# O-type Stars Stellar Parameter Estimation Using Recurrent Neural Networks

Miguel Flores R.,1 Luis J. Corral,1,2 Celia R. Fierro-Santillán3 and Silvana
G. Navarro1,2
1Centro Universitario de Ciencias Economico Administrativas (CUCEA),
Universidad de Guadalajara, 45100 Zapopan, Jal., México.
2Instituto de Astronomía y Meteorología (IAM), Universidad de Guadalajara,
44130 Guadalajara, Jal., México
3Escuela Nacional Colegio de Ciencias y Humanidades, Plantel Sur (ENCCH Sur),
Universidad Nacional Autónoma de México (UNAM), 04500, CDMX, México.  E-mail:
<EMAIL_ADDRESS>(MFR)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In this paper, we present a deep learning system approach to estimating
luminosity, effective temperature, and surface gravity of O-type stars using
the optical region of the stellar spectra. In previous work, we compare a set
of machine learning and deep learning algorithms in order to establish a
reliable way to fit a stellar model using two methods: the classification of
the stellar spectra models and the estimation of the physical parameters in a
regression-type task. Here we present the process to estimate individual
physical parameters from an artificial neural network perspective with the
capacity to handle stellar spectra with a low signal-to-noise ratio (S/N), in
the $<$20 S/N boundaries. The development of three different recurrent neural
network systems, the training process using stellar spectra models, the test
over nine different observed stellar spectra, and the comparison with
estimations in previous works are presented. Additionally, characterization
methods for stellar spectra in order to reduce the dimensionality of the input
data for the system and optimize the computational resources are discussed.

###### keywords:

methods: data analysis – techniques: miscellaneous – astronomical data bases:
miscellaneous – stars: fundamental parameters

††pubyear: 2022††pagerange: O-type Stars Stellar Parameter Estimation Using
Recurrent Neural Networks–O-type Stars Stellar Parameter Estimation Using
Recurrent Neural Networks

##  1 Introduction

Processing big astronomical databases is a current challenge that has been
approached from different perspectives Jiang, Bin, et al. (2020);
Villavicencio-Arcadia et al. (2020); Sharma et al. (2020); Bu et al. (2020);
Sharma et al. (2019); Fierro-Santillán et al. (2018); Minglei et al. (2020);
Li et al. (2017); Dafonte, C. et al. (2016); Sander, A. et al. (2015);
Navarro, S. G. et al. (2012); Teimoorinia (2012); Snider et al. (2001);
therefore, with the recent growth of the artificial intelligence methods is
natural to think in the implementation of an autonomous systems. In this
context, Artificial Neural Networks (ANNs) have proven to be a very useful
tool Li et al. (2022); Villavicencio-Arcadia et al. (2020); Bu et al. (2020);
Sharma et al. (2019); Navarro, S. G. et al. (2012).

The growing development of a diversity of artificial neural network structures
has generated a natural interest of implement it in multiple astrophysics
projects. The flexibility of the ANNs to be trained with different tensor-like
data structures, their capacity to handle large volumes of data without the
necessity of storing a big set of measures like different machine learning
algorithms, and the resistance to noise and lack in the data, has presented an
advantage in the stellar spectra classification, as well as morphological
classification, segmentation of stars and galaxies, and also, the estimation
of stellar parameters Li et al. (2022); Bu et al. (2020); Jiang, Bin, et al.
(2020); Fierro-Santillán et al. (2018); Dafonte, C. et al. (2016); Recio-
Blanco et al. (2006).

Early projects has shown highly accurate results on the determination of
physical parameters from stellar spectra using modern deep learning
techniques. In the context of ANNs, these types of algorithms have achieved
estimations over spectra with a high level of noise, near 20 S/N, and explore
different ways of improving the parameter estimation accuracy below that limit
Li et al. (2022); Villavicencio-Arcadia et al. (2020); Navarro, S. G. et al.
(2012).

This high noise limit has been a threshold where other methods indicate a
substantial increase of error as is show in Recio-Blanco et al. (2006). In
Worley, C. C. et al. (2012),is mentioned that in some cases a considerable
amount of spectra in their database was rejected from the system and physical
parameters do not estimated because of the quality in the spectra. And in
Koleva, M. et al. (2009), the impact of low S/N on the estimation over the
analyzed spectra is not addressed.

Therefore, with the increasing astronomical observations in a more automatic
way, with modern and space telescopes, generating databases in the big data
limit, the automatic computational systems are an essential tool in order to
process and analyze spectral data in a fast and reliable way. According to
this, the proposal on this work consist of an deep learning algorithm
implementation as a part of a system that estimates fundamental stellar
parameters such as $T_{eff}(K)$, $log$ $g$, $log(L/L\textsubscript{$\odot$})$
in an accurate way. Additionally, in this project, we propose a
characterization method to reduce the dimension of the training database, in
order to improve the efficiency of the system and reduce the time and
computational resources that a big database required.

This paper is organized as follows. Section 2 describes the synthetic spectra
database used in this work. Also, we introduce the method proposed for the
dimensionality reduction process, and preprocessing procedures needed. In
Section 3, we introduce the artificial neural networks and the structure we
develop for this project. Section 4 describes the estimation process, the
performance evaluation results, and the comparison of the estimation with
previous works. Finally, a brief summary of the results is made in Section 5.

##  2 Database Collection and Processing

The starting database used for this work consist on 4517 O-type star models
develop by Zsargó, J., et al. (2020). The models cover stars with mass from
near 9 to 120 $M\textsubscript{$\odot$}$, $T_{eff}(K)$ from 20,160 to 58,140,
$log(L/L\textsubscript{$\odot$})$ from 4.33 to 6.31, and $log$ $g$ from 2.56
to 4.28, with the distribution of Effective, Temperature, Surface Gravity, and
Luminosity as shown on Figures 1, 2, and 3. Each synthetic model is
represented by a spectrum in the optical region ($\lambda\lambda$ 3500-7000 Å)
of 100,001 points with uniform step of 0.035 Å. We modify each spectrum adding
white gaussian noise at different levels, in order to obtain spectra with S/N
ratio between 150 an 20. This step was done in order to provide noisy spectra
to the deep learning algorithm, and generate the capacity to deal with
observed spectra which naturally have noise.

Since the results obtained in different projects using artificial neural
networks (ANNs) (Li et al. (2022); Bu et al. (2020); Sharma et al. (2020);
Villavicencio-Arcadia et al. (2020); Sharma et al. (2019); Navarro, S. G. et
al. (2012); Snider et al. (2001)), we proposed a system based on the recurrent
network structure, and following the idea of reduce the necessity of high
computational resources, we explore the simple network structures and non-
network structures needed to complete the tasks.

For the training process, we use a final database with a total of 140,027
synthetic spectra, the 4,517 base models with 30 different levels of signal to
noise ratio (S/N) plus the clean synthetic spectrum.

Figure 1: Effective temperature distribution over the 4,517 models in current
database. Figure 2: Surface gravity distribution over the 4,517 models in
current database. Figure 3: Luminosity distribution over the 4,517 models in
current database.

###  2.1 Dimensionality Reduction

Develop an expert-type system that estimates a set of physical parameters can
start with the method that will compare a synthetic spectra generated by a
model and an observed spectra. This type of challenge can be reach using
either the full spectra, or, by establish a group of variables that
characterize the full spectra. The second method can be see as a
dimensionality reduction task, which take the advantages of the ANNs to
distinguish between a small set of characteristic variables and reduce the
computational resources compared to use a signal with a large amount of
points.

We proposed a method to reduce the amount of data that gets into the ANNs as
the input features. Starting on the system proposed in Fierro-Santillán et al.
(2018), where a set of equivalent width (EW) measures is needed, we expand the
number of absorption/emission lines measurements following the same process
presented in Miguel Flores R. et al. (2021), in order to have information of
lines in the red part of the visual spectrum, and also lines of heavier
elements such as SiIII, SiIV, etc. The lines of SiII (4128) and SiIII (4552),
which are useful to restrict the Temperature of O-type stars, were excluded
because most of the spectra are faint, with an equivalent width of thousandths
or less. Additionally, the SiII (4128) line has several neighboring lines,
which increases the uncertainty of its equivalent width measurement.

We choose 34 absorption lines that we think can be the minimal number to get
high accuracy for both approaches. These 34 lines are shown in Table 1. For
these lines, we check the possible neighbor lines that can be blended.

Wavelength Centroid (Å) | Possible Ions in the Line | Line index
---|---|---
3750 | NIV, H12, NIII, OIII (x2) | $L_{1}$
3770 | H11, SiIV, OIII, NeII, OII | $L_{2}$
3798 | NeII, OIII, SiII, H10, OII | $L_{3}$
3820 | HeI | $L_{4}$
3835 | OII H9 | $L_{5}$
3889 | OII, CII, H8 | $L_{6}$
3926 | HeI, SiIII | $L_{7}$
3934 | CaII | $L_{8}$
3970 |  HeI, HeII, CaII, H$\epsilon$ | $L_{9}$
4026 | HeI | $L_{10}$
4058 | NIV | $L_{11}$
4088 | SiIV | $L_{12}$
4101 |  CaII, H$\delta$ | $L_{13}$
4143 | HeI | $L_{14}$
4200 | HeII | $L_{15}$
4212 | SiIV | $L_{16}$
4340 |  OII (x2) H$\gamma$ | $L_{17}$
4388 | HeI | $L_{18}$
4471 | HeI | $L_{19}$
4481 | MgII | $L_{20}$
4541 | HeII | $L_{21}$
4813 | SiIII | $L_{22}$
4861 |  HeII, H$\beta$ | $L_{23}$
4943 | OII (x2) | $L_{24}$
4987 | NII | $L_{25}$
5005 | NII (x6) | $L_{26}$
5016 | HeI | $L_{27}$
5048 | NII, HeI | $L_{28}$
5801 | CIV | $L_{29}$
5812 | CIV | $L_{30}$
6381 | NIV | $L_{31}$
6527 | HeII | $L_{32}$
6563 |  HeII, H$\alpha$ | $L_{33}$
6683 | HeII | $L_{34}$
Table 1: Total lines needed as an input for the ANNs in the new approach.

Because the EW does not provide information about the morphology of the line,
wich can be very useful to accurate estimate fundamental parameters as the
rotational velocity, we create a secondary approach where the input of the
networks, are not only the EW, we also add a set of profiles of the measured
lines.

We define two ways to establish the line profiles needed to get more
information about the spectra and do not interfere with one of our statements
of avoid the necessity of super-computers. The first idea was to use all the
profiles of the non He or H Ions, having a total of 12 profile lines. The
second one, get the top $N$ line profiles with higher correlation with every
physical parameter. As a result, we will have an independent set for
$T_{eff}(K)$, $log(L/L_{o})$, and $log$ $g$. We chose the $N$ as 6 for this
work, since was observed as an average break point where the correlation start
to decrease in all the physical parameters. The 6 line profiles for every
physical parameter are shown in Table 2. Moreover, the wave length resolution
of the observed spectra we use in this work is 0.1226 Å, this is around 3.5
times less points per spectra (see Section 4). Therefore, in order to have the
same tensor lengths for the training database and the observed spectra, we use
an interpolation method to reduce the synthetic spectra in a 4 to 1 points
that will give us a similar resolution respect to the observed spectra. After
the resolution decrease process, we extract all the 34 line profiles we use to
measure the EW and then create a signal-type vector without the wavelength
relation and from this new group of line profiles we extract the group of
profiles for the two methods we mentioned previously in this section. The
final process in this dimensionality reduction, it is to match the final
length with the observed spectra, for this task we create a simple routine in
order to finally have the same number of points for each line profile. In
Figures 5, 6, we show an example of full points synthetic spectrum, and the
same spectrum after the process of resolution reduction and the extraction of
the line profiles regions for all the 34 lines shown in Table 1, and some
lines tagged for a reference on where the lines should be situated. In the
case of only using a smaller set of these 34 lines, the chosen lines will be
set in a similar way generating and smaller signal-type vector.

Physical parameter | Group of lines by Line index in Table 1
---|---
$T_{eff}(K)$ | $\\{L_{1},L_{3},L_{7},L_{8},L_{9},L_{10}\\}$
$log$ $g$ | $\\{L_{21},L_{28},L_{30},L_{31},L_{33},L_{34}\\}$
$log(L/L\textsubscript{$\odot$})$ | $\\{L_{26},L_{28},L_{29},L_{31},L_{32},L_{33}\\}$
Table 2: Top 6 line profiles with higher correlation per physical parameter
over the models data base. Figure 4: Correlation matrix between the profiles
lines of the 34 lines in 1 and the Effective Temperature, Surface Gravity, and
Luminosity. Figure 5: Example of a full synthetic spectra of 100,001 points.
Figure 6: Synthetic spectra of 5092 points (same spectra on Figure 5 after the
dimensionality reduction process). Some of the lines shown in Table 1 are
marked for reference.

##  3 Artificial Neural Networks

Because we have two different types of inputs we try to choose a network type
capable to deal with the signal-type input from the different profiles and the
numerical sequences from the equivalent width measures.

The election of this type of network structure is based on the type of input
data, which is a numerical vector with an specific sequence for each spectrum,
and the values will change in relation with the noise in the spectrum.
Additional to this, the system needs to have is the capacity to distinguish
between the nearby models. One classic solution to the systems with very
similar possible outputs is the connection between the neurons and also the
feedback connections.

###  3.1 Recurrent Neural Networks

Recurrent neural networks (RNNs) belong to a type of neural networks naturally
suited to solve sequential data problems. As a principal property, these class
of deep learning algorithms use information from previous inputs in the
current stage of input and output processing. While classical or simple deep
neural networks commonly assume independence between inputs and outputs, then,
for RNNs the output depends on the previous elements in the sequence.
Therefore, it is clear to think of this type of network as one of the best
options for our purposes.

Figure 7: Diagram of a recurrent neural network, which consists of an input
layer, a hidden layer and an output layer.

As mentioned before, for this work we established a full system where every
physical parameter will be estimated for a individual network, besides, every
network will be train using 3 different input tensor. The first one, a rank
one tensor of only a sequence of 34 EW, and for the second and third approach
a rank three tensor define by the 34 EW for the first entry, and the line
profiles for the second entry of 914 point for the non He or H lines, and for
the top 6 with higher correlation: 677 points for $log$ $g$, 588 points for
$T_{eff}(K)$, and 506 points for $logL$.

Because of the different input tensor dimension, the structure of the RNN will
be a functional-type of network, with two paths, that can see as two parallel
RNNs. The first path will process the 34 EW measures, and the second path will
process a signal-type input created by the Non He/H profile lines or the top
$N$ line profiles with higher correlation, as shown on Figure 6.

Figure 8: Diagram of a functional recurrent neural network, which consists of
an input layer, a hidden layer and an output layer for two networks, a
concatenated layer that takes the output from the last layer of the two
parallel networks, and an output layer.

##  4 Estimation Process and Results

To evaluate the performance of the system after the training process we choose
a group of O-type stars spectra observed at the National Astronomical
Observatory of San Pedro Martir over different observing seasons. These
spectra were taken using the 2.1M telescope with an Echelle spectograph. On
Table 3 the observed spectra and the SNR for every stellar spectra are shown.

Star | Type | SNR
---|---|---
HD216898 | O9V | 58.78
HD108 | O4-8f?p | 85.81
HD203064 | O7.5IIIn | 189.30
HD216532 | O8.5V | 84.05
HD15558 | O4.5III | 103.01
HD15570 | O4If | 108.47
HD15629 | O4.5V | 89.40
HD190864 | O6.5III | 18.27
HD16691 | O4If | 84.05
Table 3: Observed spectra and signal to noise ratio (SNR). The spectral type
was extracted from "The SIMBAD astronomical database", Wenger, M. et al.
(2000)

In Table 4 we show a group of estimations of the same stars shown on Table 3
published on previous works, using different methods. Additionally, it is
important to mention that, there are missing data in the observed spectra for
the stars: HD15558, HD15570, HD15629, and HD190864, on the spectral signal
related to 3,800 Å and less.

Star | $T_{eff}(K)$ |  $log$ $g$ | $log(L/L\textsubscript{$\odot$})$ | Refs.
---|---|---|---|---
HD216898 | 34000 $\pm$ 1000 | 4.00 $\pm$ 0.10 | 4.73 $\pm$ 0.25 | Marcolino, W. L. F. et al. (2009)
HD108 |  35000 $\pm$ 2000 37000 $\pm$ 2000 |  3.50 $\pm$ 0.10 3.75 $\pm$ 0.10 |  5.70 $\pm$ 0.10 5.40 $\pm$ 0.10 |  Martins, F. et al. (2012) Nazé et al. (2008)
HD203064 |  35000 $\pm$ 1500 34500 |  3.73 $\pm$ 0.15 3.50 |  5.10 +0.20/-0.30 5.50 |  Cazorla, Constantin et al. (2017) Repolust, T. et al. (2004)
HD216532 | 33000 $\pm$ 2000 | 3.70 $\pm$ 0.20 | 4.79 $\pm$ 0.25 | Marcolino, W. L. F. et al. (2009)
HD15558 |  41000 46500 |  3.80 3.71 $\pm$ 0.10 |  5.93 6.16 |  Bouret, J.-C. et al. (2012) Herrero et al. (2000)
HD15570 |  38000 $\pm$ 1000 50000 |  3.51 $\pm$ 0.10 3.51 $\pm$ 0.10 |  5.94 $\pm$ 0.10 6.44 |  Bouret, J.-C. et al. (2012) Herrero et al. (2000)
HD15629 |  40000 48000 |  3.71 3.81 $\pm$ 0.10 |  5.60 5.89 |  Repolust, T. et al. (2004) Herrero et al. (2000)
HD190864 | 37000 | 3.55 | 5.41 | Repolust, T. et al. (2004)
HD16691 | 41000 $\pm$ 1000 | 3.66 $\pm$ 0.10 | 5.94 $\pm$ 0.10 | Bouret, J.-C. et al. (2012)
Table 4: Fundamental parameters estimations from other authors.

After neural network design, training, and validation, we create 10 different
neural networks with the same structure put through the same training process
in order to establish the uncertainty of every network due to the natural
randomness of these deep learning algorithms. The results of this validation
process are shown on Figures 9, 10, and 11, where the uncertainties of the
estimations are represented as error bars similar to the reference values from
other authors.

Figure 9: Effective Temperature predictions comparison over the 9 observed
spectra using the 3 different RNNs approaches. Where the reference values
without error bars doesn’t have a uncertainty value on the published paper,
and the error bars on the estimated values are the uncertainty of the RNN.
Figure 10: Surface Gravity predictions comparison over the 9 observed spectra
using the 3 different RNNs approaches. Where the reference values without
error bars doesn’t have a uncertainty value on the published paper, and the
error bars on the estimated values are the uncertainty of the RNN. Figure 11:
Luminosity predictions comparison over the 9 observed spectra using the 3
different RNNs approaches. Where the reference values without error bars
doesn’t have a uncertainty value on the published paper, and the error bars on
the estimated values are the uncertainty of the RNN.

In order to analyze the impact of the noise over the estimations of the
different neural network structures, we compare the results with the
estimations from other authors shown on Table 4.

We show on Tables 6, 7, 8, the difference between the Effective Temperature,
Surface Gravity, and Luminosity estimated by the RNN and the estimations shown
on Table 4 made by previous authors. Every table shows the results using the
distinct input tensors, The EW of the 34 lines and the top 6 profile lines
with higher correlation, the EW of the 34 lines and the metals profile lines,
and finally, the input tensor with only the EW of the 34 lines. Where the
difference was calculated as relative to the reference values define in Eq. 1.

These tables were ordered upward by S/N (from high to low noise) and using a
gradient color where red is more noise, and most intense green, less noise.
Also, the three columns of differences have a gradient color where red is more
difference and most intense green less difference.

$\Delta_{Relative}=\frac{|Reference_{val}-RNN_{val}|}{Reference_{val}},$ (1)

Where $Reference_{val}$ is the physical parameter from other authors and
$RNN_{val}$ is the physical parameter estimated by the Recurrent Neural
Network proposed in this work.

Based on the Eq. 1, the three comparative columns on the Tables 6, 7, 8 are
define in Table 5.

Column | Input Tensor
---|---
$\Delta_{Relative}^{(1)}$ | 34 EW & Top 6 profile lines
$\Delta_{Relative}^{(2)}$ | 34 EW & Non-H/He profile lines
$\Delta_{Relative}^{(3)}$ | 34 EW
Table 5: Comparative columns definition shown on Tables 6, 7, 8. Star | S/N |  $\Delta_{Relative}^{(1)}$ $T_{eff}(K)$ |  $\Delta_{Relative}^{(2)}$ $T_{eff}(K)$ |  $\Delta_{Relative}^{(3)}$ $T_{eff}(K)$ | Reference
---|---|---|---|---|---
HD190864 | 18.27 |  0.30 |  0.08 |  0.25 | 2004
HD216898 | 58.78 |  0.15 |  0.14 |  0.07 | 2009
HD216532 | 84.05 |  0.12 |  0.21 |  0.07 | 2009
HD16691 | 84.05 |  0.28 |  0.20 |  0.01 | 2012
HD108 | 85.81 |  0.05 |  0.04 |  0.13 | 2012
HD108 | 85.81 |  0.05 |  0.04 |  0.13 | 2010
HD108 | 85.81 |  0.10 |  0.02 |  0.17 | 2008
HD15629 | 89.40 |  0.00 |  0.02 |  0.09 | 2004
HD15629 | 89.40 |  0.16 |  0.15 |  0.24 | 2000
HD15558 | 103.01 |  0.00 |  0.02 |  0.11 | 2012
HD15558 | 103.01 |  0.12 |  0.10 |  0.21 | 2000
HD15570 | 108.47 |  0.06 |  0.12 |  0.05 | 2012
HD15570 | 108.47 |  0.20 |  0.15 |  0.28 | 2000
HD203064 | 189.30 |  0.05 |  0.06 |  0.03 | 2017
HD203064 | 189.30 |  0.07 |  0.08 |  0.01 | 2004
Table 6: Difference between the Surface Gravity estimated by the RNN and the
reference from other authors (red is higher difference, green lower
difference), ordered by S/N (in ascending order).

The Table 6 shows that use the EW plus the Non-H/He profile lines the overall
difference is lower compared to the other two input tensor versions.
Additionally, non of the versions show a clear better estimation when the
noise in the signal decrease, however the $\Delta_{Relative}^{(2)}$ shows
better results on the stellar spectrum with higher noise and the
$\Delta_{Relative}^{(3)}$ shows better results on the 2nd (HD216898), 3rd
(HD216532), and 4th (HD16691) star spectrum with more noise.

Star | S/N |  $\Delta_{Relative}^{(1)}$ $log$ $g$ |  $\Delta_{Relative}^{(2)}$ $log$ $g$ |  $\Delta_{Relative}^{(3)}$ $log$ $g$ | Reference
---|---|---|---|---|---
HD190864 | 18.27 |  0.11 |  0.22 |  0.05 | 2004
HD216898 | 58.78 |  0.15 |  0.15 |  0.20 | 2009
HD216532 | 84.05 |  0.08 |  0.07 |  0.10 | 2009
HD16691 | 84.05 |  0.04 |  0.01 |  0.01 | 2012
HD108 | 85.81 |  0.07 |  0.07 |  0.08 | 2012
HD108 | 85.81 |  0.07 |  0.07 |  0.08 | 2010
HD108 | 85.81 |  0.13 |  0.13 |  0.14 | 2008
HD15629 | 89.40 |  0.11 |  0.08 |  0.01 | 2004
HD15629 | 89.40 |  0.13 |  0.10 |  0.04 | 2000
HD15558 | 103.01 |  0.13 |  0.13 |  0.10 | 2012
HD15558 | 103.01 |  0.05 |  0.06 |  0.02 | 2000
HD15570 | 108.47 |  0.03 |  0.05 |  0.11 | 2012
HD15570 | 108.47 |  0.03 |  0.05 |  0.11 | 2000
HD203064 | 189.30 |  0.02 |  0.01 |  0.08 | 2017
HD203064 | 189.30 |  0.04 |  0.06 |  0.02 | 2004
Table 7: Difference between the Surface Gravity estimated by the RNN and the reference from other authors (red is higher difference, green lower difference), ordered by S/N (in ascending order). Star | S/N |  $\Delta_{Relative}^{(1)}$ $log(L/L\textsubscript{$\odot$})$ |  $\Delta_{Relative}^{(2)}$ $log(L/L\textsubscript{$\odot$})$ |  $\Delta_{Relative}^{(3)}$ $log(L/L\textsubscript{$\odot$})$ | Reference
---|---|---|---|---|---
HD190864 | 18.27 |  0.10 |  0.12 |  0.11 | 2004
HD216898 | 58.78 |  0.16 |  0.15 |  0.16 | 2009
HD216532 | 84.05 |  0.17 |  0.12 |  0.21 | 2009
HD16691 | 84.05 |  0.06 |  0.05 |  0.02 | 2012
HD108 | 85.81 |  0.03 |  0.06 |  0.04 | 2012
HD108 | 85.81 |  0.03 |  0.06 |  0.04 | 2010
HD108 | 85.81 |  0.03 |  0.01 |  0.09 | 2008
HD15629 | 89.40 |  0.06 |  0.03 |  0.07 | 2004
HD15629 | 89.40 |  0.01 |  0.02 |  0.02 | 2000
HD15558 | 103.01 |  0.04 |  0.02 |  0.07 | 2012
HD15558 | 103.01 |  0.00 |  0.02 |  0.03 | 2000
HD15570 | 108.47 |  0.04 |  0.00 |  0.07 | 2012
HD15570 | 108.47 |  0.05 |  0.08 |  0.01 | 2000
HD203064 | 189.30 |  0.13 |  0.11 |  0.19 | 2017
HD203064 | 189.30 |  0.05 |  0.03 |  0.11 | 2004
Table 8: Difference between the Luminosity estimated by the RNN and the
reference from other authors (red is higher difference, green lower
difference), ordered by S/N (in ascending order).

After analyze the results on Figures 9, 10, and 11, and compare with the
difference on estimations with other authors shown on the gradient color
Tables 6, 7, 8, it is clear that the improvement on the system by adding
profile lines is not too high in all the observed spectra, while the
difference of the computational resources to train the networks goes from
processing data tensors with a dimension of $34\times N$, to a minimal of
$34\times 506\times N$, and a max of $34\times 904\times N$, with $N$ as the
total spectra samples. Moreover, the uncertainty for some observed spectra was
close, as we can see on HD190864 and HD15629, where the 34 EW version of the
system has very close estimations compared with the results on the 34 EW $+$
Non-H/He profiles lines on the same observed stellar spectra. Besides that,
for the Effective Temperature estimations, the addition of profile lines got
closer results to reference values over the 34 EW approach. Also, the
uncertainty was lower in this case. Additionally, the uncertainty on some
observed spectra increases for all three approaches, over the observed spectra
of HD15558, HD15570, HD15629, and HD190864 stars. A clear response to the
missing data in the spectrum. In this field, the system with 34 EW $+$ top 6
profiles lines input tensor, got lower uncertainty on the estimations of
Luminosity and Surface Gravity.

##  5 Conclusions

We have developed a deep learning system using an RNN structure able to
estimate stellar parameters with lower S/N observed spectra. In some cases,
the uncertainty of the estimations are near to the published on previous
works.

The RNNs show good results with high and low levels of S/N. For the three
input approaches the increase in the discrepancy respond to the decrease of
the S/N level, however, we have some outliers in the estimations. Look in to
with more detail, the stars mentioned before that does not have the spectrum
complete show higher uncertainty compared with the other observations.

The different input data approaches show useful information about the possible
requirements that the individual networks for every parameter could need. The
RNN that uses the 34 EW & the Non-H/He profile lines estimates closer values
of $T_{eff}(K)$, $log(L/Lo)$, while for $log$ $g$, the RNN with the input
tensor defines by the 34 EW & Top 6 profile lines more correlated to this
physical parameter, estimates closer values.

###  5.1 Future Work

We think it is crucial to extend the observed spectra database with different
levels of S/N and resolution in order to increase the analysis of the system
capabilities, and improve the accuracy after enhance the training set and
tuning the recurrent neural networks if its needed. Design and develop a final
system proposal in order to estimates more stellar parameters that the models
already handle. We are also open to defining different input tensors that can
collect the most useful information of a full stellar spectrum and stills
improve the system from a computational resources perspective.

## Acknowledgments

Based upon observations carried out at the Observatorio Astronómico Nacional
on the Sierra San Pedro Mártir (OAN-SPM), Baja California, México.

## Data Availability

A set of models for O-type stars used in this article are available at
http://www.astroscu.unam.mx/atlas/. Additionally, you can contact  Celia F.
Fierro-Santillán for the models availability. And the trained model, training
set, and test data set will be shared on reasonable request to the author
Miguel Flores R.

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11institutetext: University of Helsinki, Helsinki, Finland
11email<EMAIL_ADDRESS>CSC – IT Center for
Science Ltd, Espoo, Finland
22email<EMAIL_ADDRESS>
#  $\chi$iplot: web-first visualisation platform for multidimensional
data††thanks: Supported by the Research Council of Finland (decisions 346376
and 345704) and the Future Makers Funding Programme of Technology Industries
of Finland Centennial Foundation and Jane and Aatos Erkko Foundation.

Akihiro Tanaka 11 0000-0003-2363-9790 Juniper Tyree 11 0000-0002-7923-9609
Anton Björklund 11 0000-0002-7749-2918 Jarmo Mäkelä 1122 0000-0002-8788-3939
Kai Puolamäki 11 0000-0003-1819-1047

###### Abstract

$\chi$iplot is an HTML5-based system for interactive exploration of data and
machine learning models. A key aspect is interaction, not only for the
interactive plots but also between plots. Even though $\chi$iplot is not
restricted to any single application domain, we have developed and tested it
with domain experts in quantum chemistry to study molecular interactions and
regression models. $\chi$iplot can be run both locally and online in a web
browser (keeping the data local). The plots and data can also easily be
exported and shared. A modular structure also makes $\chi$iplot optimal for
developing machine learning and new interaction methods.

###### Keywords:

Visualisation Interactive visualisation Data visualisation Python HTML5 WASM
HCI.

This is the author’s version of the work. The definitive Version of Record
will be published in the Proceedings of ECML PKDD 2023.

##  1 Introduction and related work

This paper introduces $\chi$iplot (”kaIpl6t), a modular system for interactive
exploration of data and pre-trained machine learning models. $\chi$iplot can
be run locally on the user's computer or installation-free in a web browser.
Our motivation for writing $\chi$iplot was three-fold.

(i) First, we want a Python-based system to develop and test machine learning
and dimensionality reduction methods, such as [1], a manifold visualisation
method for explainable AI. For this purpose, we prefer a modular system that
is easy to expand and modify to test new machine learning and visualisation
methods and interaction ideas.

(ii) Second, we need a tool to facilitate collaboration with primarily domain
experts in quantum chemistry but also other domains. Ideally, we want to avoid
forcing our collaborators to install additional software. However, we also do
not want to set up and maintain server infrastructure to host a web-accessible
service.

(iii) Third, the system should be practical and usable for the end user,
including physicists and chemists, despite being built for quick prototyping
and painless implementation. We know no prior system satisfies all of these
three requirements.

Many interactive visualisation tools are available; see, e.g., [7] for a
recent survey and references. Much of our research collaboration targets
quantum chemistry; hence the system must also be capable of visualising, e.g.,
molecular structures from SMILES strings [11]. ChemInformatics Model
Explorer[5] (CIME) is another tool that explores explainable AI in small
molecule research. However, CIME has only four fixed views, and full
functionality requires a server. Another recent example is XSMILES[4], where
users can examine individual molecules in 2D diagrams and visualise
attribution scores for atoms and non-atom tokens.

##  2 Usage

The main idea of $\chi$iplot is to simultaneously show multiple plots and
visualisations to compare and contrast diverse information. Since $\chi$iplot
also targets non-technical end users, intuitive visual selection and
configuration of the plots are required.

$\chi$iplot comes with six types of plots out-of-the-box – scatterplots,
histograms, heat maps, bar plots, data tables, and SMILES plots, which render
molecules in a stick structure from a SMILES string[11] – but more can be
added with $\chi$iplot's plugin system. Users can add and remove plots to
create a layout that is the most optimal for their specific needs. The end
users have the capability to generate clusters by running a k-means algorithm
or by lasso selection on a scatterplot. Unique colours distinguish the
generated clusters. In addition, the end users can generate a 2D embedding
through Principal Component Analysis (PCA).

To use $\chi$iplot, the user may install it with pip install xiplot. The
xiplot console command is then available to host a local $\chi$iplot server.
Alternatively, an _installation-free_ WebAssembly (WASM)111WASM is supported
in most modern browsers; see https://caniuse.com/wasm. version can be used
immediately at https://edahelsinki.fi/xiplot.

Figure 1: $\chi$iplot interface when studying a regression model on a QM9
dataset.

We demonstrate the main concepts with the QM9 molecular dataset [8, 9], a
collection of quantum chemical properties calculated for small organic
molecules. Our machine-learning task is to estimate some quantum chemical
properties from their structural description. We can use physics simulators
with varying fidelity or regression models. In this example, we want to study
how the structures in the dataset relate to the estimation task. We have
precomputed a 2D Slisemap[1] embedding (revealing the structures relevant to a
regression model) and attached the embedding to the dataset file we uploaded
to $\chi$iplot.

Fig. 1 shows a view of the $\chi$iplot interface during our exploration. A
chemist can explore the Slisemap embedding in a scatter plot on the left.
There is a notable cluster structure, so we use $\chi$iplot to find the
clusters and plot their distribution in the middle. If we compare the two
clusters, we notice that the distributions of the functional groups differ.
For example, we could manually draw an additional cluster in the scatter plot
to further study the two subgroups in the rightmost cluster.

The behaviour of a molecule is not only determined by the functional groups
but also by how they are structured. However, finding good summary statistics
for structure is much more difficult. Therefore, we add a visualisation of
individual molecules on the right of Fig. 1. A chemist can then rapidly
inspect multiple molecules inside and between clusters by hovering over the
points in the scatter plot; the molecule visualisation is automatically
updated.

##  3 Description of the system

A key aspect of $\chi$iplot is interactivity, not just for a single plot but
also between plots. For example, selecting a data item in one might show you
more information about it in another, as described above. To accomplish this
interactivity, the plots of $\chi$iplot are implemented as independent
modules, communicating through shared data storage. Furthermore, to support
collaboration and sharing, the set of active plots, their configuration, and
the data can be saved to and restored from a file. Since $\chi$iplot is an
interactive system, time-consuming computations (e.g., learning the Slisemap
embedding) should be done as part of data preprocessing.

$\chi$iplot is implemented in Python using Plotly[6] for the plots and Dash
for the interactivity. Usually, this would require the users to be able to
install Python packages (see Section 2). However, we also provide a static
server-less webpage version of $\chi$iplot that runs both the Dash backend and
the Plotly frontend installation-free inside a browser using WebAssembly[10]
(WASM). This also means no data leaves the user's computer in the WASM
version.

In detail, the WASM version of $\chi$iplot uses Pyodide[3] to run Python in
the browser. The front- and backend communication is intercepted and
redirected to the in-WASM server, inspired by the WebDash prototype [2].
Crucially, neither the front- nor backend code needs to know that it runs
inside a browser.

As Pyodide does not yet support all Python packages, we use dynamic import
detection to enable certain features and fallbacks, such as additional data
file formats. Deploying the WASM version requires bundling all frontend files,
$\chi$iplot, and the scripts that bootstrap the web app in the WASM backend,
all documented in the $\chi$iplot GitHub repository.

To open up $\chi$iplot to even more use cases, $\chi$iplot has an API for
creating plugins for, e.g., new visualisations and machine learning methods.
It uses the ``entry points'' feature of Python to discover installed plugins,
which also works in the WASM version. Due to the modular design with shared
data, new plots can automatically interact with old ones.

##  4 Conclusions

We have already found $\chi$iplot helpful when collaborating with domain
experts since it lets them configure interactive plots without programming or
installing anything222Installation-free version at
https://edahelsinki.fi/xiplot. The online version also enables easy results
sharing without exposing the data to any third party. For more technical users
$\chi$iplot is easy to maintain end expand due to the modular architecture.
Finally, $\chi$iplot is available under the Open Source MIT license from
GitHub333https://github.com/edahelsinki/xiplot (which includes documentation,
usage examples, and a demonstration video).

## References

  * [1] Björklund, A., Mäkelä, J., Puolamäki, K.: SLISEMAP: Supervised dimensionality reduction through local explanations. Machine Learning 112(1), 1–43 (Jan 2023). https://doi.org/10.1007/s10994-022-06261-1
  * [2] Dafna, I., Tulop, J., Ivanov, P.: Webdash (2022), https://github.com/ibdafna/webdash
  * [3] Droettboom, M., Chatham, H., Yurchak, R., Choi, G., et al.: Pyodide/pyodide: 21.0 (Aug 2022). https://doi.org/10.5281/ZENODO.6977227
  * [4] Heberle, H., Zhao, L., Schmidt, S., Wolf, T., Heinrich, J.: XSMILES: Interactive visualization for molecules, SMILES and XAI attribution scores. Journal of Cheminformatics 15(1), 2 (Jan 2023). https://doi.org/10.1186/s13321-022-00673-w
  * [5] Humer, C., Heberle, H., Montanari, F., Wolf, T., Huber, F., Henderson, R., Heinrich, J., Streit, M.: ChemInformatics Model Explorer (CIME): Exploratory analysis of chemical model explanations. Journal of Cheminformatics 14(1), 21 (Dec 2022). https://doi.org/10.1186/s13321-022-00600-z
  * [6] Plotly: Plotly Open Source Graphing Library for Python (2023), https://plotly.com/python/
  * [7] Qin, X., Luo, Y., Tang, N., Li, G.: Making data visualization more efficient and effective: A survey. The VLDB Journal 29(1), 93–117 (Jan 2020). https://doi.org/10.1007/s00778-019-00588-3
  * [8] Ramakrishnan, R., Dral, P.O., Rupp, M., von Lilienfeld, O.A.: Quantum chemistry structures and properties of 134 kilo molecules. Scientific Data 1(1), 140022 (Aug 2014). https://doi.org/10.1038/sdata.2014.22
  * [9] Stuke, A., Todorović, M., Rupp, M., Kunkel, C., Ghosh, K., Himanen, L., Rinke, P.: Chemical diversity in molecular orbital energy predictions with kernel ridge regression. The Journal of Chemical Physics 150(20), 204121 (May 2019). https://doi.org/10.1063/1.5086105
  * [10] W3C: WebAssembly Core Specification (Apr 2022), https://www.w3.org/TR/wasm-core-2
  * [11] Weininger, D.: SMILES, a chemical language and information system. 1. Introduction to methodology and encoding rules. Journal of Chemical Information and Computer Sciences 28(1), 31–36 (Feb 1988). https://doi.org/10.1021/ci00057a005


Can one hear the full nonlinearity of a PDE from its small excitations?

Maxim Olshanii1 and Danshyl Boodhoo1$\star$

1 Department of Physics, University of Massachusetts Boston, Boston
Massachusetts 02125, USA

$\star$<EMAIL_ADDRESS>,

## Abstract

In this article, we show how one can restore an _unknown_ nonlinear partial
differential equation of a sine-Gordon type from its linearization around an
unknown stationary kink. The key idea is to regard the ground state of the
linear problem as the translation-related Goldstone mode of the nonlinear PDE
sought after.

Copyright attribution to authors.
This work is a submission to SciPost Physics Core.
License information to appear upon publication.
Publication information to appear upon publication. Received Date
Accepted Date
Published Date

##  1 Introduction

In mathematical physics, inverse problems are ones of the hardest. The most
famous one goes by the name “Can One Hear the Shape of a Drum?”[1]. There, one
attempts to infer the shape of a membrane from the spectrum of its vibrations.
Identifying a quantum scatterer from its scattering data [2] is another
example. In this article we pose the following question: “Given a linear
stability analysis equation around an unknown stationary solution of an
unknown nonlinear Partial Differential Equation (PDE), can one restore the PDE
itself?” Below, we will be using the term “inverse linearization” for such a
procedure.

##  2 Principal result: the inverse linearization protocol for PDEs of the
sine-Gordon type

Consider a linear stationary Schrödinger equation:

$\displaystyle-\delta u_{xx}+V(x)\delta u=E\delta u\quad,$ (1)

with $\delta u=\delta u(x)$. Assume that the potential $V(x)$ supports at
least one bound state. Consider its ground state $\delta u_{0}(x)$, of an
energy $E_{0}$. It is known that the ground state $\delta u_{0}(x)$ does not
have nodes and that it can be made purely real given an appropriate prefactor
[3]. Accordingly, from now on, we will assume that the ground state obeys

$\delta u_{0}\stackrel{{\scriptstyle-\infty<x<+\infty}}{{>}}0\,\,.$

Let us now rewrite the Schrödinger equation (1) as

$\displaystyle-\delta u_{xx}+(V(x)-E_{0})\delta u=\omega^{2}\delta u\,\,,$ (2)

with $\omega^{2}\equiv E-E_{0}$. (Obviously, the ground state corresponds to
$\omega=0$; it obeys

$-(\delta u_{0})_{xx}+(V(x)-E_{0})\delta u_{0}=0\,\,.)$

Imagine now that the equation (18) constitutes an eigenmode equation for small
excitations around a stationary kink-soltion of an unknown PDE of the sine-
Gordon type, further referred as a blank-Gordon equation:

$\displaystyle u_{xx}-u_{tt}=F(u)$ (3) $\displaystyle u=u(x,\,t)\,\,.$

We claim the following:

  1. 1.

The kink of (3) is related to the ground state of (1) as

$\displaystyle\bar{u}(x)=A(B+\int_{-\infty}^{x}\delta
u_{0}(x^{\prime})\,dx^{\prime})\,\,$ (4)

with $A$ and $B$ being arbitrary constants;

  2. 2.

The nonlinearity $F(u)$ is given by

$\displaystyle F(u)=f(\chi(u))\,\,,$ (5)

where $f(x)$ is proportional to the first derivative of the ground state of
(1),

$\displaystyle f(x)=A(\delta u_{0})_{x}(x)\,\,,$ (6)

and the function $\chi(u)$ is the inverse of $\bar{u}(x)$:

$\displaystyle\chi(\bar{u}(x))=x\,\,.$

The rigorous formulation of the time-dependent small excitation problem is as
follows. The small excitations dress a stationary kink $\bar{u}(x)$,

$\displaystyle\bar{u}_{xx}=F(\bar{u})\,\,.$ (7)

A general time-dependent solution $u(x,t)$ of the nonlinear PDE (3) is
represented as

$\displaystyle u(x,t)=\bar{u}(x)+\delta u(x)e^{\pm i\omega t}+{\cal O}(\delta
u(x)^{2})\,\,.$ (8)

The expansion (8) is substituted to the nonlinear equation (3), and
subsequently, only the terms of the first order in $\delta u(x)$ are kept.
This procedure results in the following eigenmode equation:

$\displaystyle-\delta u_{xx}+F^{\prime}(\bar{u}(x))\delta u=\omega^{2}\delta
u\,\,.$ (9)

##  3 Justification of the inverse linearization procedure (4), (5), (6)

First of all, in Section 2, we assumed that the inverse $\chi(\bar{u})$ of
$\bar{u}(x)$ (see (4)) exists. Thanks to the positivity of $\delta u_{0}(x)$,
the existence of this inverse is however guaranteed: $\bar{u}(x)$ is a
monotonic function of $x$.

Secondly, we need to prove that

$\displaystyle F^{\prime}(\bar{u}(x))=V(x)-E_{0}\,\,.$ (10)

Indeed

$\displaystyle F^{\prime}(\bar{u}(x))$
$\displaystyle=\frac{d}{du}F(u)\Big{|}_{u=\bar{u}(x)}$ (11)
$\displaystyle=\left(\frac{d}{dx}f(x)\Big{|}_{x=\chi(\bar{u}(x))}\right)\left(\frac{d}{du}\chi(u)\Big{|}_{u=\bar{u}(x)}\right)$
(12) $\displaystyle=\frac{\frac{d}{dx}f(x)}{\frac{d}{dx}\bar{u}(x)}$ (13)
$\displaystyle=\frac{A(\delta u_{0})_{xx}(x)}{A\delta u_{0}(x)}$ (14)
$\displaystyle=V(x)-E_{0}\,\,.$ (15)

End of proof.

However, the proof above tell one nothing about _why_ the recipe works. The
cornerstone for _understanding_ the physics behind the connection formulas
(4,5, 6) is the Goldstone theorem: if a PDE possesses a continuous symmetry,
the action of the generator of this symmetry on a stationary solution is a
small excitation around this solution, with a zero frequency. In our case, the
symmetry in question is the translational invariance of the PDE (3) in
question. Therefore the spatial derivative of a stationary solution is a
legitimate zero-frequency mode of the linear stability analysis equation
around that stationary solution. Conversely, the kink is proportional to an
integral over that eigenstate; hence the connection (4), where it is assumed
that it is the ground state of (1) that corresponds to the Golstone mode. This
is the connection between a nonlinear PDE and its linearization that we are
exploiting in this work. This connection can also be seen explicitly by
differentiating (7) with respect to $x$ and observing that the result is
identical to the linearization (9) with $\omega=0$, applied to the kink state
$\bar{u}$.

##  4 Worked examples

First of all, we would like to verify that our procedure returns the sine-
Gordon equation, if applied to its linearization around its stationary
soliton. In that case the potential of the linearized problem is the Pöschl-
Teller potential [4], $V(x)=-n(n+1)\operatorname{sech}^{2}(x)$, with $n=1$
(see row 1 of Table 1). Using its ground state, $\delta
u_{0}=\operatorname{sech}(x)$, to initialize the protocol in the protocol (4),
(5), (6), with $A=2$ and $B=0$ we indeed recover the sine-Gordon equation.

For $n=2$, a particular instance of the $\phi^{4}$ model is produced (row 2 of
the Table 1). The PDE produced by the $n=3$ Hamiltonian (row 3) does not
appear to be expressible through elementary functions. The $n=4$ case (row 4)
appears to produce a PDE that has not been studied before. Note however that
in this case, the function $F(u)$ has branching points at $u=\pm 1$. It is not
clear if the corresponding PDE will be well defined there.

The $\delta$-well and the harmonic oscillator case are also included in our
study (rows 5 and 6 respectively). In the former case, the nonlinearity $F(u)$
is a piece-wise-linear, periodic function of $u$: we named the resulting PDE a
sawtooth-Gordon equation [5]. In the latter case, a function $F(u)$ that is
expressed in terms of the inverse of the error function emerges. Similarly to
the $n=4$ Pöschl-Teller case, harmonic oscillator leads to a nonanalytic PDE.

$V(x)$ | $E_{0}$ | $A$ | $B$ | $\bar{u}(x)$ | $F(u)$
---|---|---|---|---|---
$-2\operatorname{sech}^{2}(x)$ | $-1$ | $2$ | $0$ | $4\arctan(e^{x})$ | $\begin{array}[]{c}\sin(u)\\\ \text{(sine-Gordon})\end{array}$
$-6\operatorname{sech}^{2}(x)$ | $-4$ | $1$ | $-1$ | $\tanh(x)$ | $\begin{array}[]{c}-2u+2u^{3}\\\ \text{($\phi^{4}$ model)}\end{array}$
$-12\operatorname{sech}^{2}(x)$ | $-9$ | $1$ | $0$ | $\arctan(e^{x})+\frac{\sinh(x)}{2\cosh^{2}(x)}$ | $\begin{array}[]{c}\text{no elementary}\\\ \text{functions for $F(u)$}\end{array}$
$-20\operatorname{sech}^{2}(x)$ | $-16$ | $\frac{3}{2}$ | $-\frac{2}{3}$ | $\frac{3}{2}\tanh(x)-\frac{1}{2}\tanh^{2}(x)$ | $-12\varphi(u)\left(1-4\varphi^{2}(u)\right)$
$-\delta(x)$ | $-\frac{1}{4}$ | $\frac{1}{2}$ | $-2$ | $\operatorname{sign}(x)(1-e^{-|x|/2})$ | $\begin{array}[]{c}\frac{1}{4}(u-\operatorname{sign}(u))\\\ \text{(sawtooth-Gordon)}\end{array}$
$4x^{2}$ | $2$ | $\frac{2}{\sqrt{\pi}}$ | $-\frac{\sqrt{\pi}}{2}$ | $\operatorname{erf}(x)$ | $-\frac{4e^{-\operatorname{inverf}(u)^{2}}\operatorname{inverf}(u)}{\sqrt{\pi}}$
Table 1: Worked examples of the application of the inverse linearization
procedure that allows to restore an unknown PDE from its linear stability
analysis counterpart, around a stationary kink soliton. The ground state of
the linear system (a linear Schrödinger equation in our case) is the
translational Goldstone mode of the problem. The first column corresponds to
the potential of the linear problem (1), with $E_{0}$ (second column) being
its ground state energy. The third and the forth columns are the parameters of
the ground state vs. kink connection (4). The fifth column identifies the
relevant kink soliton. The sixth column gives the revealed PDE. Here and
below, $\varphi(u)\equiv\sin\left(\frac{\arcsin(u)}{3}\right)$ . The function
$\operatorname{inverf}(u)$ is the inverse of the error function.

##  5 Conclusions and outlook

In this article, we showed that information about low amplitude excitations of
a sine-Gordon-type PDE with an unknown nonlinearity (the blank-Gordon
equation), in the vicinity of its single kink stationary state (also unknown),
is sufficient to recover that nonlinearity. If one regards the blank-Gordon
equation as a toy field theory, our finding would also imply that in a two-
vacuum state of that theory, low-energy physics contains full information
about the high-energy sector.

Next, in [5], it has been shown that integrable PDEs produce reflectionless
linear stability analysis equations. Regretfully, the converse is not true, as
witnessed by the the row 2 of Table 1 pertaining to formed the $n=2$ Pöschl-
Teller Hamiltonian (see [6] on its transparency) and a particular $\phi^{4}$
model ($F(u)=-2u+2u^{3}$) whose non-integrable behavior has been well
documented [7].

Looking ahead, several directions future research can be foreseen. First note
that the ground state of the linear Schrödinger equation (1) is not the only
eigenstate that produces a monotonic (and hence invertible) kink state
$\bar{u}(x)$ (see (4)). For $V(x)$ supporting a single bound state or no bound
states at all, the $E=0$ eigenstate can be chosen to be monotonic, leading to
a monotonic $\bar{u}(x)$. In that case the equation (18) needs to be modifies
as

$\displaystyle-\delta u_{xx}+V(x)\delta u=\omega^{2}\delta u\,\,,$ (16)

with $\omega^{2}\equiv E$. For example, the $E=0$ of the $n=1$ Pöschl-Teller
potential produces the Liouville equation [8, 5], whose kink solitons grow
linearly at $x\to\pm\infty$. Remark that the latter behavior is not generic:
it follows from the absence of a linear divergence of the $E=0$ eigenstate of
the $n=1$ Pöschl-Teller Hamiltonian, the former being specific to the quantum
supersymmetric Hamiltonians [9].

In retrospect, the monotonicity of the kink state $\bar{u}(x)$ is a sufficient
but not a necessary condition for our inverse linearization scheme to
function. As follows from the equation (7), any function $\bar{u}(x)$ whose
second spatial derivative $\bar{u}_{xx}(x)$ is a function of $\bar{u}(x)$
itself can be a candidate for a kink soliton of a nonlinear PDE. And purely
monotonic functions $\bar{u}(x)$ are not the only members of this class. It
can be proven that there exist three and only three cases when

$\displaystyle\bar{u}_{xx}(x)=F(\bar{u}(x))\,\,:$ (17)

  * Case 1.

$\bar{u}(x)$ is monotonic;

  * Case 2.

$\bar{u}(x)$ has a single extremum and it is spatially even with respect to
this extremum;

  * Case 3.

$\bar{u}(x)$ is periodic; $\bar{u}(x)$ has a single local maximum and a single
local minumum per period, the former being separated by a half-period;
$\bar{u}(x)$ is spatially even with respect to each of its extrema.

Case 1 is obvious. To prove that the remaining instances are covered by the
Cases 2 and 3, observe the following. The equation (17) can be regarded as a
second order ODE. At a point of extremum, $x_{0}$, the initial conditions on
$\bar{u}(x_{0})$ and $\bar{u}_{x}(x_{0})=0$ are spatially even, and they will
generate a function that is even about $x_{0}$. If this is the only extremum,
the Case 2 is covered. In case if there are more than one extrema, it is easy
to show inductively that any pair consisting of a minimum and a maximum, at
$x_{0,\,\text{min}}$ and $x_{0,\,\text{max}}$, with no extrema in between
generates a periodic function with a period of
$2|x_{0,\,\text{max}}-x_{0,\,\text{min}}|$. This constitutes the Case 3.

As an example of an inverse linearization pertaining to the Case 2, one may
consider the fist excited state $\delta u_{1}(x)$ (of energy $E_{1}=-1$) of
the $n=2$ Pöschl-Teller problem leading to a different kind of the $\phi^{4}$
model, with $F(u)=u-2u^{3}$. The analogue of the equation (18) will read

$\displaystyle-\delta u_{xx}+(V(x)-E_{1})\delta u=\omega^{2}\delta u\,\,,$
(18)

with $\omega^{2}\equiv E-E_{1}$.

We were not able to identify any instances corresponding to the Case 3.

It should be noted that whenever the Goldstone mode is not represented by the
ground state of the linearization problem, the corresponding kink soliton is
dynamically unstable. Indeed, in that case, the ground state, as well as any
other eigensate with energy below the Goldstone value will have an imaginary
frequency.

One more comment is in order. Rigorously speaking, the connection (4), (5),
(6) generates a _family_ of PDEs. While the constant $A$ trivially corresponds
to normalization of $\delta u_{0}$ (more generally, the state associated with
the Goldstone mode), different values of the constant $B$ may produce
different PDEs. This freedom needs to be explored further.

## Acknowledgements

We are grateful to Vanja Dunjko for his valuable comments.

### Funding information

MO was supported by the NSF Grant No. PHY-1912542.

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