##  V Conclusions

The first detection of a GW signal from a CCSN is highly anticipated as it can
provide important information about the physical processes occurring during
the gravitational collapse of massive stars. In this paper we have discussed a
method to extract information through the analysis of simulated CCSN
waveforms. In particular we have developed DL techniques to perform
classification and parameter inference of rapidly rotating CCSN using the
information encoded in the early (non-stochastic) part of their gravitational
waveforms. To do so we have trained NNs on a catalog of numerically generated
signals [20] injected into real LIGO-Virgo background noise from O2 and O3a,
both in the form of spectrograms (time-frequency diagrams) and time series.
The results presented in this work provide further support to the usefulness
of DL methods for GW data analysis. In the specific context of CCSN, the
methods discussed here seem worth of being implemented in CCSN detection
(classification) and parameter estimation pipelines.

We have first presented a classification test using a ResNet101 on a dataset
composed of $10^{4}$ spectrograms. CCSN GW signals were injected into real
noise from O2 of the LIGO Hanford, LIGO Livingston, and Virgo detectors,
making up half of the dataset. The other half contained only background noise
spectrograms, which had to be separated by the network’s model. The model
obtained after training accomplished an accuracy of 82% and a True Positive
Rate of 90%. We have also analyzed the influence of SNR and distance to the
source on the network’s performance. For SNR > 30 the accuracy of the model
becomes 95%. In addition, we attain more than 50% of success on the
classification only for distances below 20 kpc.

For the second, and main part of our study, we have considered time series. We
have found that there is a big improvement in accuracy for the results in the
time domain when compared with the ones with spectrograms and there is also a
significant decrease in the false alarm rate. All the tests performed with
time series have been done with Residual CNNs applied to datasets composed of
numerically generated signals from different waveforms. The classification
test was accomplished with a $10^{4}$ dataset consisting on equal parts of
elements with only background noise from O3a and elements of injected signals.
The network has provided a model with an accuracy of 98% with only one false
positive. These results confirm the efficiency of these networks regarding
classification and show that, even with a small dataset, high values of
accuracy can be obtained. By further inspecting the false negatives we noticed
a correlation with the SNR of the signals, as all of the misclassifications
occurred for elements with SNR < 10.

To perform parameter estimation we have used three datasets composed only of
injected signals from the catalog of [20]. These datasets differ in their time
duration: 1.00, 0.50, and 0.25 s. Considering a minimum SNR value of 15, the
best results are obtained with the 0.25 s dataset. Using the PCA function, we
have found that the slope of the distribution of the $D\cdot\Delta h$
calibration plot is 0.884 and the standard deviation $\sigma_{D\cdot\Delta
h,15}=52.6$ cm. For the peak frequency, the value of the slope obtained is
0.917 with $\sigma_{f_{\textrm{peak}},15}=18.3$ Hz standard deviation. As a
final test we have assessed our best model with 30 additional waveforms from a
new catalog of rapidly rotating CCSN models, computed recently by [32]. For
each waveform we have created a dataset composed of 100 elements with 0.25 s
duration and SNR $\geq$ 15\. The results obtained have shown that most of the
true values are within 1$\sigma$ of the prediction distribution, confirming
that our model might be able to infer parameters of fast-rotating Galactic
CCSN GW signals.

While the results reported in this work show the potential of Deep Learning
for inferring properties of rotational CCSN they can nonetheless be improved.
An obvious improvement would be accomplished by enlarging the datasets. The
datasets used for the training of our models only have $10^{4}$ elements,
which is considered a small size for the kind of classification and regression
tests with neural networks. However, enlarging the datasets would require
having available more waveforms from CCSN simulations, which are
computationally costly. We note that we are currently working in the
computation of new fast-rotating models for different progenitor masses and
expect to present an updated study of our findings elsewhere. In addition, in
this study we have used background noise from the existing network of GW
detectors (namely Advanced LIGO and Advanced Virgo). As current detectors are
improved and third-generation detectors become operational, the sensitivity
will improve and the background noise will be reduced, which will lead to
higher SNR values and easier recognition of CCSN GW signals.

## Acknowledgements

We thank Pablo Cerdá-Durán, Anastasios Theodoropoulos, José D. Martín, Nino
Villanueva, and Pedro Passos for fruitful discussions during the course of
this work. GE acknowledges the Spanish MINECO grants MINECO/FEDER Projects,
PID2021-122547NB-I00 FIS2021, MCIN with funding from European Union
NextGenerationEU PR47/21 MADQuantum-CM PRTR-CM and Ministry of Economic
Affairs Quantum ENIA project. OGF is supported by an FCT doctoral scholarship
(reference UI/BD/154358/2022). JAF and ATF are supported by the Spanish
Agencia Estatal de Investigación (grant PID2021-125485NB-C21) funded by
MCIN/AEI/10.13039/501100011033 and ERDF A way of making Europe. Further
support is provided by the Generalitat Valenciana (grant CIPROM/2022/49), by
the EU’s Horizon 2020 research and innovation (RISE) programme H2020-MSCA-
RISE-2017 (FunFICO-777740), and by the European Horizon Europe staff exchange
(SE) programme HORIZON-MSCA-2021-SE-01 (NewFunFiCo-101086251). AO in partially
supported by FCT, under the contract CERN/FIS-PAR/0037/2021. This material is
based upon work supported by NSF’s LIGO Laboratory which is a major facility
fully funded by the National Science Foundation.

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