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Citation: Saraeva, K.; Bednyakova, A.
Enhanced bi-LSTM for Modeling
Nonlinear Amplification Dynamics of
Ultra-Short Optical Pulses. Photonics
2024,11, 126. https://doi.org/
10.3390/photonics11020126
Received: 18 December 2023
Revised: 9 January 2024
Accepted: 23 January 2024
Published: 29 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
photonics
hv
Article
Enhanced bi-LSTM for Modeling Nonlinear Amplification
Dynamics of Ultra-Short Optical Pulses
Karina Saraeva †and Anastasia Bednyakova *,†
Physics Department, Novosibirsk State University, Pirogova Str. 2, Novosibirsk 630090, Russia; k.saraeva@g.nsu.ru
*Correspondence: anastasia.bednyakova@gmail.com
†These authors contributed equally to this work.
Abstract: Fiber amplifiers are essential devices for optical communication and laser physics, yet the
intricate nonlinear dynamics they exhibit pose significant challenges for numerical modeling. In this
study, we propose using a bi-LSTM neural network to predict the evolution of optical pulses along a
fiber amplifier, accounting for the dynamically changing gain profile and the Raman scattering. The
neural network can learn information from both past and future data, adhering to the fundamental
principles of physics governing pulse evolution over time. We conducted experiments with a diverse
range of initial pulse parameters, covering the variation in the ratio between dispersion and nonlinear
length, ranging from 0.25 to 250. This deliberate choice has resulted in a wide variety of propagation
regimes, ranging from smooth attractor-like to noise-like behaviors. Through a comprehensive
evaluation of the neural network performance, we demonstrated its ability to generalize across the
various propagation regimes. Notably, our results showcase a relative speedup of 2000 times for
evaluating the intensity evolution map using our proposed neural network compared to the NLSE
numerical solution employing the split-step Fourier method.
Keywords: long short-term memory; recurrent neural network; Raman scattering; gain-guiding
nonlinearity; fiber amplifier; numerical simulation
1. Introduction
A fiber amplifier is a crucial component of a laser system. The main challenges to
deal with in active fibers are managing significant nonlinear phase accumulation without
wave breaking and amplifying ultrashort pulses that are affected by strong gain shaping.
Recently, there has been growing interest in a new regime for amplifying linearly chirped
asymmetric pulses with gain-guiding nonlinearity (GGN), which was demonstrated in a
research study by a group from Cornell University [
1
]. It is worth noting that this regime
governs the pulse evolution in the symmetric arms of the Mamyshev oscillator [
2
], making
it possible to achieve record-breaking peak power levels. The GGN regime is a nonlinear
amplification process that occurs when high-power picosecond pulses are propagated,
where the width of the spectrum is comparable to or exceeds the width of the amplification
profile. This amplification process results in intricate nonlinear dynamics that lead to pulse
asymmetry and the formation of a nonlinear attractor [
3
]. The dynamically changing ampli-
fication profile plays a crucial role in shaping the nonlinear attractor. Therefore, numerical
simulations must use a complex model that considers the evolution of amplification along
the fiber and its wavelength dependence.
Conventional numerical modeling presents substantial challenges for practical ap-
plications, primarily due to the time-consuming computations required for each new
set of system parameters. Real experimental conditions are often difficult to fully pa-
rameterize, leading to the necessity of making assumptions and neglecting the physical
model description.
Given that a fiber amplifier is the most computationally demanding component of a
laser system in numerical modeling tasks, its application in real-time experimental scenarios
Photonics 2024,11, 126. https://doi.org/10.3390/photonics11020126 https://www.mdpi.com/journal/photonics
Photonics 2024,11, 126 2 of 11
becomes challenging. One potential solution involves employing neural networks to predict
the evolution of intensity profiles along the fiber [
4
–
7
]. Neural networks accelerate the
modeling process by reducing the number of computational operations and overcoming
the limitations associated with numerical simulations that rely on approximations and
discretizations. Additionally, they possess the ability to generalize information, enabling
the derivation of solutions from imperfect and noisy experimental data in cases in which a
precise consideration of all factors influencing the experiment proves unfeasible.
In most studies employing deep learning techniques for fiber optics applications,
modern architectures are rarely utilized. Instead, linear perceptrons are widely used;
these do not account for temporal context and are suitable only for classification and for
predicting the output pulse profile. Since the task of modeling pulse propagation through
the fiber is entirely equivalent to forecasting time series, a more effective solution can
be achieved by employing recurrent neural networks [
8
,
9
]. These networks, equipped
with internal memory, efficiently leverage the preceding stages of pulse evolution to
predict the subsequent steps. Importantly, when trained with a dataset generated through
comprehensive numerical modeling, the PI-RNN becomes attuned to the physical principles
governing pulse evolution in optical fibers. Other deep learning algorithms may not be
able to incorporate such physics-informed features.
Our study presents the results of employing a physically informed recurrent neural
network (PI-RNN) for forecasting the nonlinear evolution of the spectral and temporal
pulse intensity along an active optical fiber. We have chosen the range of initial pulse
parameters that covers the variation in the relation between dispersion and nonlinear
length from 0.25 to 250. This choice has led to a wide variety of propagation regimes,
from smooth attractor-like modes to noise-like ones. Training the RNN within this range of
parameters requires generalization across various propagation modes, which is a challeng-
ing task. We demonstrate that a single PI-RNN, trained on numerical simulation results,
can accurately and rapidly reproduce the intricate dynamics of a nonlinear attractor within
a fiber amplifier across a wide range of initial parameters. Building upon the findings
presented in [
8
], in which the focus was on substituting the nonlinear Schrodinger equation
(NLSE)-based numerical modeling of a passive fiber with LSTM predictions, we success-
fully developed an architecture capable of simulating the propagation through an active
fiber with a more complicated physical model involved. The novelty of our study lies
in the fact that, in contrast to the majority of existing works that employ deep learning
methods to explore dynamics in passive fibers, we have effectively trained PI-RNN to
predict nonlinear amplification, considering a dynamically changing gain profile and the
Raman scattering. Most works applying RNNs to dynamic prediction typically present
only a few evolution heatmaps, hindering an accurate assessment of real predictive ability.
To address this limitation, we provide comprehensive error maps that illustrate predictive
performance across the parameter domain. Additionally, we delve into the capabilities
and adaptability of the technique for constructing autoregressive predictions employing a
cold-start initialization.
2. Numerical Model of the Amplifier
We consider pulse evolution in a typical, highly doped, ytterbium fiber amplifier.
A Gaussian pulse at 1028 nm is launched into an Yb-doped fiber amplifier with a 6-
µ
m
core diameter, which is co-pumped at 976 nm. As the pulse propagates, it accumulates a
pronounced nonlinear phase, resulting in a significant broadening of the spectra. When the
spectrum is broadened to match the width of the gain spectrum, the proper parameters of
the input pulse facilitate an evolution towards a nonlinear attractor in the GGN amplifica-
tion regime. It is also worth noting that, apart from this, there exists a wide diversity of
different pulse propagation regimes, often accompanied by the formation of noisy Raman
pulses. For modeling highly nonlinear propagation of ultrashort pulses inside the amplifier,
a complex numerical model considering a dynamically changing amplification profile and
Raman scattering is required.
Photonics 2024,11, 126 3 of 11
The numerical model employed in simulations comprises a system of coupled equa-
tions governing pulsed signal generation and continuous-wave pump [10–13]:
∂As(z,t)
∂z=−iβ2
2
∂2As(z,t)
∂t2+Z∞
−∞
gs(ω,z)
2˜
As(z,ω)ex p(−iωt)dω+
iγ1+i
ω0
∂
∂tA(z,t)Z∞
−∞R(t′)|A(z,t−t′)|2dt′(1)
∂Pp(z)
∂z=gp(z)Pp(z), (2)
where
As(z
,
t)
is the slowly varying envelope associated with the signal,
Pp(z)
is the
average power of continuous-wave pump,
β2
is the group velocity dispersion,
γ
is the
Kerr nonlinearity,
gs
and
gp
are signal and pump gain/loss coefficients, correspondingly.
The response function
R(t) = (
1
−fR)δ(t) + fRhR(t)
includes both instantaneous electronic
and delayed Raman contributions [
14
]. We used the Hollenbeck vibrational model [
15
] to
describe the Raman response function
hR
. The spectral window considered in the model
extended from 865 to 1260 nm with the central wavelength at 1028 nm. The temporal
window was equal to 150 ps.
The wavelength dependence of the gain is considered in the frequency domain, where
the optical field
˜
A(z
,
ω)
is multiplied by the gain profile
gs(ω
,
z)
. Each spectral component
of the gain
gs(λi
,
z)
(
i=
1,
. . .
,
Nω
, where
Nω
—is the number of the discreet frequencies in
simulations) and the pump gain/loss coefficients at each step along the fiber were found
based on the rate equations in the stationary case dN2/dt =0:
gs(λi,z) = σs
21(λi)ρs(λi)N2(z)−σs
12(λi)ρs(λi)N1(z),i=1, . . . , Nω(3)
gp(z) = σp
21ρpN2(z)−σp
12ρpN1(z), (4)
dN2(z)
dt =σp
12ρpPp(z)
hνp+k
∑
k=1σs
12(λk)ρs(λk)Ps(λk,z)
hνkN1(z)−
σp
21ρpPp(z)
hνp+k
∑
k=1σs
21(λk)ρs(λk)Ps(λk,z)
hνk+1
TN2(z),
N1(z) = N−N2(z), (5)
here,
N1,2
are population densities in the ground and excited energy levels correspondingly,
N=4.8 ·1015 m−1
is the total number of Yb-ions integrated over the fiber mode cross-
section,
Ps(ωk
,
z) = |˜
A(z
,
ωk)|2
is the signal power at the frequency
ωk
and position z
along the fibre, and
T=
850
µ
s is the fluorescence lifetime. The effective pump absorption
and emission cross sections at pump wavelengths of 976 nm are
σp
12 =
2.5
·
10
−25
m
2
and
σp
21 =
2.44
·
10
−27
m
2
. The absorption and emission cross-section spectra in the considered
spectral window are described by
σs
12(λi)
and
σs
21(λi)
. The normalized pump and signal
power distributions through the fiber cross-section are marked
ρp,s=Γp,s/πa2
, where
a=
3
µ
m is the core radius of a single-mode fiber,
Γp(Γs)
corresponds to the modal overlap
factor between the pump (signal) mode and the ion distribution.
Γp=
1 for core pumping,
Γs=1−ex p(−2a2/w2),wis the 1/e electric field radius of the equivalent Gaussian spot.
We used the open-source Pyofss library [
16
] for numerical modeling; it has a newly
added module that enables parallel computing with a Raman influence [
17
]. We also added
our own modules for parallel amplification computing based on the Yb-coupled equations
described above.
3. The Architecture of a Recurrent Neural Network
The neural network should not only predict the dynamics in a particular propagation
regime but should also guess the regime to be predicted. Predicting the spectrum intensity
is complicated by the substantial broadening during propagation and its high modulation.
Photonics 2024,11, 126 4 of 11
In terms of the temporal intensity prediction, the primary challenge lies in accurately
forecasting the gain variation throughout the evolution. To our knowledge, there have
been no attempts to simulate a fiber amplifier using a physically accurate gain model with
a neural network in the field of fiber optics up to the present date.
Similarly to the study [
8
], we utilised a single-layer LSTM neural network architecture
as a baseline. Since the baseline architecture proved to be insufficient for the dataset
used, we improved the proposed architecture by incorporating stacked LSTM layers and
a bidirectional cell structure. All the outputs from the LSTM cells, instead of just the last
cell output, were then fed through several dense layers to refine the results further. We
implemented our neural network using the PyTorch Python library [18].
The recurrent neural network’s architecture is depicted in Figure 1.
Figure 1. Scheme of the recurrent neural network used consisting of a recurrent (biLSTM) part and a
fully connected part.
Here, we employ two neural networks with similar structures to predict spectral and
temporal intensity evolution independently.
4. Data Preparation and Training Process
The RNN was trained with synthetic data generated using the model described
in Section 2. The chosen range of initial pulse parameters spans the variation in the
relationship between dispersion and nonlinear length, ranging from 0.25 to 250. The initial
pulse intensity is uniformly variable, ranging from 100 W to 1000 W, while the pulse width
varies logarithmically from 0.1 to 10 ps. The explored parameter space encompasses a
wide diversity of pulse propagation regimes, ranging from GGN amplification to pulse
amplification, accompanied by the formation of noise Raman pulses in high-intensity
cases [
19
]. We selected a length of 7 m for the optical fiber, a choice deemed sufficient for
stabilizing a nonlinear attractor, as suggested by Sidorenko et al. [
1
]. The training dataset
comprises 879 examples of pulse evolution within the specified range of initial parameters.
The RNN is trained to forecast the evolution profile at fiber intervals of 46 mm, requiring
150 steps to predict the evolution along the entire length of the fiber. No preprocessing was
applied to the data, except for reducing the resolution using linear interpolation along the
temporal (spectral) coordinate, from 16384 to 500 points. The choice of this dimensionality
reduction was based on the resolution needed to display fine spectral modulation dynamics
and the computational resources available for training the neural network, and it can be
varied for different problems.
The dataset utilized for testing the model consisted of displaced grid points within the
same initial pulse parameter interval. To ensure the construction of a reliable model capable
of accurately predicting the evolution of different propagation regimes on a uniform grid, it
Photonics 2024,11, 126 5 of 11
is essential to employ test and train datasets of equal size and distribution. Therefore, main-
taining a 1:1 test-to-train ratio is imperative to effectively assess the prediction performance
of the neural network within the chosen parameter range for this task.
The data for training and testing the neural network are prepared using the sliding
window method illustrated in Figure 2a. This technique facilitates the division of the
data into smaller segments. The first ten intensity profiles are interpreted as the neural
network input, with the subsequent one serving as its target output. The window then
slides by one point over the fiber length and repeats the process until the end of the training
evolution. After creating 10+1 pairs, it is essential to shuffle these pairs from all the training
evaluations to ensure a stable learning process. The prepared data are then fed into the
neural network during the training process.
Figure 2. Train data preparation process. (a) Illustration of the sliding window approach for data
preparation: synthetic data were subdivided to packs of 10 input (X) data and one output (Y).
(b) Preparation of the cold-start data scheme.
To find the global minimum in the loss function and ensure effective training, we
employed several optimization and learning stabilization techniques. These included the
Adam optimizer, hyperparameter tuning, and a learning rate scheduler. By utilizing these
techniques, we aimed to enhance the training efficiency and stability of our model. Model
training was performed on a local server using an NVIDIA RTX 4090 graphics processing
unit (GPU).
5. Results
We used an autoregression approach to reconstruct the prediction. This method allows
the neural network to forecast the data evolution for any number of steps forward by
sequentially feeding its output back into its input.
5.1. Metrics Used for Tracking RNN Performance
Here, we outline the metrics used to assess the final predictive performance of the
trained neural network. The network is trained to predict a single pulse profile by leverag-
ing the evolution dynamics extracted from a sequence of profiles using the mean squared
error (MSE) loss:
MSE(I,ˆ
I) = 1
N
N
∑
i=1
(Ii−ˆ
Ii)2, (6)
where
I
represents the temporal (spectral) intensity array at a fixed z coordinate along the
fiber, and Ndenotes the size of the temporal (spectral) domain.
In autoregression prediction, the model effectively handles newly self-generated data
that were not part of the training sample. To evaluate the final result, a different metric
was employed. The normalized root mean square error (NRMSE) metric, widely utilized
Photonics 2024,11, 126 6 of 11
in similar tasks, provides a robust accuracy estimation for predicting errors in individual
intensity profiles.
NRMSE(I,ˆ
I) = v
u
u
t
∑Ndomain
i=1(Ii−ˆ
Ii)2
∑Ndomain
i=1(ˆ
Ii)2(7)
However, when applied to the entire evolution map, the NRMSE tends to overestimate
errors in the case of low-energy pulses with narrow spectra and underestimate errors in
high-energy propagation regimes. Additionally, interpreting the quality of the prediction
is not straightforward. In an attempt to address these issues, we propose using another
version of the normalized MSE—the peak signal-to-noise ratio (PSNR) metric [20]:
PSNR(Imap,ˆ
Imap) = 20 ·log10
max(ˆ
Imap)
qMSE(Ima p,ˆ
Imap)
, (8)
where
Imap
is a 2D array that represents the evolution of a temporal (spectral) intensity
array along the spatial coordinate z.
The PSNR is frequently employed for assessing the quality of reconstructed images
and offers several advantages over the NRMSE metrics, including a decibel scale and
normalization based on the maximum intensity.
5.2. Forecasting of Temporal and Spectral Intensity Evolution
We assessed the prediction error of the PI-RNN model using Equation (7) and depicted
the temporal and spectral intensity maps in Figures 3a and 4a, respectively. The plots
illustrate the smooth evolution of errors along the fiber for all initial pulse parameters,
with no discernible discontinuities or outliers within the prediction maps.
Figure 3. Temporal error maps for the test dataset, showing a 140-step prediction using 10 inputs for
initiation. (a) Illustration of the NRMSE evolution in predicting temporal intensity depending on the
distance along the fiber; (b) dependency of the PSNR metric on various initial parameters; points
labeled in red correspond to examples of propagation regimes shown in Figure 5.
Figures 3b and 4b, calculated using Equation (8), illustrate the comprehensive error
maps across the plane of initial pulse parameters. These maps reveal the impact of initial
pulse parameters on the overall PSNR error along the fiber. The interpretation of the
error diagrams is as follows: the prevailing trend indicating the most significant errors
corresponds to situations with numerous high-intensity modulation peaks or noise-like gen-
eration. Specifically, in the prediction of the temporal evolution, the highest accumulated
error is associated with the Raman generation. When predicting the spectral evolution,
the highest error is associated with regions exhibiting significant nonlinear phase accumu-
lation before transitioning to a smooth attractor-like regime or regions characterized by the
emergence of a noisy Raman pulse.
Photonics 2024,11, 126 7 of 11
Figure 4. Spectral error maps for the test dataset, showing a 140-step prediction using 10 inputs for
initiation. (a) Illustration of the NRMSE evolution in predicting spectral intensity depending on the
distance along the fiber; (b) Dependency of the PSNR metric on various initial parameters; points
labeled in red correspond to examples of propagation regimes shown in Figure 6.
We conducted tests on various data preprocessing methods, including normalization
and logarithmic transformation, which have demonstrated effectiveness for passive fibers,
as reported in previous works [
8
,
9
]. However, we observed that these methods diminished
the information content of the spectral intensity evolution within the active fiber. Recog-
nizing the significance of spectral intensity amplitude as an additional feature aiding the
neural network in precise predictions and evolution stage determination through autore-
gression, we opted to use the spectral intensity data in the original linear scale without
any preprocessing.
To compare pulse propagation maps predicted by the PI-RNN with numerical model-
ing using the NLSE, we picked three distinctive regimes from the data. Three main types of
behaviors include a smooth GGN regime resulting in a formation of a nonlinear attractor,
a transient regime, and a regime showing a pronounced influence of the stimulated Raman
scattering on a pulse amplification. Figure 5shows examples of the temporal evolutions,
along with their locations in the error map displayed with red labeled points in Figure 3b.
Figure 6shows typical spectral intensity propagation regimes, with their locations in the
error map in Figure 4b. The Raman scattering manifests itself as the generation of a noise-
like pulse with energy comparable to the main pulse, downshifted by about 13.2 THz in
frequency (Figure 6c) [21].
The PI-RNN demonstrates its capability to model the temporal and spectral evolution
for any point within the training parameter area.
Autoregressive reconstruction of the evolution map along the 7-meter-long fibre,
as presented in the Figures 5and 6, takes about 0.05 s with PI-RNN. This is approximately
700 times faster than the fastest paralleled numerical NLSE model when using the NVIDIA
RTX 4090 GPU and 2000 times faster than the conventional CPU-based numerical model.
This notable difference is attributed to the reduction in the number of numerical operations
required for each step along the fiber, a decrease in the total number of steps needed to ob-
tain a solution, and the lower temporal (spectral) resolution required for the computations.
The final PI-RNN model has an estimated tens of millions of trainable parameters.
Photonics 2024,11, 126 8 of 11
Figure 5. Comparison between the temporal intensity evolution calculated using the NLSE and
the prediction made with PI-RNN. The prediction is built 140 steps ahead along the fiber using
10 consecutive pulses as input. (a)
P0=
432 W,
T0=
0.16 ps, (b)
P0=
810 W,
T0=
2.19 ps,
(c)P0=460 W, T0=8.9 ps.
5.3. Resistance to Noise
The neural network, trained on undisturbed synthetic data, was found to be robust
to external noise in the test dataset. The PI-RNN can capture the pulse propagation
regime up to the signal-to-noise ratio values of 20 dB. Specifically, when given an initial
pulse with added ’white’ Gaussian noise, the neural network predicts the output pulse
with noise suppression. This feature allows for greater practical significance of this
study in the future, as it can handle realistic experimental data that may contain noise or
other imperfections.
5.4. Autoregression Problems
Overfitting poses a significant challenge in autoregressive prediction problems.
In autoregressive predictions, the neural network establishes a feedback loop, using
slightly perturbed input data influenced by its own prediction errors to make subsequent
predictions. This feedback loop complicates the monitoring and prevention of overfitting,
since the neural network is only trained to predict only one step initially and is not
explicitly trained for autoregressive prediction. Even though the model has high accuracy
Photonics 2024,11, 126 9 of 11
for one-step prediction on a validation dataset, it may face issues when estimating
autoregressive predictions on a test dataset. The slightly undertrained model seems
to be more robust to overfitting the numerical modeling data when predicting the
evolution autoregressively.
Figure 6. Comparison between the spectral intensity evolution calculated using the NLSE and
the prediction made with PI-RNN. The prediction is built 140 steps ahead along the fiber using
10 consecutive pulses as input. (a)
P0=
432 W,
T0=
0.16 ps, (b)
P0=
810 W,
T0=
2.19 ps,
(c)P0=460 W, T0=8.9 ps.
5.5. Cold Start Problems
Cold start is a method of evolution reconstruction that starts with a single initial pulse
profile, which is then fed to all RNN inputs. This method facilitates the reconstruction of
the autoregressive evolution map using only a single pulse profile, thereby simplifying its
application for various tasks.
To improve the model’s performance in cold start prediction scenarios, we used a
specific approach to prepare the training data. The main concept involves incorporating
“cold start data” into the training sample, as shown in Figure 2b. “Cold start data” are a
synthetic type of data that mimic the model’s task of reconstructing evolution from one tiled
initial profile. The basic idea is to replace the first
n
evolution profiles, where
n
is the number
of inputs for the recurrent neural network, with the initial profile. This approach aids the
model in improving predictions by learning from these artificial examples.
Photonics 2024,11, 126 10 of 11
We obtained results with comparable accuracy for predicting cold start temporal and
spectral evolutions as the prediction based on the starting pulse sequence.
6. Conclusions
In this paper, we explored the potential of using PI-RNN to predict the evolution
of spectral and temporal pulse intensity along the fiber amplifier—a computationally
challenging task in nonlinear optics. We introduced an updated PI-RNN architecture
designed to learn the complex dynamics of the optical field from a large dataset gener-
ated via numerical simulations and conducted a thorough evaluation of its performance.
We found that the PI-RNN can accurately and precisely estimate the evolution map, out-
performing the conventional split-step numerical solution of the NLSE by a significant
margin—2000 times faster. This improved speed persists even when parallelized with a
GPU, resulting in a 660-fold faster computation.
Furthermore, the PI-RNN adeptly performs interpolation and extrapolation of
the field evolution along the fiber with reasonable accuracy. It exhibits adaptability to
various grid sizes along the z-coordinate. Notably, a single neural network proved capable
of capturing diverse propagation regimes, making it a universal tool for investigating
dynamics within the chosen parameter space. The PI-RNN approach can achieve further
enhancement by integrating experimental data into the training set. This can not only
improve the time performance but also enhances its descriptive capabilities, surpassing
numerical modeling.
We have also detailed the challenges encountered during the PI-RNN training process
for autoregressive prediction tasks. One significant drawback identified in reconstructing
the entire evolution map using PI-RNN is its cold start prediction performance, heavily
dependent on the data sample provided. To address this issue, we introduce a novel
approach to preparing training data, resulting in improved cold start execution. This
modification leads to notable performance enhancements, particularly in predicting the
temporal intensity map using a single input profile.
In conclusion, we assert that the PI-RNN has proven to be a promising technique for
predicting the evolution of pulse intensity along an active fiber, demonstrating substantial
advantages over traditional numerical methods.
Author Contributions: Conceptualization, A.B.; methodology, K.S. and A.B.; software, K.S.; valida-
tion, A.B. and K.S.; investigation, K.S. and A.B.; writing—original draft preparation, K.S. and A.B.,
visualization, K.S. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Ministry of Science and Higher Education of the Russian
Federation (Project No. FSUS-2021-0015).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Conflicts of Interest: The authors declare no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
PI-RNN Physics-informed recurrent neural network.
NLSE Nonlinear Schrodinger equation.
LSTM Long short-term memory.
MSE Mean square error.
PSNR Peak signal-to-noise ratio.
GPU Graphics processing unit.
Photonics 2024,11, 126 11 of 11
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