# Detecting Fast Neutrino Flavor Conversions with Machine Learning

Sajad Abbar  0000-0001-8276-997X Max-Planck-Institut für Physik (Werner-
Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany Hiroki Nagakura
0000-0002-7205-6367 Division of Science, National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

###### Abstract

Neutrinos in dense environments like core-collapse supernovae (CCSNe) and
neutron star mergers (NSMs) can undergo fast flavor conversions (FFCs) once
the angular distribution of neutrino lepton number crosses zero along a
certain direction. Recent advancements have demonstrated the effectiveness of
machine learning (ML) in detecting these crossings. In this study, we enhance
prior research in two significant ways. Firstly, we utilize realistic data
from CCSN simulations, where neutrino transport is solved using the full
Boltzmann equation. We evaluate the ML methods’ adaptability in a real-world
context, enhancing their robustness. In particular, we demonstrate that when
working with artificial data, simpler models outperform their more complex
counterparts, a noteworthy illustration of the bias-variance tradeoff in the
context of ML. We also explore methods to improve artificial datasets for ML
training. In addition, we extend our ML techniques to detect the crossings in
the heavy-leptonic channels, accommodating scenarios where $\nu_{x}$ and
$\bar{\nu}_{x}$ may differ. Our research highlights the extensive versatility
and effectiveness of ML techniques, presenting an unparalleled opportunity to
evaluate the occurrence of FFCs in CCSN and NSM simulations.

††preprint: MPP-2023-237

##  I Introduction

Core-collapse supernovae (CCSNe) and neutron star mergers (NSMs) are
cataclysmic stellar events that represent the dramatic culmination of massive
stars’ life cycles and the collision and coalescence of incredibly dense
remnants, respectively. These events not only mark the end of massive stars
and dense objects, but also unveil some of the most energetic and enigmatic
phenomena in the universe. In the heart of these cosmic fireworks, one of the
most fascinating processes at play is the neutrino emission, which are
released in vast quantities during CCSNe and NSMs.

As they journey through the extraordinarily dense and extreme conditions
within these events, neutrinos undergo an intriguing phenomenon known as
collective neutrino oscillations. This fascinating behavior arises from their
interactions with the dense background neutrino gas, where coherent forward
scatterings play a pivotal role. This phenomenon occurs in a nonlinear and
collective manner, creating a rich tapestry of flavor transformations
pantaleone:1992eq ; sigl1993general ; Pastor:2002we ; duan:2006an ;
duan:2006jv ; duan:2010bg ; Mirizzi:2015eza  (for a recent review see Ref.
Volpe:2023met ).

Of particular interest are the so-called _fast_ flavor conversions (FFCs),
which occur on scales characterized by $\sim G_{\rm{F}}^{-1}n_{\nu}^{-1}$
(see, e.g., Refs. Sawyer:2005jk ; Sawyer:2015dsa ; Chakraborty:2016lct ;
Izaguirre:2016gsx ; Capozzi:2017gqd ; Abbar:2018beu ; Capozzi:2018clo ;
Martin:2019gxb ; Capozzi:2019lso ; Johns:2019izj ; Martin:2021xyl ;
Tamborra:2020cul ; Sigl:2021tmj ; Kato:2021cjf ; Morinaga:2021vmc ;
Nagakura:2021hyb ; Sasaki:2021zld ; Padilla-Gay:2021haz ; Abbar:2020qpi ;
Capozzi:2020syn ; DelfanAzari:2019epo ; Harada:2021ata ; Padilla-Gay:2022wck ;
Capozzi:2022dtr ; Zaizen:2022cik ; Shalgar:2022rjj ; Kato:2022vsu ;
Zaizen:2022cik ; Bhattacharyya:2020jpj ; Wu:2021uvt ; Richers:2021nbx ;
Richers:2021xtf ; Dasgupta:2021gfs ; Nagakura:2022kic ; Ehring:2023lcd ;
Ehring:2023abs ; Xiong:2023vcm ; Zaizen:2023ihz ; Xiong:2023upa ;
Fiorillo:2023hlk ; Nagakura:2023wbf ; Martin:2023gbo ; Grohs:2023pgq ). Here,
$G_{\rm{F}}$ represents the Fermi coupling constant, and $n_{\nu}$ denotes the
neutrino number density. These FFCs can take place on timescales much shorter
than what would be expected in the vacuum.

FFCs occur _iff_ the angular distribution of the neutrino lepton number,
defined as,

$\begin{split}G(\mathbf{v})=\sqrt{2}G_{\mathrm{F}}\int_{0}^{\infty}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}}{(2\pi)^{3}}&[\big{(}f_{\nu_{e}}(\mathbf{p})-f_{\nu_{x}}(\mathbf{p})\big{)}\\\
&-\big{(}f_{\bar{\nu}_{e}}(\mathbf{p})-f_{\bar{\nu}_{x}}(\mathbf{p})\big{)}],\end{split}$
(1)

crosses zero at some $\mathbf{v}=\mathbf{v}(\mu,\phi_{\nu})$, with
$\mu=\cos\theta_{\nu}$ Morinaga:2021vmc . Here, $E_{\nu}$, $\theta_{\nu}$, and
$\phi_{\nu}$ are the neutrino energy, the zenith, and azimuthal angles of the
neutrino velocity, respectively, and $f_{\nu}$’s are the neutrino occupation
numbers. When $\nu_{x}$ and $\bar{\nu}_{x}$ have similar angular
distributions, a scenario commonly observed in state-of-the-art CCSN
simulations, this expression transforms into the conventional $\nu$ELN
(neutrino electron lepton number).

Exploring $\nu$ELN crossings necessitates access to the complete angular
distributions of neutrinos. However, obtaining such detailed angular
information poses a significant challenge in most cutting-edge CCSN and NSM
simulations due to the prohibitive computational demands involved.

As a practical alternative, many simulations simplify neutrino transport by
relying on a limited set of angular distribution moments. In our specific
investigation, we focus on radial moments, defined as,

$I_{n}=\int_{-1}^{1}\mathrm{d}\mu\ \mu^{n}\
\int_{0}^{\infty}\int_{0}^{2\pi}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}\
f_{\nu}(\mathbf{p}).$ (2)

These moments capture the key aspects of the neutrino angular distribution, at
the same time allowing for its more computationally manageable treatment.

Despite the inherent loss of a significant amount of information when
considering only a select few neutrino angular moments, ingenious methods can
still be devised to harness these limited information for assessing FFCs in
CCSN and NSM simulations. In the initial stages of research in this field, the
primary focus was on analytical or semi-analytical techniques Dasgupta:2018ulw
; Abbar:2020fcl ; Johns:2021taz ; Johns:2019izj ; Nagakura:2021hyb ;
Richers:2022dqa ; Nagakura:2021suv . While these methods have demonstrated
their ability to capture ELN crossings and have found relative success in the
literature, they are constrained in their performance. This limitation arises
from their either sluggish computational speed or their inefficiency in
identifying ELN crossings which impacts their ability to efficiently detect
FFCs in real-time simulations. Specifically, the most efficient techniques
tend to be noticeably sluggish, and their development can be relatively
intricate when starting from scratch.

Recent research has demonstrated the remarkable effectiveness of machine
learning (ML) techniques in identifying FFC in CCSN and NSM simulations
Abbar:2023kta . While ML methods are data-intensive and require an initial
training phase with data, it’s important to note that once trained, they
exhibit exceptional speed and efficiency. This presents a promising avenue for
real-time detection of FFI’s within the context of CCSN and NSM simulations.
Moreover, integrating pre-trained ML models is a straightforward procedure,
significantly reducing the requirement for extensive coding work, even when
analyzing the occurrence of FFCs in a post-processing phase. In fact, ML
techniques offer the fastest and most precise approach to detect FFCs, and
their performance can be further enhanced as one encounters increasingly
complex environments.

In this paper, we advance the prior study in two pivotal directions. Firstly,
in earlier work, ML models were trained using artificial data generated from
specific parametric angular distributions. While these models showed promise,
they were only partially validated against a limited amount of realistic data
from NSM remnant simulations. It is essential for ML techniques to be trained
and tested on data that closely resembles real-world simulations. In our
study, we take a significant step forward by utilizing authentic data from a
CCSN simulation, where neutrino transport was modeled using the full Boltzmann
equation. This allows us to assess the adaptability of ML methods in a real-
world context and examine their limitations as well as their optimal
performance range. Furthermore, we acknowledge that artificial data are more
readily available and can offer broader distributions that are expected in
CCSN and NSM environments. Consequently, we also investigate methods to
enhance artificial datasets for ML training purposes.

In addition, in our previous study, we assumed that the distributions of
$\nu_{x}$ and $\bar{\nu}_{x}$ were identical. From a ML perspective, this
simplification facilitated the development of our ML module by requiring a
smaller number of features and a more efficient classification. In this work,
we develop ML techniques to detect the crossings regarding the neutrino heavy-
leptnic channel distribution ($\nu$XLN) addressing scenarios where $\nu_{x}$
and $\bar{\nu}_{x}$ may exhibit differences, a _previously unexplored_ area in
the literature. While our results may be slightly less accurate than those in
the previous scenario, ML methods still prove remarkably effective in
identifying the occurrence of FFC in this scenario..

In the upcoming section, we delve into our CCSN model, the source of our data.
We then assess the performance of ML methods in detecting FFCs in our CCSN
model, specifically focusing on the detection of $\nu$ELN crossings. Finally
and before presenting our conclusions, we analyze the performance of ML
methods in detecting the crossings in the heavy-leptnic channel.

##  II CCSN model

Here, we construct a ML technique to detect $\nu$ELN- and XLN angular
crossings based on an axisymmetric CCSN model with full Boltzmann neutrino
transport Nagakura:2019evv . Before entering into the detail of our ML, we
briefly describe our CCSN model providing neutrino dataset used for ML
training and its testing.

The numerical simulation was carried out by a Boltzmann-neutrino-radiation
hydrodynamic code Nagakura:2014nta  with some special treatments to handle
proper motions of proto-neutron star (PNS) Nagakura:2016jcl ; Nagakura:2019rdf
. In this model, a table of multi-nuclear variational method equation-of-state
Furusawa:2017auz  was used, and the nuclear abandance in the table was also
used to compute neutrino-matter interactions for the consistent treatment
between EOS and weak rates Nagakura:2018qpg . We used a 11.2 solar mass
progenitor model in Ref. Woosley:2002zz .

One of the noticeable features in the CCSN model is that large-scale
asymmetric neutrino emission emerges $>150$ms after bounce, that corresponds
to the timing of asymmetric shock expansion and the onset of PNS proper motion
(see Fig. 1 in Ref. Nagakura:2019evv ). We note that the asymmetric emission
is clearly anti-correlated between $\nu_{e}$ and $\bar{\nu}_{e}$, which is
attributed to the distribution of electron fraction ($Y_{e}$) in the vicinity
of PNS, and $\bar{\nu}_{e}$ tends to be more abundant in low $Y_{e}$
environments. As shown in Ref. Nagakura:2019sig , the increase of
$\bar{\nu}_{e}$ reduces the disparity between $\nu_{e}$ and $\bar{\nu}_{e}$
angular distributions, leading to enhance the possibility of $\nu$ELN
crossings. In the region with higher $Y_{e}$ environments, on the other hand,
$\nu_{e}$ becomes much higher than $\bar{\nu}_{e}$, indicating that FFI is
unlikely to occur.

In the CCSN model, $\nu$ELN angular crossings are observed rather stably at
$>200$ms (see Fig. 2 in Ref. Nagakura:2019sig ); hence, we employ three
different time snapshots for our ML training (200, 250, and 300ms after
bounce), in which there are both spatial regions with and without $\nu$ELN
angular crossings. It should also be mentioned that $\nu$ELN crossings are
also observed in PNS convective layer and in pre-shock regions, which are
consistent with previous studies Glas:2019ijo ; DelfanAzari:2019tez ;
Morinaga:2019wsv . For more detailed discussion about neutrino angular
distributions associated with arguments of $\nu$ELN crossings, we refer
readers to Ref. Nagakura:2019sig .

##  III ML algorithms

ML, at the crossroads of computer science and artificial intelligence, is
transforming how computers learn and make decisions from data. By unraveling
intricate patterns, ML drives progress across diverse domains, from image and
speech recognition to healthcare, finance, and autonomous vehicles.

Recent advancements have demonstrated the effective utilization of ML
algorithms for detecting $\nu$ELN crossings in CCSN and NSM simulations
Abbar:2023kta . In this context, we commence with a brief overview of the data
preparation and ML techniques employed in Ref. Abbar:2023kta . Subsequently,
we provide a comprehensive discussion of our research findings.

To effectively train and evaluate our ML algorithms, it is imperative to
possess a substantial dataset comprising labeled values for $I_{0}$ and
$I_{1}$ associated with $\nu_{e}$ and $\bar{\nu}_{e}$. These labels are
instrumental in discerning the presence or absence of $\nu$ELN crossing. It’s
worth highlighting that our current emphasis is primarily on the first two
moments. These moments are of particular interest as they are the ones
typically tracked directly in the simulation processes.

In order to train our ML algorithms, we partially employ two parametric
neutrino angular distributions which have beed widely used in the literature
Richers:2022dqa ; Wu:2021uvt ; Yi:2019hrp , namely, the maximum entropy
distribution defined as,

$f^{\rm{max-ent}}_{\nu}(\mu)=\exp[\eta+a\mu],$ (3)

and the Gaussian distribution,

$f^{\rm{Gauss}}_{\nu}(\mu)=A\exp[-\frac{(1-\mu)^{2}}{a}],$ (4)

with,

$f_{\nu}(\mu)=\int_{0}^{\infty}\int_{0}^{2\pi}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}f_{\nu}(\mathbf{p}).$
(5)

Here the parameters a, $\eta$, and $A$ determines the overall neutrino number
density and the shape of the neutrino distributions.

In addition, in order to improve the efficiency of our ML algorithms, we do
feature engineering and instead of $I_{0}$ and $I_{1}$ of $\nu_{e}$ and
$\bar{\nu}_{e}$ which are the provided information, we use,

$\alpha=I^{\bar{\nu}_{e}}_{0}/I^{\nu_{e}}_{0},\
F_{\nu_{e}}=I^{\nu_{e}}_{1}/I^{\nu_{e}}_{0},\ \mathrm{and}\
F_{\bar{\nu}_{e}}=I^{\bar{\nu}_{e}}_{1}/I^{\bar{\nu}_{e}}_{0},$ (6)

as the relevant features to be considered in the ML algorithms. This is
justified by bearing in mind that an overall normalisation factor does not
affect the occurrence of $\nu$ELN crossings.

To facilitate the training and evaluation of ML algorithms, it is essential to
partition the dataset into three distinct sets: i) The training set: This
subset is employed to train the ML algorithm, allowing it to learn patterns
and relationships within the data, ii) The development set: This set serves as
a tool for fine-tuning the algorithm’s hyperparameters, ensuring optimal
performance, and iii) The test set: This portion is dedicated to assessing the
ML method’s performance on previously unseen data, providing a reliable
measure of its effectiveness.

To comprehensively assess the effectiveness of our ML algorithm, we go beyond
mere accuracy, which can be a somewhat simplistic measure. Instead, we
consider also more detailed evaluations using precision and recall metrics,
defined as,

$\begin{split}&\mathrm{accuracy}=\frac{T_{p}+T_{n}}{T_{p}+T_{n}+F_{p}+F_{n}}\\\
&\mathrm{precision}=\frac{T_{p}}{T_{p}+F_{p}}\\\
&\mathrm{recall}=\frac{T_{p}}{T_{p}+F_{n}}\\\
&F_{1}=2\times\frac{\mathrm{precision}\times\mathrm{recall}}{\mathrm{precision}+\mathrm{recall}},\end{split}$
(7)

with $T(F)_{p(n)}$ denoting True (False) positive (negative) classifications.
A discerning reader will notice that the precision/recall metric informs us
about the reliability/detectability of classifications, while $F_{1}$ is their
harmonic mean. In this study, we opt for accuracy as the suitable metric
because we aim for equal sensitivity to the presence or absence of $\nu$ELN
crossings.

In this study we consider the following ML algorithms: i) Logistic Regression
(LR): a statistical classification algorithm that models the probability of a
binary outcome. LR turns out to be one of the most promising ML algorithm to
be used in detecting $\nu$ELN crossings Abbar:2023kta , ii) k-Nearest
Neighbors (KNN): an intuitive learning algorithm that classifies data points
based on the majority class of their k-nearest neighbors in the feature space,
iii) Support Vector Machine (SVM): a powerful algorithm that separates data
into classes by finding the hyperplane that maximizes the margin between them
in a high-dimensional space, and iv) Decision Tree (DT): a tree-like model
used for both classification and regression tasks, where the data is split
into subsets based on feature conditions, ultimately leading to a decision or
prediction.

There are two final aspect of the LR and SVM algorithms that requires a bit of
clarification. While LR incorporates the nonlinear logistic function, it
fundamentally operates as a linear classifier. Consequently, it cannot be
directly applied to the detection of $\nu$ELN crossings, which inherently
represents a non-linear problem Abbar:2023kta . To overcome this limitation,
it becomes necessary to undertake a preprocessing step involving non-linear
transformations and the creation of new features based on the original three
features involved in the problem. The degree of polynomial transformation,
being a hyper-parameter of this algorithm, plays a crucial role in this
process.

Regarding the SVM algorithm, we should mention that we employ the radial basis
function (RBF) kernel, defined as,
$\mathcal{K}(x,x^{\prime})=\exp(-\gamma||x-x^{\prime}||^{2})$. Here $\gamma$
is a hyper-parameter which is set to be $\gamma=100$ Abbar:2023kta .

In the next part, we present our results. In addition, to promote transparency
and collaboration, we have made our ML methodologies available on GitHub.

###  III.1 ML-based detection of $\nu$ELN crossings using the SN data

Table 1: A summary of the metric scores of the previously-trained ML algorithms (using artificial data) tested on the realistic dataset. This is to be compared with TABLE I. in Ref. Abbar:2023kta . Alongside each algorithm, one can find its corresponding accuracy score. | LR (n = 9) (68%) |  |
---|---|---|---
| precision | recall |  $F_{1}$-score
no crossing | 72% | 82% | 77%
crossing | 59% | 43% | 50%
| KNN (n=3) (77%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 77% | 89% | 83%
crossing | 75% | 55% | 63%
| SVM (87%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 98% | 81% | 89%
crossing | 75% | 98% | 85%
| Decision tree (71%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 74% | 84% | 79%
crossing | 63% | 48% | 55%
Figure 1:  The accuracy of the LR algorithms evaluated on the realistic
dataset for models trained on various training sets, with a focus on the
impact of polynomial degree of the nonlinear transformations. It is noteworthy
that LR models trained on the improved artificial training sets can achieve
comparable accuracies to those trained on realistic data at lower polynomial
degrees, implying simpler model structures. The black dashed line represents
the performance of the old LR model trained on artificial data when tested on
its dedicated test set.

We initially assess the performance of the pre-trained ML models of Ref.
Abbar:2023kta  (using artificial data) on the realistic dataset obtained from
the simulation. The metric scores for the performance of these pre-trained ML
algorithms are presented in TABLE. 1. It becomes evident that the performance
of ML models, which were initially trained on artificial data, degrades when
applied to realistic data. Notably, the accuracies of LR and SVM are of
paramount concern. Specifically, the LR model (here was considered with a
polynomial degree of 9) exhibits a significant deviation from its accuracy on
artificial data (as indicated in TABLE I of Ref. Abbar:2023kta ). This
indicates poor generalization of the old LR algorithm with a polynomial degree
$n=9$, a known issue referred to as high variance in the context of ML.

Another enlightening aspect centers on the SVM model’s exceptional
performance, consistently achieving high accuracy even when confronted with
previously unseen data. This serves as yet another compelling example of the
effectiveness of maximizing margins to separate classes, underscoring its
conceptual soundness and its ability to enhance the model’s overall
generalizability in ML.

However and as depicted in Fig. 1, the former LR model (red-curve) trained on
synthetic data, attains its peak performance around $n\simeq 2$ achieving an
accuracy of approximately 85%. In simpler terms, the more straightforward
models outperform their counterparts when it comes to $\nu$ELN detection on
previously unseen data. This serves as a prime illustration of the bias-
variance tradeoff within the realm of ML. As we see later on, it seems to be a
general observation that when considering LR, opting for nonlinear
transformations with lower polynomial degrees tends to yield better results.

While achieving an accuracy of $\sim 85\%$ is commendable, it’s important to
note that the performance of models developed using artificial datasets can
still be enhanced. The initial ML models were trained on data with $\alpha$
values ranging from 0.03 to 2.5, alongside random selections of $F_{\nu_{e}}$
and $F_{\bar{\nu}_{e}}$ in the (0, 1) range. However, this approach, while
suitable for an initial step, doesn’t align with realistic conditions. In
realistic simulations of CCSNe and NSM’s, it’s anticipated that
$F_{\nu_{e}}\lesssim F_{\bar{\nu}_{e}}$.

To address this issue, we enhance our ML model by training it with artificial
data while considering $F_{\nu_{e}}$ within the range of
$(0.6F_{\bar{\nu}_{e}},F_{\bar{\nu}_{e}})$. The performance of such a ML model
on realistic data is illustrated in Fig. 1 (red-dotted curve), revealing two
significant insights. Firstly, high variance is observed at large polynomial
degrees, indicating poor generalizability of the ML performance at those
polynomial degrees. Secondly, the ML model, trained on this improved
artificial dataset, achieves notably higher accuracy compared to the previous
version. This reaffirms the significance of a well-representative training
dataset for accurate testing.

In the final step, we enhance our ML model by incorporating real-world data
into the training set. It’s important to highlight that we don’t exclusively
rely on real data for training. This approach allows us to maintain
variability in the angular distributions of neutrinos. The performance of this
ML model is depicted by the blue curve in Fig. 1. Notably, it exhibits
exceptionally high accuracy across all polynomial degrees. However, drawing
from our previous experiences, we favor selecting $n=2$ due to its strong
potential for effective generalization to unseen data. Beyond its
generalization capabilities, opting for $n=2$ also brings the advantage of
reduced computational intensity when implementing the LR model in CCSN and NSM
simulations on the fly. In addition, In Fig. 2, we present the comprehensive
set of metric scores obtained from our improved LR model. It’s worth
highlighting that all these scores attain satisfactory values when the
polynomial degree is set to 2.

Figure 2:  The metric scores of the LR algorithm trained on a combination of the artificial and realistic datasets, as a function of the polynomial degree of the nonlinear transformations. It’s worth noting that the precision and recall scores typically exhibit opposing trends, a phenomenon commonly referred to as the precision-recall tradeoff within the field of ML. Table 2: A summary of the metric scores of ML algorithms trained on the combination of the realistic and artificial datasets, and then tested with the realistic data. Alongside each algorithm, one can find its corresponding accuracy score. | LR (n = 2) (94%) |  |
---|---|---|---
| precision | recall |  $F_{1}$-score
no crossing | 96% | 95% | 95%
crossing | 91% | 93% | 92%
| KNN (n=3) (98%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 98% | 99% | 99%
crossing | 98% | 97% | 98%
| SVM (97%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 98% | 98% | 98%
crossing | 96% | 97% | 97%
| Decision tree (99%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 99% | 99% | 99%
crossing | 98% | 98% | 98%

In Table 2, we present the performance results of our enhanced ML models,
which were trained using a combination of both realistic and artificial
datasets. The outcomes demonstrate that our ML models achieve exceptionally
high performance metrics. Notably, these scores surpass those obtained by the
ML model trained exclusively on artificial data, as shown in Table I of Ref.
Abbar:2023kta . This improvement can be attributed to the fact that the
artificial data includes noisy labels, as discussed in Ref. Abbar:2023kta ,
which was identified as a primary source of inaccuracies once the ML model is
tested on artificial data.

###  III.2 Detection of $\nu$ELN-XLN crossings

So far, our discussion has focused exclusively on detecting $\nu$ELN
crossings. However, in a broader context, it’s important to acknowledge that
the angular distributions of $\nu_{x}$ and $\bar{\nu}_{x}$ can be different in
CCSN and NSM environments. This difference becomes particularly pronounced
when we account for the potential creation of muons at the core of these
extreme astrophysical objects Bollig:2017lki ; Fischer:2020vie ; Guo:2020tgx .
Consequently, to accurately identify the occurrence of FFCs under the most
realistic conditions, we must shift our attention towards detecting $\nu$ELN-
XLN crossings rather than confining ourselves to the $\nu$ELN ones.

The distinction between the detection of $\nu$ELN-XLN crossings and $\nu$ELN
crossings presents several key differences. One of the most significant
distinctions lies in the increased complexity of required information. In the
context of our ML methods, this translates to an expansion in the number of
essential features. Specifically, we now necessitate seven features instead of
the previous three, namely $\alpha_{\nu_{e}}$, $\alpha_{\nu_{x}}$,
$\alpha_{\bar{\nu}_{x}}$, $F_{\nu_{e}}$, $F_{\bar{\nu}_{e}}$, $F_{\nu_{x}}$,
and $F_{\bar{\nu}_{x}}$, with
$\alpha_{\nu_{\beta}}=n_{\nu_{\beta}}/n_{\nu_{e}}$.

The increase of the number of features and the greater demand for information
significantly contribute to an elevated classification error in this context.
Notably, when $\nu_{x}$ and $\bar{\nu}_{x}$ exhibit disparities, the $\nu$ELN-
XLN profile can exhibit more intricate characteristics, e.g., it is
conceivable that even multiple crossings occur.

Table 3: A summary of the metric scores of the ML algorithms for $\nu$ELN-XLN crossing detection. Alongside each algorithm, one can find its corresponding accuracy score. | LR (n = 2) (88%) |  |
---|---|---|---
| precision | recall |  $F_{1}$-score
no crossing | 87% | 89% | 88%
crossing | 88% | 86% | 87%
| KNN (n=3) (88%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 89% | 88% | 89%
crossing | 88% | 89% | 88%
| SVM (88%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 92% | 84% | 88%
crossing | 85% | 93% | 89%
| Decision tree (87%) |  |
| precision | recall |  $F_{1}$-score
no crossing | 87% | 87% | 87%
crossing | 87% | 87% | 87%

In order to train our ML models, we use artificial distributions for
neutrinos, given the fact that labeled data regarding the existence of
$\nu$ELN-XLN crossings are not available. In order to prepare our data, we
consider $\alpha_{\nu_{e}}$ $\alpha_{\nu_{x}(\bar{\nu}_{x})}$ to be in the
range of (0., 2.5) and (0., 3.), respectively. Also we assume an allowed
maximum 40% difference between $\nu_{x}$ and $\bar{\nu}_{x}$ quantities. This
is consistent with the observation that the difference between $\nu_{x}$and
$\bar{\nu}_{x}$ should be subdominant in realistic simulations Bollig:2017lki
. In addition, we keep in mind what we learned previously that the data should
be enough representative of the realistic data. Thus, we also respect the
hierarchy $F_{\nu_{e}}\lesssim F_{\bar{\nu}_{e}}\lesssim
F_{\nu_{x}(\bar{\nu}_{x})}$.

The performance of our ML models in detecting $\nu$ELN-XLN crossings is
presented in Table 3. Notably, the overall performance lags behind that of
$\nu$ELN crossing detection. This disparity can be attributed to the presence
of intricate patterns governing the crossings and an increase in label noise.

##  IV DISCUSSION AND OUTLOOK

Recent advancements have showcased the remarkable capabilities of ML in
identifying the $\nu$ELN crossings in the CCSN and NSM simulations
Abbar:2023kta . In this study, we have propelled prior research in two pivotal
and distinctive directions. Firstly, we have subjected ML models to the
rigorous test of real-world data acquired from CCSN simulations, where the
intricate problem of neutrino transport is addressed through the comprehensive
Boltzmann equation. Secondly, we have expanded our ML techniques to encompass
the detection of $\nu$ELN-XLN crossings, accommodating situations where there
may exist distinctions between $\nu_{x}$ and $\bar{\nu}_{x}$.

Using realistic CCSN data, we have demonstrated that the simpler models
consistently outperform their more complex counterparts in the context of
$\nu$ELN detection when applied to previously unseen data. This provides a
clear and compelling example of the bias-variance tradeoff within the domain
of ML. Specifically, it was observed that the LR model performs most
effectively when a polynomial transformation of degree $n=2$ is applied, as
opposed to the previously suggested degree of $n=9$ which was based on
artificial data. This underscores the importance of considering the complexity
of models and their suitability for real-world data, highlighting that
sometimes, a simpler approach can yield superior results.

We demonstrate a significant enhancement in model performance when utilizing
artificial datasets by aligning the parameter space of the synthetic data with
the realism expected in CCSN and NSM simulations. Specifically, we adhere to
the hierarchy $F_{\nu_{e}}\lesssim F_{\bar{\nu}_{e}}\lesssim
F_{\nu_{x}(\bar{\nu}_{x})}$, which mirrors the conditions anticipated in these
astrophysical events. This deliberate consideration in the preparation of
artificial data results in model performance that rivals that of ML models
trained on realistic data, at least within a certain parameter range.

We have further fortified our ML models by integrating real-world data into
the training set, and this enhancement has yielded remarkable generic accuracy
in out ML models. Based on our meticulous observations, we are inclined to
assert that the LR model with a polynomial degree of $n=2$ stands out as the
optimal choice for detecting FFCs in CCSN and NSM simulations. This choice not
only exhibits exceptional generalization capabilities for unseen data but also
offers the distinct advantage of reducing computational overhead when
deploying the LR model in real-time CCSN and NSM simulations.

We have also developed ML models to identify neutrino flavor crossings in the
$\nu$ELN-XLN distributions. This is particularly relevant because the angular
distributions of $\nu_{x}$ and $\bar{\nu}_{x}$ can exhibit variations in CCSN
and NSM environments. Unlike the simpler task of detecting $\nu$ELN crossings,
detecting $\nu$ELN-XLN crossings introduces a higher level of complexity due
to the need for more information. In the context of our ML methods, this
complexity manifests as an increase in the number of essential features. This
augmented feature set substantially contribute to an elevated classification
error in this specific context.

In summary, our study significantly expands upon prior research, allowing for
more confident utilization of ML methods in detecting FFCs. However, there
remain crucial avenues for exploration. Specifically, our current analysis
focuses on crossings occurring in the zenith angle ($\mu$), when the angular
distribution is integrated over $\phi_{\nu}$. Nevertheless, as demonstrated in
previous references, a substantial fraction of $\nu$ELN crossings may be
exclusively in $\phi_{\nu}$, exhibiting non-axisymmetric behavior
Nagakura:2019sig . Thus, it is imperative to develop ML techniques capable of
capturing these non-axisymmetric crossings. Closely related to this issue is
the exploration of FFCs in rotating CCSN models Harada:2021ata , as the
prominence of non-axisymmetric features in such models could influence the
performance of ML algorithms. Given the proven and remarkable versatility and
effectiveness of ML in this context, implementing these measures will further
improve the detection of FFCs in CCSN and NSM simulations.

## Acknowledgments

We would like to thank Georg Raffelt for useful discussions. We would also
like to express our sincere gratitude to the Institute of Physics of Academia
Sinica for their warm hospitality and support during the _Focus Workshop on
Collective Oscillations and Chiral Transport of Neutrinos_ , where the
inception of this project took place. Their gracious hosting and collaborative
environment played a pivotal role in shaping the foundation of our work. S.A.
was supported by the German Research Foundation (DFG) through the
Collaborative Research Centre “Neutrinos and Dark Matter in Astro- and
Particle Physics (NDM),” Grant SFB-1258, and under Germany’s Excellence
Strategy through the Cluster of Excellence ORIGINS EXC-2094-390783311. H. N.
was supported by Grant-inAid for Scientific Research (23K03468). The numerical
simulations for CCSNe were carried out by using ”K”, ”Fugaku”, and the
highperformance computing resources of ”Flow” at Nagoya University ICTS
through the HPCI System Research Project (Project ID: 220173, 220047, 220223,
230033, 230204, 230270). We would also like to acknowledge the use of the
following softwares: Scikit-learn pedregosa2011scikit , Matplotlib Matplotlib
, Numpy Numpy , SciPy SciPy , and IPython IPython .

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