Further author information: (Send correspondence to G.G.)
G.G.: E-mail<EMAIL_ADDRESS>Telephone: +31 71 565 4625

# Optical throughput and sensitivity of JWST NIRSpec

Giovanna Giardino  ATG Europe for the European Space Agency, ESTEC, The
Netherlands  Rachana Bhatawdekar  European Space Agency, ESTEC, Noordwijk, The
Netherlands  Stephan M. Birkmann  European Space Agency, STScI, Baltimore, USA
Pierre Ferruit  European Space Agency, ESAC, Madrid, Spain  Timothy Rawle
European Space Agency, STScI, Baltimore, USA  Catarina Alves de Oliveira
European Space Agency, ESAC, Madrid, Spain  Torsten Böker  European Space
Agency, STScI, Baltimore, USA  Peter Jakobsen  Cosmic Dawn Center, Niels Bohr
Institute, Copenhagen, Denmark  Nimisha Kumari  AURA for the European Space
Agency, STScI, Baltimore, USA  Marcos López-Caniego  Aurora Technology for the
European Space Agency, ESAC, Madrid, Spain  Nora Lützgendorf  European Space
Agency, STScI, Baltimore, USA  Elena Manjavacas  AURA for the European Space
Agency, STScI, Baltimore, USA  Charles Proffitt  Space Telescope Science
Institute, Baltimore, USA  Marco Sirianni  European Space Agency, STScI,
Baltimore, USA  Maurice Te Plate  European Space Agency, STScI, Baltimore, USA
Peter Zeidler  AURA for the European Space Agency, STScI, Baltimore, USA

###### Abstract

To achieve its ambitious scientific goals, the Near-Infrared Spectrograph,
NIRSpec, on board the Webb Space Telescope, needs to meet very demanding
throughput requirements, here quantified in terms of photon-conversion
efficiency (PCE). During the calibration activities performed for the
instrument commissioning, we have obtained the first in-flight measurements of
its PCE and also updated the modeling of the light losses occurring in the
NIRSpec slit devices.

The measured PCE of NIRSpec fixed-slit and multi-object spectroscopy modes
overall meets or exceeds the pre-launch model predictions. The results are
more contrasted for the integral-field spectroscopy mode, where the
differences with the model can reach $-$20%, above 4 $\mu$m, and exceed
$+$30%, below 2 $\mu$m. Additionally, thanks to the high quality of the JWST
point-spread function, our slit-losses, at the shorter wavelength, are
significantly decreased with respect to the pre-flight modeling.

These results, combined with the confirmed low noise performance of the
detectors, make of NIRSpec an exceptionally sensitive spectrograph.

###### keywords:

JWST, NIRSpec, near-infrared spectrograph

##  1 INTRODUCTION

The Near-Infrared Spectrograph (NIRSpec) is one the four focal plane
instruments on the Webb Space Telescope; its main scientific goal is to enable
multi-object spectroscopy of faint high redshift galaxies and thus advance our
understanding of the early stage of galaxy formation at the dawn of our
universe. Nevertheless its innovative design allows sensitive observations of
a wide range of targets to be performed, including detailed spectroscopic
studies of the atmospheres of exoplanets, to choose an example at the opposite
end of the Universe distance scale[1].

To support these studies, NIRSpec is capable of carrying out low, medium and
high resolution spectroscopy, both in single- object mode employing any one of
five fixed slits (FS), or a 3.1$\times$3.2 arcsec2 integral field unit (IFU)
for Integral Field Spectroscopy observations (IFS), or in multi-object (MOS)
mode employing a novel programmable micro-shutter array (MSA) covering a
3.6$\times$3.4 arcmin2 field of view[1]. NIRSpec is equipped with a double
pass-prism for low-resolution spectroscopy over the entire wavelength-range,
0.6 $-$ 5.3 $\mu$m, and two sets of three diffraction gratings providing
spectral resolutions of $R{\simeq}1000$ and $R{\simeq}2700$, respectively. The
two sets of gratings are formally specified to cover the 1.0-5.0 $\mu$m
wavelength region in each of three overlapping bands: 1.0-1.8 $\mu$m (Band
I), 1.7-3.0 $\mu$m (Band II) and 2.9-5.0 $\mu$m (Band III). The nominal
instrument configurations, and corresponding combination of dispersive
elements and filters, are listed in Table 1. The shorter wavelength F070LP
long-pass filter listed is included to enable observations below 1 $\mu$m
with the G140M and G140H gratings, which are free of contaminating second-
order light only up to a wavelength of 1.27 $\mu$m.

Table 1: Nominal NIRSpec science configurations Band | Disperser element | Resolution $\lambda/\Delta\lambda$ | Filter | Spectral range / $\mu$m
---|---|---|---|---
0 | G140M, G140H | 1000, 2700 | F070LP | 0.7–1.2
I | G140M, G140H | 1000, 2700 | F100LP | 1.0–1.8
II | G235M, G235H | 1000, 2700 | F170LP | 1.7–3.1
III | G395M, G395H | 1000, 2700 | F290LP | 2.9–5.2
n/a | PRISM |  100 | CLEAR | 0.6–5.3

###  1.1 Photon-conversion efficiency

A key performance parameter in every optical instrument, either a telescope or
microscope, is its efficiency: how many photons will make it through the
optical chain the instrument is comprised of. Barring systematics, the
throughput and the noise performance of the detectors are the parameters
driving the photometric sensitivity of an instrument.

NIRSpec is a complex instrument that supports many different modes of
observations. Its optical train is reflective throughout, save for the order-
separation filters and the low resolution dispersive prism. The three primary
optical modules of NIRSpec are each implemented in the form of three-mirror
anastigmats employing high-order aspherical surfaces[2]. Counting the five
plane fold mirrors and the disperser, the light entering NIRSpec undergoes a
total of 15 reflections (in the FS/MOS mode) before reaching the detector
array. Coupled with the telescope element of JWST, the total number of
reflections a photon goes thorough in its path from the primary mirror to the
detectors at the NIRSpec focal plane is 19! The IFU mode adds a further 8
gold-coated reflections to the light path. Ultimately, though, we do not count
photons but the electrons generated in the semi-conductor substrate of the
detectors, with their own performance in terms of quantum efficiency (QE). For
this reason, we describe the throughput of our instrument in terms of Photon
Conversion Efficiency, PCE.

To achieve its ambitious science goals, NIRSpec was designed with demanding
requirements in terms of optical efficiency across its entire wavelength range
- and therefore particular attention was placed in the design and
manufacturing of its reflective elements, making sure that all the optical
surfaces received the highest quality optical coatings and were kept extremely
clean during instrument assembly. With the aim of achieving high optical
throughput at the blue end, all mirrors are coated with protected silver.
Additionally, the two Teledyne H2RG focal plane detectors have excellent QE,
overall exceeding 70% , as illustrated by Rauscher et al. [3].

Preliminary measurements of the NIRSpec PCE were obtained during the NIRSpec
ground-testing campaign that took place in 2013. The measurements, though,
were affected by significant uncertainties (due to limitations of the ground-
testing equipment), and, in addition, in 2015 the detector arrays were
replaced with new devices. Therefore, before flight, the PCE performance of
NIRSpec was based on a radiometric model of the instrument obtained by
combining the measured absolute QE of the replaced arrays with individual-
component measurements made on each optical element or appropriate witness
samples, together with the nominal reflectivity of the telescope optics[4].
The PCE values predicted before flight were very high, reaching in excess of
50% in prism configuration, and at peak blaze in all six gratings - see
Jakobsen et al.[1].

The moment of truth finally came during the commissioning of the instrument,
with the first observations of a spectro-photometric standard star.

##  2 OBSERVATIONS AND PROCESSING

JWST was launched on the 25th of December of 2021. After deployment and
reaching the target halo-orbit around the Sun-Earth L2 point, the telescope
and instrumentation underwent a six months period of commissioning activities
to prepare the observatory for its scientific observations. During this phase,
NIRSpec commissioning team commanded NIRSpec to perform internal calibration
exposures and on-sky observations, acquiring the data necessary to assess the
instrument’s in-flight performance and generate calibration reference data[5].

For the absolute spectro-photometric calibration of the instrument, we
observed the standard stars 1808347 (TYC 4433-1800-1), an A3 star selected
from STScI Calibration database CALSPEC, as well as two other reference stars
during one of the first instrument check-out exposures on sky: wd1057+719 (a
white dwarf) and p177d (G0), using all the dispersers, for a total of 9
filter-disperser combinations111Proposal ID (PID) 1128. To asses the PCE for
the MOS and FS mode, the star was observed through the square 1.6 arcsec-wide
slit (S1600A1), which has minimal geometrical loss, with typical values of
3–9% (see Sect. 3.1). To enable the subtraction of the background emission a
two-point 1-arcsec dither in the spatial direction was executed within the
slit, in each configuration, while, in the case of the IFS mode, a 4-point nod
was executed across the IFU aperture field-of-view.

The exposures were processed with our NIRSpec ramp-to-slope pipeline that
performs the following basic data reduction steps: bias subtraction, reference
pixel subtraction, linearity correction, dark subtraction and finally count-
rate estimation, including jump detection and cosmic-ray rejection – see
Birkmann et al.[6] for more details on this last step. From the count-rate
images the wavelength calibrated spectra were obtained using the Stage 2 of
the NIRSpec pipeline to perform the following operations for the FS: subtract
background (combining the exposures from the two-nods); extract sub-image
containing the spectral trace and assign wavelength and spatial coordinates to
each pixel therein; generate a rectified spectrum re-sampled on regular
2D-grid; compute the 1D-spectrum obtained by spatial integration. Figure 1
shows the spectra for configurations F070LP-G140H and F290-G395M, as examples.
For the IFS exposures, after background subtractions and rectification of the
30 IFU traces, Stage 2 of NIRSpec pipeline builds the three-dimensional data
cube from where the 1D-spectrum is obtained by spatial integration of a
circular area.

The PCE of the instrument through the given aperture is given simply by
dividing the observed number of electron rate per wavelength element by the
value of photons flux collected by the primary mirror (per wavelength
element).

---
Figure 1:  Example of extracted spectra in units of electron rate per unit
wavelength for two of the disperser-filter configurations. The gap in the
high-resolution spectrum (left-hand panel) is due to the gap between the two
NIRSpec detectors. Minimal processing was applied to these data – see text.

##  3 RESULTS

The PCE for the MSA/FS mode and the IFS mode are shown in Fig. 2 for the low-
resolution configuration, together with predictions from the model of the
instrument response that we developed and used throughout to estimate the
sensitivity of NIRSpec before flight. In the figure, the throughput response
of the different components entering the model are also shown. The model does
not include the light losses due to the finite size of the aperture, which
however do affect the data presented here - in the case of the square-aperture
these are expected to be relatively small. The PCE values, in MSA/FS mode, for
the medium and high resolution configurations are shown in Fig. 3 \- also
compared to the model. For the IFU, the PCE values for the grating
configurations available are shown in Fig. 4.

---
Figure 2:  The measured PCE as a function of wavelength of the low resolution
configurations for the FS/MOS mode on the left and the IFS mode on the right.
The data are here compared with the predictions by our radiometric model of
the instrument. The model includes the reflectivity of the telescope optics
(marked OTE) and the responsive quantum efficiency of the NIRSpec detector
array, but excludes the light losses occurring in the slit device employed.
The measured values on the other-hand are affected by path-losses (for which
we have not applied any correction).
---
Figure 3:  The measured PCE (coloured lines) as a function of wavelength of
the medium and high resolution configurations, for the FS/MOS mode, compared
with the values predicted by our radiometric model (shaded areas). The model
excludes light loss occurring in the slit (S1600A1), while the measured values
are affected by this (for which we have not applied any correction)

Although not presented here, we note that for the configurations for which
observations of more than one standard star was available we obtained results
completely consistent with each other.

In general the efficiency of NIRSpec meets or exceeds the predictions of our
pre-flight model. Significantly higher than predicted performances are
achieved in particular for the high-resolution configurations, for the MOS/FS
mode, and for all configurations below $\sim 2.6~{}\mu$m, for the IFS mode. We
notice a dip in efficiency with regard to predictions above $\sim 4~{}\mu$m –
in particular for the IFS mode, where we see 10–20% less throughput than
predicted. In the case of the FS/MOS, this difference is (at least partly)
explained by the effect of the slit losses and the fact that they become more
significant at these wavelength, due to the geometrical truncation of the
larger point-spread function (PSF) - see next section. In the case of IFS
mode, the differences between data and predictions are likely due, partly, to
diffraction losses at the level of the IFU 100 mas-slice, which affect the
observations (prevalently in the red) but are not included in the model and,
partly, to slight inaccurate assumptions in the gold-reflectivity as a
function of wavelength, at cryogenic temperatures (recall that the IFU adds 8
gold-coated reflections to the light-path).

---
Figure 4:  The measured PCE (coloured lines) as a function of wavelength of
the medium and high resolution configurations, for the IFS mode, compared with
the values predicted by our radiometric model (shaded areas). The model
excludes diffraction loss occurring at the IFU slices, while the measured
values are affected by this (for which we have not applied any correction)

###  3.1 Path-losses

Ultimately, in the case of NIRSpec, the total amount of light registered when
observing an object is also affected by the light losses occurring in the slit
device employed. These depend on a variety of factors: the nature and shape of
the target, the slit employed, the positioning of the object within the
aperture, and the optical quality of the image formed within the slit. For
simplicity we will limit our discussion here to the case of a centered point
source. In this case, when employing the relatively wide square aperture
(S1600A1), the losses are small, but for the narrower slits and the micro-
shutters these are substantial. In the IFS mode, where the flux of the
scientific target in the image plane can be recovered by summing over multiple
slices, the path losses are limited to their diffraction component and can be
in excess of 10% at the longest wavelengths.

Before flight, we computed the expected path losses for a point source, in the
various apertures, using a Fourier-optics approach based on the available
measured wavefront error maps of all surfaces in the light path[1], including
predictions for the telescope. Indeed, because the major source of losses is
due to the truncation of the PSF in the silt plane by the geometry of the
aperture, the quality of the image is of fundamental importance.

---
Figure 5:  Left-panel: prediction of slit transmission pre-flight and in-
flight after having updated the wavefront-error map of the telescope used in
the computation. Right-panel: estimated in-flight slit transmission for the
available slit sizes

Fig. 5 shows the predicted transmission coefficient as a function of
wavelength for slit S200A1/A2, for the pre-flight computation and the updated
values using a representative map of the wavefront error obtained during
commissioning – see [7]. Since the PSF achieved in flight is of such a high
quality, in particular with a higher Strehl ratio, the transmission from our
narrowest slit is predicted to improve significantly up to $\sim 4$ $\mu$m,
where the truncation of the PSF core starts to be the dominant factor in the
loss of flux.

The computed transmission coefficients for all the fixed slits as a function
of wavelength based on the in-flight PSF are also shown in Fig. 5 \- right
panel; the transmission coefficients shown here include both the geometrical
losses due to the finite size of the aperture, the dominant term, and the
diffraction losses in the collimator pupil, contributing by a few percent[1].
While the relative improvement of the transmission, with respect to the pre-
flight prediction, is most noticeable in the blue, the absolute level of
losses are higher on the red-side of NIRSpec wavelength range where the PSF
full-width half-maximum is larger: indeed, this is partly the reason for an
apparent drop in performance in the red when comparing the observed PCE with
the model predictions (which does not include path losses), for the FS/MOS
mode. Using the transmission coefficients for slit S1600A1, to correct for
this effect for the case of the F290LP filter, both M and H gratings, brings
the PCE values right in line with the predictions also for these
configurations, as shown in Fig. 6.

We have currently not yet updated our estimates of path losses in the case of
the IFS mode.

---
Figure 6:  PCE values for Band III, through slit S1600A1 as directly observed
and after correction by the estimated slit transmission coefficients. The pre-
flight model-based estimate is also shown.

##  4 DISCUSSION AND CONCLUSION

For many years during the development of NIRSpec, we had to rely on a model of
the instrument PCE, anchored to measurements obtained during the instrument
level testing that took place in 2013 - when the instrument was not yet in its
final flight configurations. For the MOS/FS mode, the anticipated PCE,
excluding slit losses, was very high, reaching in excess of 50% for the prism,
and at peak blaze in all six gratings [1]. For the IFS the PCE performance was
predicted to be approximately 50% to 10% lower, going from the blue- to the
red-end of NIRSpec operating wavelength range[8], since the IFS introduces
eight further reflections.

The observations performed during commissioning demonstrates that NIRSpec is
equipped with extraordinarily efficient optics and detectors. In the MOS/FS
mode, the observed PCE overall meets or exceeds the predictions; for the IFS
mode, the performance in the blue are significantly higher than anticipated
($+$30% below 2 $\mu$m) and lower than expected above 4 $\mu$m (up to $-$20%).
In this case, the reasons for the discrepancy between the model and the
measurements could be a combination of unaccounted for diffraction losses at
the level of the IFU 100 mas-slices and inaccurate assumptions in terms of the
gold-reflectivity as a function of wavelength, at cryogenic temperatures.

Beside the optical efficiency, the other fundamental parameter driving the
photometric sensitivity of an instrument such as NIRSpec is the noise
performance of the detector. NIRSpec is equipped with extremely low-noise
Teledyne H2RG arrays and, as illustrated by S. Birkmann et al.[6] at this
conference, the behaviour of the two sensors is excellent, with noise figures
that meet the anticipated in-flight performances. Hence, NIRSpec achieves the
sensitivity projected before launch by Ferruit et al. [9], for the FS and MOS
mode, and Böker et al. [8], for the IFS mode.

JWST/NIRSpec is confirmed to be the most sensitive near-IR spectrograph
currently available for astronomical studies ready to deliver exciting new
observations.

## References

  * [1] Jakobsen, P., Ferruit, P., Alves de Oliveira, C., et al., “The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope. I. Overview of the instrument and its capabilities,” A&A 661, A80 (May 2022).
  * [2] Geyl, R., Ruch, E., Vayssade, H., Leplan, H., and Rodolfo, J., “NIRSpec optics development: the final report,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 8146, 130 (2011).
  * [3] Rauscher, B. J., Boehm, N., Cagiano, S., et al., “New and Better Detectors for the JWST Near-Infrared Spectrograph,” PASP 126, 739 (Aug. 2014).
  * [4] Lightsey, P. A., “Stray light field dependence for the James Webb Space Telescope,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 9904, 81, SPIE (2016).
  * [5] Böker, T., Abul-Huda, Y., Altenburg, M., et al., “Status of the JWST/NIRSpec Commissioning Campaign,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series This conference, SPIE (July 2022).
  * [6] Birkmann, S., Giardino, G., Sirianni, M., et al., “The In-Flight Noise Performance of the JWST/NIRSpec Detector System,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series This conference, SPIE (July 2022).
  * [7] Scott Knight, J., Lightsey, P.A., “Webb Telescope Imaging Performance,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series This conference, SPIE (July 2022).
  * [8] Böker, T., Arribas, S., Lützgendorf, N., et al., “The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope. III. Integral-field spectroscopy,” A&A 661, A82 (May 2022).
  * [9] Ferruit, P., Jakobsen, P., Giardino, G., et al., “The Near-Infrared Spectrograph (NIRSpec) on the James Webb Space Telescope. II. Multi-object spectroscopy (MOS),” A&A 661, A81 (May 2022).


Data-driven rogue waves and parameter discovery in the defocusing NLS equation
with a potential using the PINN deep learning

Li Wang and Zhenya Yan∗ ∗Email address<EMAIL_ADDRESS>(Corresponding
author)

Key Laboratory of Mathematics Mechanization, Academy of Mathematics and
Systems Science,
Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

The physics-informed neural networks (PINNs) can be used to deep learn the
nonlinear partial differential equations and other types of physical models.
In this paper, we use the multi-layer PINN deep learning method to study the
data-driven rogue wave solutions of the defocusing nonlinear Schrödinger (NLS)
equation with the time-dependent potential by considering several initial
conditions such as the rogue wave, Jacobi elliptic cosine function, two-
Gaussian function, or three-hyperbolic-secant function, and periodic boundary
conditions. Moreover, the multi-layer PINN algorithm can also be used to learn
the parameter in the defocusing NLS equation with the time-dependent potential
under the sense of the rogue wave solution. These results will be useful to
further discuss the rogue wave solutions of the defocusing NLS equation with a
potential in the study of deep learning neural networks.

Key words: Defocusing NLS equation with the time-dependent potential; initial-
boundary value conditions; physics-informed neural networks; deep learning;
data-driven rogue waves and parameter discovery

##  1 Introduction

With the quick development in computational capabilities and big data
generation, the distinct machine learning (ML) approaches have been proposed
to explore the effective and accurate results in diverse applications, such as
the data mining, computer vision, natural language processing, biomedical
analysis, product recommendations, optical text/character recognition, speech
recognition, handwriting recognition, data assimilation, genomics, traffic
prediction, self-driving cars, medical diagnosis, and stock market trading
(see, e.g., Refs. [1, 2, 3, 4, 5, 6] and references therein). However, with
the challenges of acquiring data from the distinct application fields, how to
excerpt information effectively and accurately under the condition of partial
data missing has becomes an urgent and longstanding problem [4, 5, 6].
Moreover, many results obtaining from the ML techniques may be incorrect due
to the few samples and lack of robustness. In fact, it seems unconvincing for
us to draw the conclusions relating to physical laws competently by means of
the only ML techniques with the input-output pairs of data, especially for
problem with high-dimensional conditions [4]. The deficiency of the prior laws
pertaining to the physical systems may be one of the main reasons. Hence, many
researchers tried to use the ML algorithms along with some physical laws to
improve the accuracy of the unknown solutions of physical models [7, 8].

Recently, with the aid of sufficient information pertaining to the physical
systems, the ML techniques were put forward to establish a both data-efficient
and linear physical-informed neural networks (PINNs) [9, 10, 11]. Moreover,
the deduced solutions seemed to be efficient from the ML methods utilizing
Gaussian process regression in combination with a linear operator [12].
Furthermore, the linear problem was also extended to the nonlinear ones, which
supplied a novel idea to recognize and deduce physical models efficiently [13,
14]. However, both the local linearizations of nonlinear problems in the time
direction and theoretical assumptions for physical systems, especially for the
Bayesian features induced from Gaussian process regression, hinder the
promotion and accuracy performance of the algorithm. In contrast, the method
tackling with the nonlinear problems [15] can avoid the limitations [13, 14],
and used a deep neural network and a PINN to approximate the observation
solution $\widehat{w}(x,t)$ and

$\displaystyle f(x,t):=\widehat{w}_{t}-{\cal
N}[x,t,\widehat{w},\widehat{w}_{x},\widehat{w}_{xx},...],$ (1)

respectively, based on the considered nonlinear evolution partial differential
equation (PDE):

$\displaystyle w_{t}={\cal N}[x,t,w,w_{x},w_{xx},...],\quad x\in\Omega,\quad
t\in[0,T],$

where the two considered deep neural networks $\widehat{w}(x,t)$ and $f(x,t)$
share the same parameters (e.g., weights and biases), and are able to be
trained by minimizing the mean squared error (MSE) loss arising from the
initial-boundary value conditions related to the hidden function
$\widehat{w}(x,t)$ and PINN $f(x,t)$. The automatic differentiation technique
[16, 17] can be used to perform the partial derivatives of $\widehat{w}(x,t)$
with respect to the input variables for the sake of avoiding any additional
constraints. It should be pointed out that the deep learning approach with the
PINNs is powerful since the physical laws arising from the nonlinear evolution
PDEs were considered in the PINNs [15]. This similar idea has been applied to
the diverse models with more complicated dynamical behaviors such as soliton
solutions of nonlinear evolution PDEs inspired by the effective methods [7, 8,
15, 18, 19, 20, 21, 22, 23].

It is well-known that the usual defocusing NLS equation is not capable of
admitting the Benjamin-Feir instability (or modulation instability (MI)) [24,
25], which is now regarded as one main reason of the rogue wave (RW)
generation in the nonlinear wave systems. However, recently an intriguing idea
was put forward to find that the defocusing NLS equation with some time-
dependent external potential could support the stable RW solution [26]. Of
course, this idea can also be extended to other related linear or nonlinear
wave systems in the mathematical and physical fields [27, 28]. In this paper,
motivated by the similar ideas about the PINNs [7, 8, 15, 29, 30, 31, 32], we
would like to make use of the multi-layer PINN to focus on the RW solutions of
the Cauchy problem of the defocusing NLS equation with a time-dependent
potential and initial-boundary value conditions [33, 34, 26]

$\displaystyle\left\\{\begin{array}[]{l}i\psi_{t}=-\dfrac{1}{2}\psi_{xx}+V(x,t)\psi+\left|\psi\right|^{2}\psi,~{}~{}~{}x\in(-L,L),~{}~{}~{}t\in(-T,T),\vspace{0.1in}\\\
\psi(x,-T)=\phi_{0}(x),\quad x\in[-L,L],\vspace{0.1in}\\\
\psi(-L,t)=\psi(L,t),\quad t\in[-T,T],\end{array}\right.$ (5)

where the subscripts denote the partial derivatives of the complex field
$\psi$ with respect to the space $x$ and time $t$, and $V(x,t)$ denotes the
spatio-temporal potential in the form [26]

$\displaystyle
V(x,t)=\frac{4\left(x^{2}-t^{2}\right)-1}{\left(x^{2}+t^{2}+0.25\right)^{2}}-2,$
(6)

where $V(x,t)\leq 0$ and $V(x,t)\to-2$ as $|x|,|t|\to\infty$. Eq. (5) can be
written as a variational form $i\partial\psi/\partial
t=\delta\mathcal{H}/\delta\psi^{*}$, with the generalized Hamiltonian

$\displaystyle\mathcal{H}=\int_{-\infty}^{+\infty}\left[\frac{1}{2}|\psi_{x}|^{2}+V(x,t)|\psi|^{2}+\frac{1}{2}|\psi|^{4}\right]dx.$
(7)

The rest of the paper is arranged as follows. Section 2 devotes to demonstrate
the data-driven RW solutions of system (5) in the sense of distinct initial
conditions such as the rogue wave, Jacobi elliptic cosine function, two-
Gaussian function, or three-hyperbolic-secant function, and the periodic
boundary conditions. In Sec. 3, we use the PINN to discuss the identification
of the parameter in Eq. (5) with the potential (6). Finally, we give the
conclusions and discussions in Sec. 4.

##  2 The data-driven rogue wave scheme and application

In this section, we would like to study the data-driven RW solutions of Eq.
(5) by means of the PINN deep learning method [15]. The PINN involving the
physical information $f(x,t)$ pertaining to the defocusing NLS equation with
the potential (6) is chosen as

$f(x,t):=i\widehat{\psi}_{t}+\frac{1}{2}\widehat{\psi}_{xx}-V(x,t)\widehat{\psi}-|\widehat{\psi}|^{2}\widehat{\psi},$
(8)

where $\widehat{\psi}(x,t)=\widehat{\psi}(x,t;W,b)$ with weights $W$ and
biases $b$ denotes the training latent solution generated by utilizing the
PINN. Since the two NNs $\widehat{\psi}(x,t)$ and $f(x,t)$ are both complex-
valued functions, thus one can rewrite them as
$\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\,\widehat{v}(x,t)$ and
$f(x,t)=f_{R}(x,t)+if_{I}(x,t)$, where both
$\widehat{u},\,\widehat{v},\,f_{R}(x,t)$ and $f_{I}(x,t)$ are all real-valued
functions, and $f_{R}$ and $f_{I}$ satisfy

$\displaystyle\left\\{\begin{array}[]{l}f_{R}(x,t):=-\widehat{v}_{t}+\dfrac{1}{2}\widehat{u}_{xx}-V(x,t)\widehat{u}-\widehat{u}(\widehat{u}^{2}+\widehat{v}^{2}),\vspace{0.1in}\\\
f_{I}(x,t):=\widehat{u}_{t}+\dfrac{1}{2}\widehat{v}_{xx}-V(x,t)\widehat{v}-\widehat{v}(\widehat{u}^{2}+\widehat{v}^{2}).\end{array}\right.$
(11)

The expression (8) provides one with some physical laws in the framework of
the defocusing NLS equation with a potential, based on which, we bulid a
neural networks $f(x,t)$ full of physical information. Making use of the
automatic differentiation technique [16, 17], the arbitrary-order partial
derivatives of hidden solution $\widehat{\psi}(x,t)$ in the condition of the
deep neural network (NN) can be gained by using the chain rule. Furthermore,
the PINN $f(x,t)$ shares the same parameters with the NN $\widehat{\psi}(x,t)$
even though there exits a variety of activation functions during the
differentiation process in the nonlinear system (8).

Therefore, with the aid of L-BFGS optimization approach [35], the common
parameters in the hidden function
$\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$ and PINN
$f(x,t)=f_{R}(x,t)+if_{I}(x,t)$ can be trained by minimizing the whole MSE
loss [15]

$MSE=MSE_{int}+MSE_{b}+MSE_{f},$ (12)

where the $MSE_{int},\,MSE_{b},\,MSE_{f}$ are defined by

$\displaystyle MSE_{int}$
$\displaystyle=\frac{1}{N_{int}}\sum_{j=1}^{N_{int}}\left|\widehat{\psi}(x_{-T}^{j},-T)-\psi(x_{-T}^{j},-T)\right|^{2}$
$\displaystyle=\frac{1}{N_{int}}\sum_{j=1}^{N_{int}}\left(\widehat{u}(x_{-T}^{j},-T)-\left|u(x_{-T}^{j},-T)\right|^{2}+\left|\widehat{v}(x_{-T}^{j},-T)-v(x_{-T}^{j},-T)\right|^{2}\right),$
(13) $\displaystyle MSE_{b}$
$\displaystyle=\frac{1}{N_{b}}\sum_{j=1}^{N_{b}}\left|\widehat{\psi}(-L,t_{b}^{j})-\widehat{\psi}(L,t_{b}^{j})\right|^{2}$
$\displaystyle=\frac{1}{N_{b}}\sum_{j=1}^{N_{b}}\left(\left|\widehat{u}(-L,t_{b}^{j})-\widehat{u}(L,t_{b}^{j})\right|^{2}+\left|\widehat{v}(-L,t_{b}^{j})-\widehat{v}(L,t_{b}^{j})\right|^{2}\right),$
(14) $\displaystyle MSE_{f}$
$\displaystyle=\frac{1}{N_{f}}\sum_{j=1}^{N_{f}}\left|f(x_{f}^{j},t_{f}^{j})\right|^{2}=\frac{1}{N_{f}}\sum_{j=1}^{N_{f}}\left(\left|f_{R}(x_{f}^{j},t_{f}^{j})\right|^{2}+\left|f_{I}(x_{f}^{j},t_{f}^{j})\right|^{2}\right),$
(15)

the observed measurements
$\\{\widehat{\psi}(x_{-T}^{j},-T)=\widehat{u}(x_{-T}^{j},-T)+i\widehat{v}(x_{-T}^{j},-T)\\}_{1}^{N_{int}}$
of the hidden field $\widehat{\psi}(x,t)$ is linked with the sampled initial
training data
$\\{x_{-T}^{j},\,\psi(x_{-T}^{j},-T)=u(x_{-T}^{j},-T)+iv(x_{-T}^{j},-T)\\}_{1}^{N_{int}}$
at time $t=-T$, $\\{\widehat{\psi}(\pm L,t_{b}^{j}\\}_{1}^{N_{b}}$ is relevant
to the selected boundary training data, and
$\\{x_{f}^{j},t_{f}^{j}\\}^{N_{f}}_{1}$ is connected with the marked points
for the PINN $f(x,t)$. As a result, for the randomly chosen points,
$MSE_{int}$ and $MSE_{b}$ represent the MSE losses of initial and periodic
boundary data, respectively, and $MSE_{f}$ is associated with the MSE loss of
the PINN (8).

In the following subsections, we would like to consider the distinct initial
conditions and the periodic boundary conditions to study the data-driven RW
solutions of Eq. (5) by using the above-mentioned PINN $f(x,t)$.

###  2.1 The rogue-wave initial condition

It is known that Eq. (5) with the time-dependent potential (6) admits the
exact RW solution [26]

$\psi(x,t)=\left[1-\frac{4\left(1+2it\right)}{4\left(x^{2}+t^{2}\right)+1}\right]e^{it}.$
(16)

In what follows, we will consider the initial condition $\psi(x,-T)$ of Eq.
(5) arising from the RW solution (16). To use the PINN deep learning to
investigate the Cauchy problem of Eq. (5) with the initial condition given by
Eq. (16)

$\psi(x,-2)=\left(1-\frac{4-16i}{4x^{2}+17}\right)e^{-2i},\quad
x\in[-2.5\pi,\,2.5\pi],$ (17)

and the Dirichlet periodic boundary condition
$\psi(-2.5\pi,t)=\psi(2.5\pi,t)$, we imitate Eq. (5) with the time-dependent
potential (6) by means of the Fourier pseudo-spectral method [36, 37] to
produce the training data for the PINN. The implementation of pseudo-spectral
algorithm is to use the discrete Fourier transform in the spatial interval
$x\in[-2.5\pi,2.5\pi]$ with $512$ Fourier modes, and a fourth-order Runge-
Kutta time-stepping scheme in the temporal interval $[-2,2]$ with time-step
$\triangle t=2.4\times 10^{-3}$.

Based on the sampled initial and Dirichlet periodic boundary data, the
training data-set used in the 10-layer PINN is comprised of the randomly
sampled $N_{int}=100$ points from the initial data $\psi(x,-2)$ given by Eq.
(17), $N_{b}=100$ points from the periodic boundary data, and $N_{f}=5,000$
collocation points for the PINN $f(x,t)$ given by Eq. (8) within the
considered spatio-temporal region of the hidden solution
$\widehat{\psi}(x,t)$. Notice that all the marked data in the NN is chosen
randomly by mens of the Latin Hypercube Sampling idea [38]. Moreover, the
hidden solution $\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$ can
be learned by means of the $8$-hidden-layer deep PINN along with $80$ neurons
per layer, and a hyperbolic tangent activation function by minimizing the MSE
loss given by Eq. (12).

Figure 1 displays the results of the PINN related to the Cauchy problem of the
defocusing NLS equation with the time-dependent potential given by Eqs. (5),
(6), and (17), as well as the periodic boundary condition. Fig. 1(a) depicts
the magnitude of the hidden RW solution
$\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$, together with the
locations of some initial-boundary training data. Fig. 1 (b) exhibits the
better matches between the learning solution and one derived from the MATLAB
at three distinct times $t=-0.64,\,0.03$, and $0.34$, where the
$\mathbb{L}_{2}$-norm error of $\psi(x,t)$ is 1.3636e-01. Fig. 1(c) displays
the three-dimensional (3D) profile of the learning RW solution.

Figure 1: (Color online). (a) The data-driven RW solution resulted from the
PINN with the randomly chosen $N_{int}=100$ initial points, $N_{b}=100$
boundary points, and $N_{f}=5,000$ points for Eq. (8) in the hidden solution
region, and three distinct tested times $t=-0.75$, $0.01$, and $0.45$ (black
dashed lines); (b) The comparisons between the learning and exact RW solutions
at the distinct times. The $\mathbb{L}_{2}$-norm error of RW solution
$\widehat{\psi}(x,t)$ is 1.3636e-01; (c) The 3D profile of the learning RW
solution.

Figure 2: (Color online). (a) The learning RW solution arising from the PINN
with the randomly chosen $N_{int}=100$ initial points, $N_{b}=100$ boundary
points, and $N_{f}=50,000$ points for Eq. (8) in the hidden solution region,
and the three different tested times $t=-0.64,\,0.03$, and $0.34$ (black
dashed lines); (b) The comparison between the learning and exact solutions at
the distinct times, where the $\mathbb{L}_{2}$-norm error of the hidden
solution is 9.4481e-02; (c) The 3D profile of the learning RW solution.

###  2.2 The double-periodic initial condition

We here consider another initial condition in the form of the double-periodic
function

$\psi(x,-1)=1+{\rm cn}(x,0.8),\quad x\in[-8,8],$ (18)

and the Direchlet-Neumann periodic boundary conditions

$\displaystyle\psi(-8,t)=\psi(8,t),\qquad\psi_{x}(-8,t)=\psi_{x}(8,t),\quad
t\in[-1,1]$ (19)

to study the RW generation in the context of Eq. (5), where ${\rm cn}(x,m)$ is
a Jacobi elliptic cosine function of modulus $m\in(0,1)$.

By means of the Fourier pseudo-spectral method [36, 37] in MATLAB, we exert
the double-periodic initial condition (18) and periodic boundary conditions
(19) to acquire the corresponding data for the PINN in the physical framework
(8). We choose the spatial region $x\in[-8,8]$ with the space-step $\triangle
x=0.0312$, and the time region $t\in[-1,1]$ with time-step $\triangle
t=0.004$. The sampled points used in the multi-layer neural network are
consist of randomly distributed $N_{int}=100$ points from the initial
condition $\psi(x,-1)$ given by Eq. (18), $N_{b}=100$ points from the periodic
boundary conditions (19), and $N_{f}=50,000$ points for Eq. (8) within the
spatio-temporal region of the hidden solution $\widehat{\psi}(x,t)$.
Similarly, by minimizing the MSE loss given by Eq. (12) with Eq. (14) replaced
by

$\displaystyle\begin{array}[]{rl}MSE_{b}&=\displaystyle\dfrac{1}{N_{b}}\sum_{j=1}^{N_{b}}\left(\left|\widehat{\psi}(-L,t_{b}^{j})-\widehat{\psi}(L,t_{b}^{j})\right|^{2}+\left|\widehat{\psi}_{x}(-L,t_{b}^{j})-\widehat{\psi}_{x}(L,t_{b}^{j})\right|^{2}\right)\\\
&=\displaystyle\dfrac{1}{N_{b}}\sum_{j=1}^{N_{b}}\left(\left|\widehat{u}(-L,t_{b}^{j})-\widehat{u}(L,t_{b}^{j})\right|^{2}+\left|\widehat{v}(-L,t_{b}^{j})-\widehat{v}(L,t_{b}^{j})\right|^{2}\right.\\\
&\qquad\qquad+\left.\left|\widehat{u}_{x}(-L,t_{b}^{j})-\widehat{u}_{x}(L,t_{b}^{j})\right|^{2}+\left|\widehat{v}_{x}(-L,t_{b}^{j})-\widehat{v}_{x}(L,t_{b}^{j})\right|^{2}\right),\end{array}$
(23)

we use the $9$-hidden-layer deep PINN with $80$ neurons per layer, and a
hyperbolic tangent activation function to study the Cauchy problem of Eq. (5)
with Eqs. (18) and (19).

Figure 2 displays the deep learning results of the PINN related to the Cauchy
problem of the defocusing NLS equation with the time-dependent potential given
by Eqs. (5), (6), (18), and (19). Fig. 2(a) depicts the magnitude of the
hidden RW solution $\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$ in
the spatio-temporal region $(x,t)\in[-8,8]\times[-1,1]$, together with the
locations of some initial-boundary training data. Fig. 2 (b) exhibits the
better matches between the learning solution and one derived from the MATLAB
at the three distinct times $t=-0.64,\,-0.37,\,0.20$, where the
$\mathbb{L}_{2}$-norm error of $\widehat{\psi}(x,t)$ is 9.4481e-02. Fig. 2(c)
exhibits the three-dimensional profile of the learning RW solution.

###  2.3 The initial condition consisting of two Gaussian functions and one
constant

We here consider the initial condition consisting of two Gaussian functions
and a constant function

$\displaystyle\psi(x,-2)=1+e^{-(x-2.5)^{2}}+e^{(x-2.5)^{2}},\quad x\in[-8,8]$
(24)

and the same periodic boundary conditions (19), which may also contribute to
the RW generation in the defocusing NLS equation (5) with the potential (6).

Similarly, we exploit the pseudo-spectral method [36] in MATLAB to simulate
Eqs. (5) and (6) with the initial-boundary value conditions given by Eqs. (24)
and (19), where the spatial region $[-8,8]$ is separated into $256$ Fourier
modes, and the temporal region $[-2,2]$ is split into with $501$ steps with
time-step $\triangle t=0.008$. As a result, we can obtain the corresponding
data for the PINN in the physical framework (8). The training points are made
up of the randomly chosen $N_{int}=100$ points from initial data $\psi(x,-2)$,
$N_{b}=100$ points from the boundary data, and $N_{f}=10,000$ points for the
PINN $f(x,t)$ given by Eq. (8) within the considered spatio-temporal region of
the hidden solution $\widehat{\psi}(x,t)$. Particularly, all the selected
collocation points utilized in the PINN are deduced via the Latin Hypercube
Sampling method [38]. Therefore, by minimizing the MSE loss given by Eq. (12)
with Eq. (14) replaced by Eq. (23), we exert a $7$-hidden-layer deep PINN with
$60$ neurons per layer, and a hyperbolic tangent activation function to study
the initial-boundary problem of Eq. (5) with Eqs. (24) and (19).

Figure 3 displays the learning results of the deep PINN related to the Cauchy
problem of the defocusing NLS equation with the time-dependent potential given
by Eqs. (5), (6), (24), and (19). Fig. 3(a) depicts the magnitude of the
hidden RW solution $\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$ in
the spatio-temporal region $(x,t)\in[-8,8]\times[-2,2]$, in combination with
the locations of some initial-boundary training data. Fig. 3(b) exhibits the
better matches between the learning solution and one deduced from the MATLAB
at the three distinct times $t=-0.51,\,-0.02,\,0.50$, where the
$\mathbb{L}_{2}$-norm error of $\widehat{\psi}(x,t)$ is 9.6152e-02. Fig. 3(c)
exhibits the three-dimensional profile of the learning RW solution.

Figure 3: (Color online) (a) The learning RW solution of the deep PINN with
randomly sampled $N_{int}=100$ initial points, $N_{b}=100$ boundary points,
and $N_{f}=10,000$ points for the $f(x,t)$ given by Eq. (8), and three
different tested times $t=-0.51,-0.02$, and $0.50$ (black dashed lines); (b)
The comparisons between the learning and exact solutions at the distinct
times. Moreover, the $\mathbb{L}_{2}$-norm error of the hidden solution is
9.6152e-02. (c) The 3D profile of the learning RW solution.

###  2.4 The initial condition consisting of three hyperbolic secant functions
and one constant

In this subsection, our aim is to choose the initial condition consisting of
three hyperbolic secant functions and one constant described as

$\psi(x,-1)=1+\sum_{j=-5,0,5}\mathrm{sech}(x+j),\quad x\in[10.5,10.5],$ (25)

and the Dirichlet-Neumann periodic boundary conditions

$\displaystyle\psi(-10.5,t)=\psi(10.5,t),\quad\psi_{x}(-10.5,t)=\psi_{x}(10.5,t),\quad
t\in[-1,1]$ (26)

to demonstrate the RW generation in the defocusing NLS equation (5) with the
potential (6) via the deep PINN. We make use of the pseudo-spectral method
[36] to generate the training data for the PINN. We here consider the spatial
region $x\in[10.5,10.5]$ with $337$ Fourier modes, and temporal domain
$t\in[-1,1]$ with time-step $\triangle t=0.004$.

The training data are composed of the randomly sampled $N_{int}=100$ initial
points from $\psi(x,-1)$, $N_{b}=100$ boundary points from the periodic
boundary conditions (26), and $N_{f}=10,000$ collocation points for the PINN
$f(x,t)$ given by Eq. (8) within the spatio-temporal region of the hidden
solution $\widehat{\psi}(x,t)$. Moreover, all the considered training points
are produced by means of the Latin Hypercube Sampling method [38]. Therefore,
by minimizing the MSE loss given by Eq. (12) with Eq. (14) replaced by Eq.
(23), the deep PINN can be trained to study the RW solution of Eq. (5) with
the initial-boundary value conditions given by Eqs. (25) and (26) by a
$8$-hidden-layer PINN with $60$ neurons per layer, and hyperbolic tangent
activation function.

Figure 4 displays the learning results of the deep PINN related to the Cauchy
problem of the defocusing NLS equation with the time-dependent potential given
by Eqs. (5), (6), (25), and (26). Fig. 4(a) depicts the magnitude of the
hidden RW solution $\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$ in
the spatio-temporal region $(x,t)\in[10.5,10.5]\times[-1,1]$, in combination
with the locations of some initial-boundary training data. Fig. 4(b) exhibits
the better matches between the learning solution and one deduced from the
MATLAB at the three distinct times $t=-0.56,\,0.04$, and $0.26$, where the
$\mathbb{L}_{2}$-norm error of $\widehat{\psi}(x,t)$ is 1.0832e-02. Fig. 4(c)
exhibits the three-dimensional profile of the learning RW solution. Note that,
the neural network in the specific restriction of physical information
supplies one way to predict a hidden solution $\widehat{\psi}(x,t)$ with the
highest amplitude $3.636$ under the small sample points.

Figure 4: (Color online) (a) The learning RW solution of the deep PINN with
randomly sampled $N_{int}=100$ initial points, $N_{b}=100$ boundary points,
and $N_{f}=10,000$ points for the $f(x,t)$ given by Eq. (8), and three
different tested times $t=-0.56,\,0.04$, and $0.26$ (black dashed lines); (b)
The comparisons between the learning and exact solutions at the distinct
times. Moreover, the $\mathbb{L}_{2}$-norm error of the hidden solution is
1.0832e-02. (c) The 3D profile of the learning RW solution.

Figure 5: (Color online) (a) The learning RW solution from the PINN with the
data in the absence of noise; (b1)-(b3) The comparison between the learning
and exact RW solutions at the distinct times $t=-0.1224,-0.0024$, and
$0.1176$; (c) The data-driven parameter ($\lambda$) discovery: correct and
identified defocusing NLS equation with the potential.

##  3 Data-driven parameter discovery of the physical model

In this section, we would like to use the PINN to explore the data-driven
parameter discovery of the defocusing NLS equation with the time-depdent
external potential

$i\psi_{t}=-\lambda\psi_{xx}+V(x,t)\psi+|\psi|^{2}\psi,$ (27)

where $\psi(x,t)=u(x,t)+iv(x,t)$ ($u,\,v$ are the real and imaginary parts,
respectively) is a complex field, and the potential $V(x,t)$ is given by Eq.
(6), the coefficient $\lambda$ of the second-order dispersive term indicates
the unknown parameter for training in a multi-layer deep PINN. The complex
PINN $f(x,t)$ is defined as

$f(x,t):=i\widehat{\psi}_{t}+\lambda\widehat{\psi}_{xx}-V(x,t)\widehat{\psi}-|\widehat{\psi}|^{2}\widehat{\psi},$
(28)

where the hidden function
$\widehat{\psi}(x,t)=\widehat{u}(x,t)+i\widehat{v}(x,t)$ with
$\widehat{u},\,\widehat{v}$ being the real and imaginary parts, respectively,
and $f(x,t)=f_{u}(x,t)+if_{v}(x,t)$ with $f_{u},\,f_{v}\in\mathbb{R}[x,t]$
satisfying

$\displaystyle\left\\{\begin{array}[]{l}f_{u}(x,t):=-\widehat{v}_{t}+\lambda\widehat{u}_{xx}-V(x,t)\widehat{u}-\widehat{u}(\widehat{u}^{2}+\widehat{v}^{2}),\vspace{0.1in}\\\
f_{v}(x,t):=\widehat{u}_{t}+\lambda\widehat{v}_{xx}-V(x,t)\widehat{v}-\widehat{v}(\widehat{u}^{2}+\widehat{v}^{2}),\end{array}\right.$
(31)

which can be learned by training the NN $[\widehat{u}(x,t),\widehat{v}(x,t)]$
considered as the outputs. Note that, the parameter $\lambda$ in Eq. (27) and
residual construction $[f_{u},f_{v}]$ is same as underlying constants in
unknown solution $[\widehat{u}(x,t),\widehat{v}(x,t)]$, which can be derived
in the PINN by minimizing the MSE loss function

$MSE=\frac{1}{N_{p}}\sum_{j=1}^{N_{p}}\left(\left|\widehat{u}(x_{p}^{j},t_{p}^{j})-u(x_{p}^{j},t_{p}^{j})\right|^{2}+\left|\widehat{v}(x_{p}^{j},t_{p}^{j})-v(x_{p}^{j},t_{p}^{j})\right|^{2}+\left|f_{u}(x_{p}^{j},t_{p}^{j})\right|^{2}+\left|f_{v}(x_{p}^{j},t_{p}^{j})\right|^{2}\right).$
(32)

To learn the parameter $\lambda$ in Eq. (27) with the aid of the PINN, we need
to randomly choose $N_{p}=1,800$ points from a training data-set for
$\psi(x,t)$ in the spatio-temporal region
$(x,t)\in[-2.5\pi,2.5\pi]\times[-0.6,0.6]$, which is obtained by using the
pseudo-spectral method to simulate the Eq. (27) with $\lambda=0.5$ and
initial-boundary conditions given by Eq. (16) with $t=-0.6$ and
$\psi(-2.5\pi,t)=\psi(2.5\pi,t)$ in MATLAB. The purpose of pseudo-spectral
method is to employ the discrete Fourier transform to calculate the spatial
derives in $x$ direction and to use a fourth-order Runge-Kutta method in the
time interval $[-0.6,0.6]$ with time-step $\Delta t=0.0024$. The neural
network $[\widehat{u}(x,t),\widehat{v}(x,t)]$ and residual network
$[f_{u},f_{v}]$ can be generated from a 9-layer deep PINN with 50 neurons per
hidden layer, and the hyperbolic tangent activation function by minimizing the
MSE loss (32) via the L-BFGS optimizer [35].

Figure 5(a) describes the magnitude of the hidden function,
$|\widehat{\psi}(x,t)|=\sqrt{\widehat{u}^{2}(x,t)+\widehat{v}^{2}(x,t)}$, and
three vertical solid lines at the diverse times $t=-0.1224,\,-0.0024$, and
$0.1776$. Figs. 5(b1)-(b3) display the comparisons between the exact and
learning solutions at the three distinct times $t=-0.1224,\,-0.0024$ and
$0.1776$, which show that the learning and exact solutions match better in the
PINN with the small sample points. Fig. 5(c) exhibits the value of the
training parameter $\lambda$ is almost identical to the exact one, where the
error of $\lambda$ is $1.7658\%$ in the absence of noise, and $0.265139\%$ in
the presence of $1\%$ noise.

##  4 Conclusions and discussions

In conclusion, we have used the multi-layer PINN deep learning method to study
the data-driven rogue wave solutions of the defocusing NLS equation with the
spatio-temporal potential under the distinct initial conditions (e.g., the
rogue wave, Jacobi elliptic cosine function, two-Gaussian function, or three-
hyperbolic-secant function) and periodic boundary conditions. In particular,
when we choose the exact RW solution as the initial condition we find that the
PINN can better learn the rogue wave structures. This results imply that the
proper initial condition will be useful to improve the learning results of the
PINN. The PINN can also be used to deep learn rogue waves of other nonlinear
wave equations in many fields.

Moreover, the data-driven parameter discovery of the defocusing NLS equation
with the time-dependent potential under the sense of the rogue wave solution
is also studied. It should be pointed out that the considered periodic
boundary conditions are non-zero, which differ from the other learning
solitons with zero boundary conditions [15], and seem to be difficultly
learned. In brief, these results show that the PINNs can be used to learn the
rogue waves of the defocusing NLS equation with a spatio-temporal potential
even though the small sampled points are applied. However, there are many
unknown issues such as i) there is no the theoretical analysis of the PINNs
with different activations, weights and bias functions for indicating latent
solutions; ii) what is the more suitable error loss for the different physical
models ? iii) whether can the more physical laws make the deep PINNs better
learn the corresponding nonlinear physical models ? These problems will be
considered in future.

Acknowledgements

This work is partially supported by the NSFC under Grant Nos. 11731014 and
11925108.

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