# Recognition of Cardiac MRI Orientation via Deep Neural Networks and a Method
to Improve Prediction Accuracy

Houxin Zhou

###### Abstract

In most medical image processing tasks, the orientation of an image would
affect computing result. However, manually reorienting images wastes time and
effort. In this paper, we study the problem of recognizing orientation in
cardiac MRI and using deep neural network to solve this problem. For multiple
sequences and modalities of MRI, we propose a transfer learning strategy,
which adapts our proposed model from a single modality to multiple modalities.
We also propose a prediction method that uses voting. The results shows that
deep neural network is an effective way in recognition of cardiac MRI
orientation and the voting prediction method could improve accuracy.

##  1 Introduction

When medical images were stored, they may have different image orientations.
In the further segmentation or computing, this difference may affect the
results, since current deep neural network (DNN) systems generally only take
the input and output of images as matrices or tensors, without considering the
imaging orientation and real world coordinate. So it is crucial to recognize
it before the further computing. This work is aimed to provide a study of the
Cardiac Magnetic Resonance (CMR) image orientation, for fitting the coordinate
system of human reality, and to develop an efficient method for recognition of
the orientation.

Deep neural network has performed outstandingly in computer vision and
gradually replaced the traditional methods. DNN also take a important role in
medical image processing, such as image segmentation[3] and myocardial
pathology analysis[2]. For CMR images, standardization of all the images is a
prerequisite for further computing tasks based on DNN-based methodologies.

Most studies in the field of medical image processing have only focused on the
further computing, so they have to spend a lot of manpower to do the
preprocess. If we can auto adjust the images, it will save lots of time.
Nevertheless, recognizing the orientation of different modality CMR images and
adjusting them into standard format could be as challenging as the further
computing tasks[1]. In a broad sense, recognition orientation is also a kind
of image classification task, so DNN is of no doubts an effective way to solve
this problem. In this work, we still use DNN as our main method.

In most image classification problem like ImageNet, we do some transformation
to the image but these transformation do not change the label, for example we
rotate a dog image and it’s still a dog image. However, the orientation could
be changed if we do transformation like flipping to the images. In this work,
we utilize this character to built a predicting model. Combining DNN and
predicting model, we built a framework for recognition of Orientation.

This work is aimed at designing a DNN-based approach to achieve orientation
recognition for multiple CMR modalities. Figure 1 presents the pipeline of our
proposed method. The main contributions of this work are summarized as
follows:

Figure 1: The pipeline of our proposed method. Firstly, we do some
transformatioin to the image(See 2.1 and 2.3). Then, the images are taken as
input to the DNN model and generate orientations. Finally, we apply inverse
transformation to these orientations and vote to get the result.

  1. 1.

We propose a scheme to standardize the CMR image orientation and categorize
all the orientations for classification.

  2. 2.

We present a DNN-based orientation recognition method for CMR image and
transfer it to other modalities.

  3. 3.

We propose a predicting method to improve the accuracy for orientation
recognition.

##  2 Method

In this section, we introduce our proposed method for orientation recognition.
Our proposed method is based on Deep Neural Network which was proved effective
in image classification. In CMR Image Orientation Categorization, we improved
the predicting accuracy by the following four steps. Firstly, we apply
invertible operators to the image to get another 7 images. Then we predict
these images and get 8 orientations. Finally, we use inverse transformation to
these orientations and then vote to get the result.

###  2.1 CMR Image Orientation Categorization

Due to different data sources and scanning habits, the orientation of
different CMR images may be different, and the orientation vector
corresponding to the image itself may not correspond correctly. This may cause
problems in tasks such as image segmentation or registration. Taking a 2D
image as an example, we set the orientation of an image as the initial image
and set the four corners of the image to $\boxed{\begin{array}[]{cc}1&2\\\
3&4\\\ \end{array}}$ , Then the orientation of the 2D MR image may have the
following 8 variations, which is listed in Table 1.

For each image $X_{t}$ from dataset, the target is to find the correct
orientation from 8 classes. We denote the correct orientation of image $X_{t}$
as $i_{t}$ and denote the correctly adjusted image $X_{t}$ as $Y_{t}$ If we
view each orientation as a function $f$, we can get a function set
$\\{f_{i},i=0\dots 7\\}$ and $(X_{t},Y_{t},i_{t})$ satisfy the equation
$f_{i_{t}}(Y_{t})=X_{t}$. In the following, function set $\\{f_{i},i=0\dots
7\\}$ is referred to as $F$.

Table 1: Orientation categorization of 2D CMR Images. Here, sx, sy and sz
respectively denote the size of image in X-axis, Y-axis and Z-axis.

No. | Operation | Image | Correspondence of coordinates
---|---|---|---
0 | initial state | $\boxed{\begin{array}[]{cc}1&2\\\ 3&4\\\ \end{array}}$ | Target[x,y,z]=Source[x,y,z]
1 | horizontal flip | $\boxed{\begin{array}[]{cc}2&1\\\ 4&3\\\ \end{array}}$ | Target[x,y,z]=Source[sx-x,y,z]
2 | vertical flip | $\boxed{\begin{array}[]{cc}3&4\\\ 1&2\\\ \end{array}}$ | Target[x,y,z]=Source[x,sy-y,z]
3 | Rotate $180^{\circ}$ clockwise | $\boxed{\begin{array}[]{cc}4&3\\\ 2&1\\\ \end{array}}$ | Target[x,y,z]=Source[sx-x,sy-y,z]
4 | Flip along the upper left-lower right corner | $\boxed{\begin{array}[]{cc}1&3\\\ 2&4\\\ \end{array}}$ | Target[x,y,z]=Source[y,x,z]
5 | Rotate $90^{\circ}$ clockwise | $\boxed{\begin{array}[]{cc}3&1\\\ 4&2\\\ \end{array}}$ | Target[x,y,z]=Source[sx-y,x,z]
6 | Rotate $270^{\circ}$ clockwise | $\boxed{\begin{array}[]{cc}2&4\\\ 1&3\\\ \end{array}}$ | Target[x,y,z]=Source[y,sy-x,z]
7 | Flip along the bottom left-top right corner | $\boxed{\begin{array}[]{cc}4&2\\\ 3&1\\\ \end{array}}$ | Target[x,y,z]=Source[sx-y,sy-x,z]

###  2.2 Deep Neural Network

Suppose given image $X_{t}$, $X_{t}$ is then normalized. We denote the
processed $X_{t}$ as $X_{t}^{\prime}$. CNN take the $(X_{t}^{\prime},i_{t})$
as input. In the proposed framework, the orientation recognition network
consists of 3 convolution layers and 2 fully connected layers. The orientation
predicted is denoted as $\hat{O_{i}}$. We use the standard categorical loss to
calculate the loss between predicted orientation $\hat{O_{i}}$ and orientation
label $O_{i}$. The orientation loss is formulated as below:

$L_{orientation}=\sum_{i=1}^{8}O_{i}log(\hat{O_{i}})$ (1)

###  2.3 Improved Prediction Method

As we can see in 2.1, we regard label $i_{t}$ as function $f_{i_{t}}$. It can
be easily proved that $f_{i}\circ f_{j}\in F$ for any $i$ and $j$, so we can
not only view $f_{i}$ as a function but an operator whose define domain and
value domain are both $F$. For convenience, we denote the operator $f_{i}$ as
$g_{i}$. Surpprisingly, we can prove that operator $g_{i}$ is a surjection in
$F$ for any $i$, which can be simply explained by the following matrix $A$. In
matrix $A$, $A_{ij}=k$ means $g_{i}(f_{j})=f_{k}$.

$A=\left(\begin{array}[]{cccccccc}0&1&2&3&4&5&6&7\\\ 1&0&3&2&5&4&7&6\\\
2&3&0&1&6&7&4&5\\\ 3&2&1&0&7&6&5&4\\\ 4&6&5&7&0&2&1&3\\\ 5&7&4&6&1&3&0&2\\\
6&4&7&5&2&0&3&1\\\ 7&5&6&4&3&1&2&0\end{array}\right)$

Because $F$ is a finate set, $g_{i}$ is a injection and invertible. For any
operator $g_{i}$, it exist an inverse operator and we denote it as
$g^{-}_{i}$. Operator $g_{i}^{-}$ is also a surjection and injection in $F$,
which can be simply explained by the following matrix $A^{-}$. In matrix
$A^{-}$, $A^{-}_{ij}=k$ means $g_{i}^{-}(f_{j})=f_{k}$. For simplification, we
omit $f$ and use $g_{i}^{-}(j)=k$ to express the results above.

$A^{-}=\left(\begin{array}[]{cccccccc}0&1&2&3&4&5&6&7\\\ 1&0&3&2&5&4&7&6\\\
2&3&0&1&6&7&4&5\\\ 3&2&1&0&7&6&5&4\\\ 4&6&5&7&0&2&1&3\\\ 6&4&7&5&2&0&3&1\\\
5&7&4&6&1&3&0&2\\\ 7&5&6&4&3&1&2&0\end{array}\right)$

Based on the above premise, we built the predicting method by the follow 4
steps. Figure 1 shows the method by a flow chart.

  1. 1.

Apply $(f_{0},f_{1}\dots,f_{7})$ to the image $X_{t}$ to get 8 images. We
denote these images as $(X_{t0},X_{t1},\dots,X_{t7})$

  2. 2.

Take these 8 images as input to DNN and get 8 orientations
$(i_{t0},i_{t1},\dots,i_{t7})$.

  3. 3.

Apply $(g^{-}_{0},g^{-}_{1},\dots,g^{-}_{7})$ to these 8 labels and get
another 8 labels
$(g^{-}_{0}(i_{t_{0}}),g^{-}_{1}(i_{t1}),\dots,g^{-}_{7}(i_{t7}))$.

  4. 4.

The labels which occur most in
$(g^{-}_{1}(i_{t_{0}}),g^{-}_{2}(i_{t1}),\dots,g^{-}_{7}(i_{t7}))$ is the
final result.

##  3 Experiment

###  3.1 Experiment Setup

We evaluate orientation recognition network on the MyoPS dataset[3, 2]. The
MyoPS dataset provides the three-sequence Cardiac Magnetic Resonance (LGE, T2
and C0) and three anatomy masks, including myocardium (Myo), left ventricle
(LV), and right ventricle (RV), some of the three-sequence Cardiac Magnetic
Resonance is shown as Figure 2 . MyoPS further provides two pathology masks
(myocardial infarct and edema) from the 45 patients. For the simplified
orientation recognition network, we train model for single modality on the
MyoPS dataset, then transfer the model to other modalities. For each sequence,
we resample each slice of each 3d image and the corresponding labels to an in-
plane resolution of 1.367 × 1.367 mm.

(a) C0

(b) LGE

(c) T2

Figure 2: the three-sequence Cardiac Magnetic Resonance

We divide slices into three sub-sets, i.e., the training set, validation set
and test set, at the ratio of 50%, 30% and 20%. Three sub-sets don’t have
slices from same patient. Then, for each standard 2d image, we apply all
function from $F$ to it to expand dataset. For training set, image slices are
cropped or padded to $256\times 256$ for the orientation recognition network,
and apply random augmentation. For test set and validation set, the images are
only resized to $256\times 256$.

###  3.2 Orientation Recognition Network

During each training iteration, a batch of the three-channel images X’ is fed
into the orientation recognition network. Then, the network outputs the
predicted orientation network.

Figure 3 and Table 2 shows the training process and accuracy of the three
sequences on test set. The results show that the model get quite high accuracy
in three modalities. Howeverm the size of test data is small, random factors
influence the result heavily. In the following, we redivide the data, retrain
the model, and analyses sensitivity of the model.

(a) C0

(b) LGE

(c) T2

Figure 3: Training images Table 2: Accuracy of the three Modalities Modality | Accuracy | Description
---|---|---
C0 | 1.0 | Pre-train
LGE | 1.0 | Transfer learning
T2 | 1.0 | Transfer learning

###  3.3 Sensitivity analysis

When using deep learning method to solve the problem, the volume of data is
always the most important factor. In medical image processing, it is difficult
to get a lot of data, so analysing the sensitivity of model is necessary. We
redivided the data set into training set and test set 5 times, and the
proportion of training set was 60%, 50%, 40%, 30% and 20% respectively. In
each divided data set, we retrain the model in training set and compute
accuracy in test set.

The accuracy variation is shown in Table 3. There is not a distinct difference
among the accuracy while the volume of data decrease. Therefore DNN is a
suitable method in CMR orientation recognition with high accuracy and low
sensitivity.

Table 3: Accuracy of different training data size, modalities and prediction methods. The left column is the ratio of training data. Improved prediction means the method we proposed in 2.3 while Direct prediction means inputing the resized image to the DNN directly | Improved prediction | Direct prediction
---|---|---
| C0 | LGE | T2 | C0 | LGE | T2
60% | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.998
50% | 1.0 | 1.0 | 1.0 | 1.0 | 0.991 | 0.999
40% | 0.974 | 1.0 | 1.0 | 0.967 | 0.996 | 0.996
30% | 1.0 | 0.979 | 1.0 | 1.0 | 0.975 | 0.997
20% | 0.982 | 0.960 | 0.986 | 0.978 | 0.953 | 0.983

###  3.4 Comparison between improved prediction and direct prediction

Table 3 shows the difference of accuracy between improved prediction and
direct prediction. We can find that improved prediction always have higher
accuracy in our experiment. However, it is not inevitable, because the vote
may make the original correct decision wrong. Sometimes, the improved
prediction have a lower accuracy, but in an average sense, the improved
prediction is better than direct prediction.

##  4 Conclusion

DNN model get quite high accuracy in recognition of CMR image orientation and
transfer learning make it easy to be transferred to other modalities . Thanks
to the data expansion and augmentation, the model only need a few data. The
improved prediction we proposed further increase the accuracy. We are sure
that DNN model combining with transfer learning and improved prediction can be
used in other recognition of orientation tasks.

## References

  * [1] Ke Zhang and Xiahai Zhuang.  Recognition and Standardization of Cardiac MRI Orientation via Multi-tasking Learning and Deep Neural Networks.  In Xiahai Zhuang and Lei Li, editors, Myocardial Pathology Segmentation Combining Multi-Sequence Cardiac Magnetic Resonance Images, Lecture Notes in Computer Science, pages 167–176, Cham, 2020. Springer International Publishing.
  * [2] Xiahai Zhuang.  Multivariate mixture model for cardiac segmentation from multi-sequence mri.  In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 581–588. Springer, 2016.
  * [3] Xiahai Zhuang.  Multivariate mixture model for myocardial segmentation combining multi-source images.  IEEE transactions on pattern analysis and machine intelligence, 41(12):2933–2946, 2019.


# Application of Neural Networks for the Reconstruction of Supernova Neutrino
Energy Spectra Following Fast Neutrino Flavor Conversions

Sajad Abbar  0000-0001-8276-997X Max-Planck-Institut für Physik (Werner-
Heisenberg-Institut), Boltzmannstr. 8, 85748 Garching, Germany Meng-Ru Wu
0000-0003-4960-8706 Institute of Physics, Academia Sinica, Taipei, 11529,
Taiwan  Institute of Astronomy and Astrophysics, Academia Sinica, Taipei,
10617, Taiwan  Physics Division, National Center for Theoretical Sciences,
Taipei 10617, Taiwan  Zewei Xiong  0000-0002-2385-6771 GSI Helmholtzzentrum
für Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt, Germany

###### Abstract

Neutrinos can undergo fast flavor conversions (FFCs) within extremely dense
astrophysical environments such as core-collapse supernovae (CCSNe) and
neutron star mergers (NSMs). In this study, we explore FFCs in a _multi-
energy_ neutrino gas, revealing that when the FFC growth rate significantly
exceeds that of the vacuum Hamiltonian, all neutrinos (regardless of energy)
share a common survival probability dictated by the energy-integrated neutrino
spectrum. We then employ physics-informed neural networks (PINNs) to predict
the asymptotic outcomes of FFCs within such a multi-energy neutrino gas. These
predictions are based on the first two moments of neutrino angular
distributions for each energy bin, typically available in state-of-the-art
CCSN and NSM simulations. Our PINNs achieve errors as low as $\lesssim 6\%$
and $\lesssim 18\%$ for predicting the number of neutrinos in the electron
channel and the relative absolute error in the neutrino moments, respectively.

††preprint: MPP-2024-12

##  I Introduction

Core-collapse supernovae (CCSNe) and neutron star mergers (NSMs) represent
some of the most extreme astrophysical settings, where neutrino emission plays
a crucial role Burrows:2020qrp ; Janka:2012wk ; Foucart:2022bth ;
Kyutoku:2021icp ; Colgate:1966ax ; Lattimer:1974slx ; mirizzi2016supernova .
Within the dense and extreme conditions of these environments, neutrinos
undergo a complex flavor conversion process triggered by their coherent
forward scatterings with the dense background neutrino gas pantaleone:1992eq ;
sigl1993general ; Pastor:2002we ; duan:2006an ; duan:2006jv ; duan:2010bg ;
Mirizzi:2015eza ; volpe2023neutrinos .

One of the latest advancements in this field involves the discovery of _fast_
flavor conversions (FFCs), which can take place on scales significantly
shorter than those anticipated in the vacuum Sawyer:2005jk ; Sawyer:2015dsa ;
Chakraborty:2016lct ; Izaguirre:2016gsx ; Capozzi:2017gqd ; Wu:2017qpc ;
Wu:2017drk ; Abbar:2017pkh ; Abbar:2018beu ; Capozzi:2018clo ; Martin:2019gxb
; Abbar:2018shq ; Abbar:2019zoq ; Capozzi:2019lso ; George:2020veu ;
Johns:2019izj ; Martin:2021xyl ; Tamborra:2020cul ; Sigl:2021tmj ;
Morinaga:2021vmc ; Nagakura:2021hyb ; Sasaki:2021zld ; Padilla-Gay:2021haz ;
Xiong:2021dex ; Capozzi:2020kge ; Abbar:2020qpi ; Capozzi:2020syn ;
DelfanAzari:2019epo ; Harada:2021ata ; Abbar:2021lmm ; Just:2022flt ; Padilla-
Gay:2022wck ; Capozzi:2022dtr ; Zaizen:2022cik ; Shalgar:2022rjj ;
Kato:2022vsu ; Zaizen:2022cik ; Bhattacharyya:2020jpj ; Wu:2021uvt ;
Richers:2021nbx ; Richers:2021xtf ; Dasgupta:2021gfs ; Capozzi:2022dtr ;
Nagakura:2022kic ; Ehring:2023lcd ; Ehring:2023abs ; Xiong:2023upa ;
Fiorillo:2023hlk ; Nagakura:2023wbf ; Martin:2023gbo ; Fiorillo:2023mze ;
Grohs:2023pgq . A condition both necessary and sufficient for the occurrence
of FFCs is that 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, $G_{\rm{F}}$ represents the
Fermi coupling constant, $E_{\nu}$, $\theta_{\nu}$, and $\phi_{\nu}$ are the
neutrino energy, the zenith, and azimuthal angles of the neutrino velocity,
respectively. The $f_{\nu}$’s are the neutrino occupation numbers of different
flavors, with $\nu_{x}$ and $\bar{\nu}_{x}$ denoting the heavy-lepton flavor
of neutrinos and antineutrinos. In this study, as also commonly observed in
state-of-the-art CCSN and NSM simulations, we assume that $\nu_{x}$ and
$\bar{\nu}_{x}$ have similar angular distributions. The expression in Eq. (1)
then transforms into the conventional definition of the neutrino electron
lepton number, $\nu$ELN.

The occurrence of FFCs on much shorter scales compared to typical
hydrodynamical simulations of CCSNe and NSMs, makes their integration into the
simulations a formidable task. One prospective approach includes performing
short scale simulations of FFCs and then extrapolating the insights gained to
inform broader hydrodynamic simulations xiong2020potential ; George:2020veu ;
Li2021a ; Just:2022flt ; Fernandez:2022yyv ; Ehring:2023abs ; Ehring:2023lcd .
Given this, there has been a body of research on the assessment of FFC
outcomes in local dynamical simulations with periodic boundary conditions
Bhattacharyya:2020dhu ; Bhattacharyya:2020jpj ; Wu:2021uvt ; Richers:2021nbx ;
Zaizen:2021wwl ; Richers:2021xtf ; Bhattacharyya:2022eed ; Grohs:2022fyq ;
Abbar:2021lmm ; Richers:2022bkd ; Zaizen:2022cik ; Xiong:2023vcm , where
quasistationary flavor states have been observed in the neutrino gas. In
particular, it has been demonstrated that such states can be accurately
described by analytical formulations Xiong:2023vcm .

Despite this, incorporating FFCs into CCSN and NSM simulations is still
challenging. The obstacle arises from the need for complete angular
distributions of neutrinos, a demanding task in computationally intensive
simulations.

As an alternative, advanced simulations often simplify neutrino transport
using a limited set of angular distribution moments Shibata:2011kx ;
Cardall:2012at ; thorne1981relativistic . In a multi-energy neutrino gas, one
can define the radial moments (with a focus on axisymmetric crossings) for
each energy bin as,

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

with $E_{\nu,i}$ and $\Delta E_{\nu,i}$ being the mean energy and the width of
the $i$-th energy bin, where $I_{0,i}=n_{\nu,i}$ is the number of neutrinos in
that specific bin. Following these moments allows for a computationally more
manageable treatment of the neutrino transport. In practice, simulations
typically offer only $I_{0,i}$ and $I_{1,i}$. The challenge is then to
determine the ultimate values of $I_{0,i}$ and $I_{1,i}$ following FFCs. Note
that the energy-integrated $I$’s can be simply obtained by a summation over
$I_{i}$’s.

Recently, we showed that the asymptotic outcomes of FFCs in a single-energy
neutrino gas in the moments scenario can be successfully predicted by using
artificial neural networks (NNs) Abbar:2023ltx . In particular, we employed
physics-informed neural networks (PINNs), where the learning and performance
of the NN can be enhanced with the utilization of the domain knowledge
karniadakis2021physics ; cuomo2022scientific ; raissi2019physics . Our
findings demonstrated the efficacy of a single hidden layer PINN, achieving a
remarkable accuracy for the prediction of the asymptotic values of $I_{0}$ and
$I_{1}$ in a single-energy neutrino gas.

In this paper, we extend our prior research by considering a multi-energy
neutrino gas, which is considered a more realistic scenario. We demonstrate
that when the FFC growth rate surpasses that of the vacuum Hamiltonian
significantly, all neutrinos, irrespective of energy, share a common survival
probability as dictated by the energy-integrated neutrino spectrum, consistent
with the findings of Ref. Martin:2021xyl .

To predict the asymptotic outcome of FFCs, we employ a PINN. This PINN
utilizes critical information derived from the initial (anti)neutrino zeroth
and first moments, considering both the energy-integrated neutrino spectra and
a specific neutrino energy bin. Consequently, it produces the corresponding
moments specific to the asymptotic outcome of FFCs for that energy bin. Our
findings highlight the effectiveness of a single hidden layer PINN, achieving
remarkable accuracy in predicting the asymptotic values of $I_{0}$ and $I_{1}$
for each neutrino energy bin.

This paper is organized as follows. In Sec. II, we begin by providing an
overview of our simulations concerning FFCs in a multi-energy neutrino gas. We
also elaborate on the assumptions made in deriving the outcomes of FFCs.
Moving to Sec. III, we describe the architecture of our NNs, elaborating on
the necessary feature engineering and the implementation of a tailored loss
function. Furthermore, we present and discuss our results in this section.
Finally, our findings are summarized, and conclusions are presented in Sec.
IV.

##  II FFCs in a multi-energy neutrino gas

In this section, we present the results of our simulations of FFCs in a multi-
energy neutrino gas. Essentially, when the growth rate of FFCs, $\kappa$,
significantly surpasses the vacuum frequency, $\omega$, i.e.
$\kappa\gg\omega$, one anticipates that neutrino energy becomes
inconsequential to their flavor evolution. Here, $\omega\equiv\delta
m^{2}/(4E_{\nu})$ with $\delta m^{2}$ being the squared neutrino mass
difference. This suggests that in such circumstances, all neutrinos should
experience identical survival probabilities dictated solely by the energy-
integrated neutrino spectrum, effectively making the energy irrelevant.

The condition $\kappa\gg\omega$ could be expected to be met in a dense
neutrino gas provided that $\lambda\gg\omega$ 111Note that this is not true if
one considers an oversimplified model (that constraints the activation of
inhomogeneous unstable modes), or if the $\nu$ELN crossing is too
narrow/shallow. In such situations, the growth rate of FFCs could be
significantly suppressed so that $\kappa\gg\omega$ does not hold anymore
DedinNeto:2023ykt ., where $\lambda=\sqrt{2}G_{\rm F}n_{\nu_{e}}$ with
$n_{\nu_{e}}$ being here the initial $\nu_{e}$ number density of the neutrino
gas. A crucial exception arises when the neutrino gas lepton asymmetry ratio
defined as,

$\alpha=n_{\bar{\nu}_{e}}/n_{\nu_{e}},$ (3)

is extremely close to unity, which already implies that flavor equipartition
should occur on short scales in the neutrino gas, even in the absence of FFCs
(see the discussion in Ref. abbar2020fast ).

In the following, we first demonstrate that when $\lambda\gg\omega$, all
neutrinos (with different energies) experience identical survival
probabilities. Subsequently, we discuss an analytical representation of the
survival probabilities, which will be useful for our PINN calculations.

###  II.1 Results of the simulations

We consider a multi-energy and multi-angle neutrino gas in a 1D box, extending
the framework outlined in Ref. Wu:2021uvt . Our model assumes translation
symmetry along the $x$ and $y$ axes, axial symmetry around the $z$ axis, and
employs periodic boundary conditions in the $z$ direction. We also take two
flavor approximation, exclude the consideration of neutrino-matter forward
scattering, and assume that the system consists of (anti)neutrinos of electron
flavor whose energy and angular distribution is spatially homogeneous in the
beginning for simplicity. Under these assumptions, the evolution of the
normalized neutrino and antineutrino density matrices,
$\varrho(t,z,\omega,\mu)$ and $\bar{\varrho}(t,z,\omega,\mu)$ is governed by
the following equations of motion,

$\displaystyle(\partial_{t}+\mu\partial_{z})\varrho(t,z,\omega,\mu)=-i[H(t,z,\omega,\mu),\varrho(t,z,\omega,\mu)],$
$\displaystyle(\partial_{t}+\mu\partial_{z})\bar{\varrho}(t,z,\omega,\mu)=-i[\bar{H}(t,z,\omega,\mu),\bar{\varrho}(t,z,\omega,\mu)],$
(4)

where $\mu$ here represents the neutrino velocity in the $z$ direction. The
Hamiltonian $H(t,z,\omega,\mu)$ and $\bar{H}(t,z,\omega,\mu)$ are given by
$H(t,z,\omega,\mu)=H_{\rm vac}(\omega)+H_{\nu\nu}(t,z,\mu)$ and
$\bar{H}(t,z,\omega,\mu)=H_{\rm vac}(\omega)-H^{*}_{\nu\nu}(t,z,\mu)$, where

$H_{\rm vac}=\omega\left[\begin{array}[]{cc}-\cos 2\theta_{\rm eff}&\sin
2\theta_{\rm eff}\\\ \sin 2\theta_{\rm eff}&\cos 2\theta_{\rm eff}\\\
\end{array}\right],$ (5)

with $\theta_{\rm eff}$ the effective vacuum mixing angle, and

$\displaystyle H_{\nu\nu}$ $\displaystyle(t,z,\mu)=\leavevmode\nobreak\
\lambda\int_{-1}^{1}\mathrm{d}\mu^{\prime}\int_{0}^{\infty}\mathrm{d}\omega^{\prime}(1-\mu\mu^{\prime})\times$
(6)
$\displaystyle\left[f_{\nu}(\omega^{\prime},\mu^{\prime})\varrho(t,z,\omega^{\prime},\mu^{\prime})-\alpha
f_{\bar{\nu}}(\omega^{\prime},\mu^{\prime})\bar{\varrho}^{*}(t,z,\omega^{\prime},\mu^{\prime})\right].$

Here, the neutrino distribution functions are similar to those introduced in
Eq. (1) except that they are integrated over $\phi_{\nu}$ and now normalized
by $\int\mathrm{d}\mu\mathrm{d}\omega f_{\nu(\bar{\nu})}(\omega,\mu)=1$.

For the specific simulation discussed below, we take a 1D box of size
$L=1200\lambda^{-1}$ with $\alpha=0.9$. The neutrino distribution function is
parameterized by

$f_{\nu}(\omega,\mu)\propto\omega^{-(\chi_{\nu}+2)}\exp{\left[-\frac{(\mu-1)^{2}}{2\sigma_{\nu}^{2}}-\frac{(1+\chi_{\nu})\delta
m^{2}}{4\omega\langle E_{\nu}\rangle}\right]},$ (7)

with $\sigma_{\nu}=0.6$, $\sigma_{\bar{\nu}}=0.5$, $\chi_{\nu}=3.2$,
$\chi_{\bar{\nu}}=4.5$, $\langle E_{\nu}\rangle=10$ MeV, and $\langle
E_{\bar{\nu}}\rangle=12$ MeV. For the vacuum mixing parameters, we set
$\omega/\lambda=10^{-4}$ for $E_{\nu}=1$ MeV with $\theta_{\rm eff}=10^{-5}$.
In order to introduce small inhomogeneity to the system, we follow Ref.
Wu:2021uvt  to assign perturbations to $\varrho$ and $\bar{\varrho}$ at $t=0$
by

$\displaystyle\varrho_{ee}(z,\mu)=\bar{\varrho}_{ee}(z,\mu)=(1+\sqrt{1-\epsilon^{2}(z)})/2,$
$\displaystyle\varrho_{xx}(z,\mu)=\bar{\varrho}_{xx}(z,\mu)=(1-\sqrt{1-\epsilon^{2}(z)})/2,$
$\displaystyle\varrho_{ex}(z,\mu)=\bar{\varrho}_{ex}(z,\mu)=\epsilon(z)/2,$
(8)

where $\epsilon(z)$ is a real number randomly generated between $0$ and
$0.01$. We discretize the simulation domain with 6000, 50, and 20 uniform
grids in $-600\leq z\leq 600$ $\lambda^{-1}$, $-1\leq\mu\leq 1$, and $0\leq
4\omega/\delta m^{2}\leq 0.5$ MeV-1, and use the finite difference scheme of
cose$\nu$ george2023cosenu  to conduct the simulation until
$t=2000\lambda^{-1}$ when the system has settled into the asymptotic state.

Figure 1:  Comparison of the spanned range of spatially averaged survival
probabilities for neutrinos with different $\omega$ (grey shaded area; see Eq.
(10)), with the survival probability averaged over space and $\omega$ (black
dashed curve; see Eq. (9)), at the final time of the simulation for a system
with $\omega\ll\lambda$. Also shown is the analytical prescription for
$\langle P_{\rm sur}(\mu,\omega)\rangle_{f}$ (blue dotted curve) described in
Eq. (16). Note that the grey-shaded area basically overlaps with the black
dashed line, implying that the survival probabilities are nearly independent
of the neutrino energy.

Fig. 1 shows the survival probability of electron neutrinos as a function of
$\mu$, averaged over $z$ and $\omega$ (black dashed curve) at the end of the
simulation, computed as,

$\displaystyle\langle P_{\rm
sur}(\mu)\rangle_{f}=\int\mathrm{d}z\mathrm{d}\omega
f_{\nu}(\mu,\omega)\varrho_{ee}\Big{/}\int\mathrm{d}z\mathrm{d}\omega
f_{\nu}(\mu,\omega),$ (9)

as well as the spanned range of the spatially averaged survival probabilities
(shaded grey area) by all different $\omega$ values calculated by:

$\displaystyle\langle P_{\rm
sur}(\mu,\omega)\rangle_{f}=\int\mathrm{d}zf_{\nu}(\mu,\omega)\varrho_{ee}\Big{/}\int\mathrm{d}zf_{\nu}(\mu,\omega).$
(10)

This comparison clearly shows that $\langle P_{\rm
sur}(\mu,\omega)\rangle_{f}$ is nearly independent of $\omega$ as the grey-
shaded area basically overlaps with the black dashed line. Also shown in the
plot is the analytical prescription for $\langle P_{\rm
sur}(\mu,\omega)\rangle_{f}$ (blue dotted curve) following Ref. Xiong:2023vcm
for two-flavor scenario described in Eq. (16).

While $\omega/\lambda\lesssim 10^{-4}$ is a reasonable assumption regarding
the SN neutrino decoupling region, we conducted additional calculations with
$\omega/\lambda=10^{-3}$. Although the spanned range of spatially averaged
survival probabilities turned out to be more noticeable and the analytical
formula was less precise in that case (compared to the former case), we
noticed that assuming an energy-independent survival probability remains a
justified assumption also for $\omega/\lambda\gtrsim 10^{-3}$.

###  II.2 Survival probability function

To effectively train and evaluate our PINN, we require the survival
probabilities derived from energy-integrated neutrino distributions. Our
approach involves utilizing two parametric distributions for the initial
neutrino angular distributions, as previously explored in our work
Abbar:2023ltx : the maximum entropy distribution and the Gaussian
distributions Cernohorsky:1994yg ; Richers:2022dqa ; abbar2023applications ;
Abbar:2023zkm ; Wu:2021uvt ; Yi:2019hrp , defined as,

$\begin{split}f^{\rm{max-ent}}_{\nu}(\mu)&=A\exp(a\mu),\\\
f^{\rm{Gauss}}_{\nu}(\mu)&=A\exp\big{(}-\frac{(1-\mu)^{2}}{\xi}\big{)},\end{split}$
(11)

respectively, where,

$f_{\nu}(\mu)=\int_{0}^{\infty}\int_{0}^{2\pi}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}f_{\nu}(\mathbf{p}).$
(12)

Note that here $A$, $a$ and $\xi$ are arbitrary parameters which determine the
overall neutrino number and the shape of the distributions. Allowing for two
distinct forms of angular distributions takes into consideration potential
deviations in the shape of neutrino angular distributions in realistic
simulations, which can occur, e.g., due to the use of different closure
relations.

In our analytical treatment of the survival probability, we follow closely our
recent works in Refs. Xiong:2023vcm ; Abbar:2023ltx . We assume that $G(\mu)\
(=\int_{0}^{2\pi}\mathrm{d}\phi_{\nu}G(\mathbf{v}))$ has only one zero
crossing. In the three-flavor scenario, the survival probability can then be
defined as,

$P_{\mathrm{sur}}(\mu)=\begin{cases}\frac{1}{3}&{\rm for\leavevmode\nobreak\
}\mu^{<},\\\ \mathcal{S}_{3}(\mu)&{\rm for\leavevmode\nobreak\
}\mu^{>},\end{cases}$ (13)

with,

$\mathcal{S}_{3}(\mu)=1-\frac{2}{3}h(|\mu-\mu_{c}|/\zeta).$ (14)

where $h(x)=(x^{2}+1)^{-1/2}$ and $\zeta$ can be found such that the survival
probability function is continuous. Here, $\mu^{<}(\mu^{>})$ are defined as
the $\mu$ range over which the following integral is smaller (larger):

$\begin{split}\Gamma_{+}&=\bigg{|}\int_{-1}^{1}\mathrm{d}\mu\
G(\mu)\Theta[G(\mu)]\bigg{|},\\\
\Gamma_{-}&=\bigg{|}\int_{-1}^{1}\mathrm{d}\mu\
G(\mu)\Theta[-G(\mu)]\bigg{|},\end{split}$ (15)

where $\Theta$ is the Heaviside theta function. In the case of the two-flavor
scenario, the survival probability can be obtained using:

$P_{\mathrm{sur}}(\mu)=\begin{cases}\frac{1}{2}&{\rm for\leavevmode\nobreak\
}\mu^{<},\\\ \mathcal{S}_{2}(\mu)&{\rm for\leavevmode\nobreak\
}\mu^{>},\end{cases}$ (16)

with,

$\mathcal{S}_{2}(\mu)=1-\frac{1}{2}h(|\mu-\mu_{c}|/\zeta).$ (17)

We refer an interested reader to Refs. Xiong:2023vcm  for more details.

Using the neutrino angular distributions in Eq. (11) and the survival
probability function defined in Eq. (13), one can obtain the asymptotic
outcomes of FFCs given the initial distributions.

##  III Applications of Neural Networks

To effectively train our PINN, we require information on two fronts: the
energy-integrated moments of the neutrino gas and the moments within a
specific energy bin. The former implicitly contains the necessary information
for the survival probability (dictated only by the energy-integrated
quantities), while the latter supplies the bin-specific information to which
the survival probability must be applied.

To prepare our datasets, we begin with the initial energy-integrated angular
distributions of neutrinos, which can follow either a maximum entropy or a
Gaussian distribution. With these distributions for $\nu_{e}$ and
$\bar{\nu}_{e}$, we derive analytical survival probabilities. Next, we apply
these analytical distributions to the neutrino angular distributions within a
particular energy bin (again either maximum entropy or a Gaussian). This
process helps us determine the eventual outcomes of FFCs for that specific
bin. By performing integration over the neutrino angular distributions, we can
then obtain the initial and final values of $I_{0}$’s and $I_{1}$’s for that
specific energy bin.

Before discussing our findings, it’s crucial to emphasize that to ensure high
performance in our NN models on the test set, it’s essential to divide the
dataset into three distinct sets: Training set for foundational learning,
development set for optimizing hyper-parameters, and test set for evaluating
the model’s generalization to novel data.

###  III.1 The architecture of NNs

For a given multi-energy neutrino gas, one is provided with the initial values
of energy-integrated $I_{0}$’s and $I_{1}$’s of $\nu_{e}$, $\bar{\nu}_{e}$
(also of $\nu_{x}$, which is irrelevant here since it has no effect on the
survival probability). In addition, for each specific energy bin, one has
$I_{0}$’s and $I_{1}$’s of $\nu_{e}$, $\bar{\nu}_{e}$, and $\nu_{x}$. In this
context, we make the assumption that the initial distributions of
$\bar{\nu}_{x}$ and $\nu_{x}$ are identical (though their final ones following
FFCs could be different), a simplification that aligns with the majority of
state-of-the-art CCSN and NSM simulations.

Though in total 10 $I$’s are available (which could be, in principle, the
inputs of NNs), we here introduce a layer of feature engineering to enhance
the performance of our NNs, namely we define the new features:

$\alpha,F_{\nu_{e}},F_{\bar{\nu}_{e}},n_{\nu_{e},i},n_{\bar{\nu}_{e},i},n_{\nu_{x},i},F_{\nu_{e},i},F_{\bar{\nu}_{e},i},F_{\nu_{x},i},$
(18)

with $F_{\nu}=(I_{1}/I_{0})_{\nu}$. Here, the quantities without/with
subscript $i$ indicate the quantities belonging to the energy-integrated
spectrum/specific energy bin. Note that the neutrino number densities in the
particular energy bin must be smaller than the corresponding energy-integrated
values.

The selection of these features offers explicit insights into the
configuration of neutrino angular distributions, which plays a crucial role in
understanding the asymptotic outcome of FFCs. Furthermore, it is worth
highlighting that all quantities in this context are normalized by the initial
energy-integrated $\nu_{e}$ number density, allowing the convenient choice of
setting it to $n^{\rm{initial}}_{\nu_{e}}=1$. This simplification reduces the
number of inputs to our NNs, and notably, there is no input parameter related
to $n_{\nu_{e}}$.

As we also discussed in our previous work Abbar:2023ltx , there is still the
possibility of improving our NNs through more advanced feature engineering. By
considering the neutrino survival probability’s shape, as expressed in Eq.
(13), one observes that a significant amount of information about the shape of
the survival probability can be derived by learning the position of $\mu_{c}$.
Another crucial piece of information, given $\mu_{c}$, is determining the side
where equipartition happens. The behavior of the survival probability on the
opposite side is regulated by conservation laws. This side’s determination is
described by the quantity $E_{RL}$, a binary value that equals 1 if
equipartition happens for $\mu_{c}\leq\mu$ and 0 otherwise.

Figure 2:  Schematic architecture of our NNs. The green zone shows the
implementation of the extra features, $\mu_{c}$, and $E_{RL}$, which are
obtained through an extra layer of regression, using linear and logistic
regressions, respectively. Here, $\mu_{c}$ is the crossing direction and
$E_{RL}$ is a binary, which is 1 if the equilibrium occurs for
$\mu_{c}\leq\mu$, and 0 otherwise. The blue zone represents energy-integrated
inputs, while the orange zone displays inputs for specific energy bins. Note
that the neutrino number densities in the particular energy bin must be
smaller than the corresponding energy-integrated values. In our basic NN,
referred to as the NN with no extra features, the NN only takes the inputs
highlighted in Eq. (18). However and in our PINN, we provide our NN with the
extra features $\mu_{c}$ and $E_{RL}$.

As in Ref. Abbar:2023ltx  and as illustrated in Fig. 2, we explore two
distinct architectures in our NN framework. In the foundational architecture,
we integrate only
$\alpha,F_{\nu_{e}},F_{\bar{\nu}_{e}},n_{\nu_{e},i},n_{\bar{\nu}_{e},i},n_{\nu_{x},i},F_{\nu_{e},i},F_{\bar{\nu}_{e},i},\mathrm{and}\
F_{\nu_{x},i}$ into our NN. On the other hand, our alternative NN includes
also information coming from $\mu_{c}$ and $E_{RL}$. Our feedforward NN has a
single hidden layer containing 150 neurons, as justified in Fig. 4 and the
text around it.

Regarding the output layer, our NNs return $I_{0,i}$ and $I_{1,i}$ for
$\nu_{e}$ and $\bar{\nu}_{e}$, employing a total of 4 neurons. Deriving
$I_{i}$’s for $\nu_{x}$ and $\bar{\nu}_{x}$ is achieved by applying principles
of neutrino and antineutrino number density, as well as momentum conservation.
Put simply, our NN ensures neutrino conservation laws.

Apart from the inputs, we also consider modifying the loss function of our
NN’s. In particular, we introduce an additional loss term in the optimization
of the NN model with the extra features, defined as,

$\mathcal{L}_{\rm{extra}}=\frac{1}{N_{\rm{sample}}}\Sigma_{k}(\Delta
N_{{\nu_{e}}+{\bar{\nu}_{e}},k})^{2},$ (19)

which tends to penalize any deviation in the number of neutrinos in the
electron channel, i.e.,
$N_{{\nu_{e}}+{\bar{\nu}_{e}}}=n_{\nu_{e},i}+n_{\bar{\nu}_{e},i}$, a critical
parameter of utmost significance in CCSNe and NSMs. Here $\Delta$,
$N_{\rm{sample}}$, and $\Sigma_{k}$ denote the difference between the true and
predicted values, the number of samples in the training set, and the summation
over the training samples, respectively.

The specific inclusion of the domain knowledge allows one to consider this
particular NN architecture as a PINN karniadakis2021physics ;
cuomo2022scientific ; raissi2019physics . The PINN should be compared with our
basic NN, referred to as NN with _no extra features_ , for which the loss term
only includes the ordinary mean squared errors of the output parameters.

###  III.2 The NN’s performance

Figure 3:  Performance evaluation of the PINN and the basic NN with no extra
features. We present the relative absolute error in the output parameters,
along with the relative error in the total number of neutrinos within the
electron channel, $N_{{\nu_{e}}+{\bar{\nu}_{e}}}$. It is evident that the PINN
can well outperform the basic NN with no extra features. Here, an epoch refers
to a single pass through the entire training dataset during the training
phase.

In this section, we present and discuss the performance of our NNs in
predicting the asymptotic outcome of FFCs in a multi-energy three-flavor
neutrino gas. For training/testing our NNs, we generate a dataset comprising a
well-balanced combination of maximum entropy and Gaussian initial neutrino
angular distributions. The ultimate outcome of FFCs is determined through a
three-flavor survival probability, as detailed in Eq. (13). We also set
$\alpha\in(0,2.5)$, $F_{\nu_{x},(i)}\in(0,1)$,
$F_{\bar{\nu}_{e},(i)}\in(0.4F_{\nu_{x},(i)},F_{\nu_{x},(i)})$, and
$F_{\nu_{e},(i)}\in(0.4F_{\bar{\nu}_{e},(i)},F_{\bar{\nu}_{e},(i)})$, which is
consistent with the expected hierarchy $F_{\nu_{e}}\lesssim
F_{\bar{\nu}_{e}}\lesssim F_{\nu_{x}}$. Regarding $n_{\nu,i}$’s, we take them
from a half-normal distribution with zero mean and a standard deviation of
$0.1n_{\nu}$, with $n_{\nu}$ being the energy-integrated neutrino number
density. This choice can enhance the performance of our NNs in the energy bins
with fewer neutrinos. Also note that since our NNs process only a single
energy bin at a time, the hierarchy of flux factors among different neutrino
energies is irrelevant here.

In Fig. 3, we illustrate the performance of our PINN and the basic NN without
extra features. The relative error in the electron neutrino number density
within our PINN, quantified by
$|\Delta(n_{\nu_{e},i}+n_{\bar{\nu}_{e},i})|/(n_{\nu_{e},i}+n_{\bar{\nu}_{e},i})$,
achieves a minimum of 6%. Additionally, the mean absolute relative error in
the output variables, computed as the mean of $|\Delta I_{i}|/I_{i}$, attains
values $\sim 16\%$. In contrast, when considering the basic NN, these errors
increase to $\sim 12\%$ and $\sim 22\%$, respectively, showing higher
discrepancies. The noticeable performance improvement within our PINN can be
primarily attributed to the inclusion of extra features, which provide extra
information on the shape of the survival probability distribution.

Comparing the findings illustrated in Fig. 3 with those discussed in Ref.
Abbar:2023ltx , a discerning reader will observe a substantial discrepancy in
the impact of employing PINN. Specifically, the application of PINN results in
a significantly more pronounced enhancement in performance in the former case.
While the utilization of PINN can almost reduce the error by a factor of two
in the multi-energy neutrino gas, its application to a single-energy neutrino
gas only yields a modest $\lesssim 1\%$ improvement in the error.

This discrepancy can be attributed to the substantial difference in the amount
of input information between the two cases. In the former scenario, the volume
of input information is notably larger, leading to a higher degree of
degeneracy in the input data. The introduction of PINN in this context is
remarkably effective in mitigating this degeneracy and, consequently,
substantially reducing the error.

Figure 4:  Performance evaluation of our PINN and the basic NN with no extra
features (on the validation set) as a function of the number of neurons in the
hidden layer. It is evident that the NNs have achieved their best performance
on the validation set once $n_{h}\gtrsim 150$. The labels and NN models are
the same as those in Fig. 3.

The computations conducted here have utilized a feedforward neural network
featuring a single hidden layer having $n_{h}=150$ neurons. The reasoning
behind selecting this specific number of neurons is depicted in Fig. 4, which
shows errors for different NN architectures. The optimal performance on the
validation set is observed when $n_{h}\gtrsim 50$.

Figure 5:  Absolute relative error in the output of our PINN (red curve) vs
the relative error in the number of neutrinos in electron channel, i.e.,
$N_{{\nu_{e}}+{\bar{\nu}_{e}}}$ (blue curve). The inclusion of a few thousand
data points in the training set leads to the disappearance of error variations
between the validation and training sets. Note that we do not display the
absolute relative error in the training set. This is due to the presence of
small $I^{\prime}s$ in the training set (which are removed from the test set),
causing a significantly larger relative error. Hence, any direct comparisons
between the absolute relative errors in the training and test sets would be
unfair.

In Fig. 5, we analyze our PINN’s performance concerning the training set size.
The red curve indicates the absolute relative error in the PINN’s output,
while the blue curve shows the relative error in
$N_{{\nu_{e}}+{\bar{\nu}_{e}}}$. As the training dataset expands to include
several thousand data points, the errors rapidly decrease to approximately
$\sim 6\%$ and $\sim 18\%$, respectively, and additionally, the disparity
between the validation and training set errors diminishes. These findings
align with results observed in single-energy scenario calculations
Abbar:2023ltx . This underscores the crucial minimum number of data points
required for reliable calculations using NNs.

There remains a crucial aspect regarding the assessment of the absolute
relative error that needs discussion. In our prior study concerning FFCs
within a single-energy neutrino gas Abbar:2023ltx , we primarily regarded the
absolute error as the relevant metric. However, when addressing the
complexities of a multi-energy neutrino gas, the absolute error falls short.
The problem arises because, despite normalizing all quantities by
$n_{\nu_{e}}$, the values of $I$’s within a specific energy bin are expected
to constitute a minor fraction of one. Hence, achieving a low absolute error
does not inherently guarantee accurate prediction due to the relatively small
magnitudes involved.

Hence and for the multi-energy neutrino gas, we’ve adopted the absolute
relative error as the informative metric. Nonetheless, this choice comes with
a notable drawback: extremely small $I$’s yield disproportionately large
absolute relative errors. While these cases may not include the most
intriguing aspects of the parameter space, their associated errors tend to
dominate over the rest of the parameter space. To resolve this issue, in our
performance evaluation on the test set, we have chosen to exclude data points
where $|I_{i}|\leq 5\times 10^{-3}$ (while retaining them in the training
set). In Sec.III.3, we come back to this problem and devise a solution to this
challenge.

###  III.3 Reconstruction of neutrino energy spectra

Figure 6:  Performance evaluation of our PINNs in reconstructing neutrino
energy spectra. The upper panels exhibit the initial spectra characteristics,
including the neutrino initial energy spectra, angular distributions of the
energy-integrated neutrino spectra as a function of $\mu$, and the
corresponding FFC survival probability. Here, we have assumed
$F_{\nu_{e}}=0.5$, $F_{\bar{\nu}_{e}}=0.7$, and $F_{\bar{\nu}_{e}}=0.8$. The
lower panels illustrate the post-FFCs final neutrino spectra for $\nu_{e}$,
$\bar{\nu}_{e}$, and $\nu_{x}$ ($\bar{\nu}_{x}$), respectively. The results
are shown for both the PINN approach and an exact method, assuming having
access to the full neutrino angular distributions. As one can see, the
prediction errors for antineutrinos are much smaller than those concerning
neutrinos.

In the preceding part, we engineered our NNs to process the energy-integrated
$I$’s and their values within an specific bin, $I_{i}$’s. To elaborate, our
methodology works on an energy bin-based approach, focusing solely on
predicting the final outcomes within individual energy bins. This approach
eliminates unnecessary complexity associated with attempting to reconstruct
the entire neutrino energy spectra following FFCs at once.

In this part, we explore the performance of our PINN’s to reconstruct the
complete neutrino energy spectra following FFCs. While this application of our
NNs might appear straightforward initially, it presents significant
challenges. Specifically, when our PINNs are employed to analyze the tail of
the energy spectrum, where the count of neutrinos is notably low, we encounter
a considerable obstacle. The relative error in these instances might surpass
the total values of $n_{\nu(\bar{\nu})}$, potentially leading to a scenario
where the conservation laws cannot be satisfied, as discussed in the
following. In the high-energy tail of the spectra, where neutrino number
densities can reach very low values, the predicted value of
$n_{\nu_{e}(\bar{\nu}_{e})}$ may even surpass the total (anti)neutrino number
density. This is attributed to the expected large relative errors associated
with the output of neural networks when dealing with such small values, as
discussed before. Consequently, while the conservation law for neutrino number
remains mathematically valid in principle, it loses its practical
significance. This is because adhering to the conservation laws would now
imply a negative number density for $n_{\nu_{x}(\bar{\nu}_{x})}$, which is not
physically meaningful.

To address this challenge, we devised a solution by simultaneously adjusting
the number of $\nu_{e}$ ($\bar{\nu}_{e}$) and $\nu_{x}$ ($\bar{\nu}_{x}$)
while ensuring the conservation of total neutrino numbers. Our approach indeed
involves the utilization of two PINNs. The first PINN is designed to compute
$I_{0,i}$’s and $I_{1,i}$’s for $\nu_{e}$ and $\bar{\nu}_{e}$, following the
standard architecture we discussed in previous part. Meanwhile, the second
PINN shares a similar structure but deals with $\nu_{x}$ and $\bar{\nu}_{x}$,
calculating their respective $I_{0,i}$’s and $I_{1,i}$’s. In practice, this
implies that the outputs of such a PINN are:

$I^{\nu_{x}}_{0,i},I^{\nu_{x}}_{1,i},I^{\bar{\nu}_{x}}_{0,i},I^{\bar{\nu}_{x}}_{1,i},$
(20)

while the input features are not different in the two PINNs. Importantly, in
this PINN, the $I$’s of $\nu_{e}$ and $\bar{\nu}_{e}$ are then derived through
neutrino conservation laws. In essence, the difference between these two PINNs
lies in the information they provide—while the first PINN provides $\nu_{e}$
$I$’s, the second one focuses on $\nu_{x}$, with the retrieval of $\nu_{e}$
quantities governed by conservation principles.

Given these two PINNs, the final neutrino number densities for each energy bin
can be calculated as:

$n^{\mathrm{fin}}_{\nu_{\beta}(\bar{\nu}_{\beta}),i}=\frac{N^{\mathrm{ini}}_{\nu(\bar{\nu}),i}}{N^{\mathrm{pred}}_{\nu(\bar{\nu}),i}}\
n^{\mathrm{pred}}_{\nu_{\beta}(\bar{\nu}_{\beta}),i},$ (21)

where $n^{\mathrm{pred}}_{\nu_{\beta}(\bar{\nu}_{\beta}),i}$ represents the
corresponding PINN predicted value for the neutrino species $\beta$. Here,
$N^{\mathrm{pred}}_{\nu(\bar{\nu}),i}$ denotes the predicted total
(anti)neutrino number density, and $N^{\mathrm{ini}}_{\nu(\bar{\nu}),i}$ is
its initial value. It’s important to note that the prediction for each
neutrino species is conducted by the relevant PINN model discussed earlier,
i.e., the former PINN is employed for the electron species, while the latter
is used for heavy-lepton flavors. Note that the errors in the predictions of
$n_{\nu}$ can now be automatically adjusted to ensure respecting the
conservation laws, preventing negative number densities. Furthermore, a fair
treatment is now applied to electron and heavy-lepton flavors, preventing one
from becoming unreasonably small when the error in the prediction of the other
is unreasonably large.

In Fig. 6, we present the performance evaluation of our PINNs in
reconstructing the neutrino energy spectra. The upper panels show the initial
neutrino distributions, which are prepared as follows. We describe the energy-
differential number flux of a specific neutrino species, $\nu_{\beta}$, as
Mirizzi:2015eza ,

$\mathcal{F_{\nu_{\beta}}}(E_{\nu})\propto\frac{L_{\nu_{\beta}}}{\langle
E_{\nu_{\beta}}\rangle}f_{\nu_{\beta}}(E_{\nu})$ (22)

with

$f_{\nu_{\beta}}(E_{\nu})=\frac{1}{T_{\nu_{\beta}}\Gamma(1+\eta_{\nu_{\beta}})}\bigg{(}\frac{E_{\nu}}{T_{\nu_{\beta}}}\bigg{)}^{\eta_{\nu_{\beta}}}\exp(-E_{\nu}/T_{\nu_{\beta}})$
(23)

being the normalized $\nu_{\beta}$ spectrum, where $E_{\nu}$ is the neutrino
energy. Here, $\eta_{\nu_{\beta}}$ and $\langle E_{\nu_{\beta}}\rangle$ are
the pinching parameter and the neutrino average energy which describe the
normalized spectrum and $T_{\nu_{\beta}}=\langle
E_{\nu_{\beta}}\rangle/(1+\eta_{\nu_{\beta}})$. In addition, $L_{\nu_{\beta}}$
is the neutrino luminosity. To be specific, we use the following values which
could be expected during the SN accretion phase mirizzi2016supernova :

$\begin{split}L_{\nu_{e}}:L_{\bar{\nu}_{e}}:L_{\nu_{x}}&=1:1:0.33,\\\ \langle
E_{\nu_{e}}\rangle:\langle E_{\bar{\nu}_{e}}\rangle:\langle
E_{\nu_{x}}\rangle&=9:12:16.5,\\\
{\eta}_{\nu_{e}}:{\eta}_{\bar{\nu}_{e}}:{\eta}_{\nu_{x}}&=3.2:4.5:2.3.\\\
\end{split}$ (24)

Note that for the average energies and luminosity’s only the ratios matter.
Moreover, we assumed these values for the energy-integrated flux factors:
$F_{\nu_{e}}=0.5$, $F_{\bar{\nu}_{e}}=0.7$, and $F_{\bar{\nu}_{e}}=0.8$. We
here take 14 energy bins and for each energy bin, we then adopted an
assumption describing the relationship as $F_{\nu,i}=F_{\nu}\
(70-E_{\nu,i})^{2}/60^{2}$. Here we have assumed that the spectra approach
zero for $E_{\nu,i}\gtrsim 60$ MeV. Note that we anticipate a decrease in the
flux factor as neutrino energy increases. This reduction is expected to be
nonlinear, attributed to the nonlinear scaling of the neutrino scattering
cross-section with matter in the SN environment. While our assumption about
the energy-dependent nature of the flux factor is speculative, it aligns with
the expected conditions mentioned above.

In the lower panels of Fig. 6, we present the final neutrino energy spectra
following FFCs. It is evident that a notable difference in the spectra
reconstruction error emerges between neutrinos and antineutrinos. This
disparity can be indeed quantified by an absolute spectral relative error,
defined as,

$\delta_{\nu}=\Sigma_{i}\frac{n_{\nu,i}}{n_{\nu}}\frac{|\Delta
n_{\nu,i}|}{n_{\nu,i}},$ (25)

where the prediction error for $n_{\nu,i}$ is weighted by the relative
distribution across energy bins. For the results presented in the lower panels
of Fig. 6, we observed $\delta_{\nu_{e}}=0.20$, $\delta_{\bar{\nu}_{e}}=0.06$,
$\delta_{\nu_{x}}=0.23$, and $\delta_{\bar{\nu}_{x}}=0.05$.

Despite the clear difference between neutrinos and antineutrinos in this
particular example, we’ve noticed that this observation depends notably on the
specific example and it can fluctuate across different calculations and
models. This variability highlights the need for more sophisticated neural
network architectures, such as Bayesian neural networks. These specialized
networks offer the capability to provide uncertainty estimates for predicted
quantities, addressing the intricacies and fluctuations observed in these
calculations.

##  IV DISCUSSION AND OUTLOOK

We have employed a single hidden layer Physics-Informed Neural Network (PINN)
to predict the asymptotic outcome of FFCs within a three-flavor multi-energy
neutrino gas. Our approach focuses on utilizing the first two moments of
neutrino angular distributions, making our PINNs highly relevant to state-of-
the-art CCSN and NSM simulations. We have demonstrated that our PINNs can
achieve remarkable accuracy, with errors reaching $\lesssim 6\%$ for the
number of neutrinos in the electron channel, and $\lesssim 18\%$ for the
relative absolute error in the neutrino moments.

By conducting simulations of FFCs in a 1D box with periodic boundary
conditions, we first demonstrated that in scenarios where the FFC growth rate
notably exceeds that of the vacuum Hamiltonian, a uniform survival probability
is experienced by all neutrinos, regardless of their energy. This common
survival probability is solely determined by the energy-integrated neutrino
spectrum.

In our PINNs, we incorporated novel features to effectively capture the shape
of the expected neutrino survival probability distributions. Our improvements
involve incorporating the position of the zero crossing in the distribution of
$\nu$ELN, $\mu_{c}$, and also information about the side of $\mu_{c}$ where
the expected equipartition occurs. Our research demonstrates that this
advanced feature engineering significantly improves the performance of our
PINN.

Moreover, we demonstrated that the variance between the training and
validation sets decreases significantly with a minimum of a few thousand data
points. This underscores the necessity for datasets of (at least) this size
when developing more realistic models based on simulation data in future
studies.

We also highlighted a significant challenge in applying NNs to predict the
whole neutrino energy spectrum. This challenge arises from the fact that
predicting the tail of the spectrum may lead to an error of such magnitude
that it violates the preservation of neutrino conservation laws. To address
this issue, we propose the development of two separate models: one dedicated
to predicting electron (anti)neutrino quantities and another for heavy-lepton
flavor of (anti)neutrino quantities, respectively. By scaling the numbers in
accordance with conservation laws, we could overcome this challenge. Our
demonstrated approach showed that PINNs can accurately enough reconstruct the
entire neutrino spectrum, particularly in a typical neutrino spectra scenario
during the SN accretion phase.

In summary, our research highlights the effectiveness of PINNs in predicting
the asymptotic outcomes of FFCs within a multi-energy neutrino gas.
Nevertheless, there are crucial avenues for further exploration. An important
consideration is extending our study to encompass more realistic neutrino
gases characterized by non-axisymmetric distributions, where $\nu_{x}$ and
$\bar{\nu}_{x}$ can also exhibit dissimilar patterns. Such refinements will
improve the feasibility of incorporating FFCs into CCSN and NSM simulations,
thereby advancing our capacity to model and predict accurately these extreme
astrophysical phenomena.

## Acknowledgments

We are deeply grateful to Georg Raffelt, Gabriel Martínez-Pinedo, and Oliver
Just for fruitful discussions. 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. M.-R. W. acknowledges supports from the National Science
and Technology Council under Grant No. 111-2628-M-001-003-MY4, the Academia
Sinica under Project No. AS-CDA-109-M11, and the Physics Division, National
Center for Theoretical Sciences, as well as the resource of the Academia
Sinica Grid-computing Center (ASGC). Z.X. was supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (ERC Advanced Grant KILONOVA No. 885281). We would also
like to acknowledge the use of the following softwares: Scikit-learn
pedregosa2011scikit , Keras chollet2015keras , Matplotlib Matplotlib , Numpy
Numpy , SciPy SciPy , and IPython IPython .

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URL https://ipython.org