Enhanced bi-LSTM for Modeling Nonlinear Amplification Dynamics of Ultra-Short Optical Pulses
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this paper, the authors 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. PI-RNNs was exhibited to accurately and precisely estimate evolutionary maps, outperforming the nonlinear Schrödinger equation (NLSE) by a considerable margin by traditional stepwise numerical solutions. But I have some questions:
1. Abstracts are not focused and do not express the purpose and results of the study clearly enough;
2. Did the author tried the other deep learning algorithms, except for PI-RNN?
3. What challenges does traditional numerical simulation face so as that the authors have to seek deep learning as the alternative?
4. In line 263, the author states in the conclusion that the PI-RNN model outperforms the NLSE model by a factor of 2,000, so where exactly does this difference manifest itself, and please explain clearly.
5. Conclusion section need to be rewritten.
Comments on the Quality of English Language
english is OK
Author Response
We would like to thank the referees for time spent on reading the manuscript and for their valuable comments and suggestions. We are happy to clarify points raised by the referees and improve manuscript accordingly. Please find below a detailed point-by-point response to all comments. Corresponding changes in the manuscript are marked with the red color.
Reviewer 1 question 1 (R1Q1). Abstracts are not focused and do not express the purpose and results of the study clearly enough.
Our response to question 1.1 (OR1.1) We agree with the comment. We have revised the abstract to make it clearer.
R1Q2. Did the author tried the other deep learning algorithms, except for PI-RNN?
OR1.2. We exclusively employed the PI-RNN among deep learning algorithms in our study, as we firmly believe that recurrent neural networks are the most suitable for our specific task. Our objective revolves around predicting time series data, specifically the spectral and temporal intensity profiles along an active optical fiber. Recurrent neural networks excel in handling such tasks due to their internal memory, efficiently utilizing information from previous steps. Unlike feedforward or convolutional neural networks, recurrent networks possess the essential capability to capture temporal dependencies.
Our task is not only a time series prediction problem but also a physics-informed one. Leveraging prior knowledge of the underlying physical laws governing pulse propagation in the fiber, we utilized the dataset generated through numerical modeling as a training dataset, incorporating these physical principles into the PI-RNN. Incorporating physics-informed features is a unique advantage that other deep learning algorithms may not easily provide.
Recurrent neural networks exhibit robustness and generalizability, making them a versatile tool for predicting pulses within a chosen range of parameters. This versatility extends to predicting pulses for arbitrary fiber lengths within the training parameter range. We clarified this point in the revised text (Introduction, page 2).
R1Q3. What challenges does traditional numerical simulation face so as that the authors have to seek deep learning as the alternative?
OR1.3. 1. Numerical simulation of the NLSE is based on approximations and discretizations that may introduce errors and artifacts. Moreover, real experimental problems are not fully parameterizable, as there are always additional factors affecting the system, such as temperature fluctuations, mechanical disturbances, noise, etc. These factors may not be captured by the numerical model, leading to discrepancies between the simulation and the experiment. The possibility to additionally train a model pretrained on synthetic data on experimental data can make the PI-RNN to exceed this limitation.
2. Numerical simulation of the reverse NLSE problem is more difficult than the forward problem. Proposed PI-RNN offers easy and fast reversed problem solution by reversing the training process.
3. Numerical simulation of the pulse propagation along the fiber requires a lot of computational resources and time, especially for long fibers, high power pulses, or complex gain profiles. Our PI-RNN, once trained in the required range of parameters, can provide the pulse evolution along the fiber in a matter of seconds, without the need to perform any calculations.
We addressed the most important points in the revised Introduction.
R1Q4. In line 263, the author states in the conclusion that the PI-RNN model outperforms the NLSE model by a factor of 2,000, so where exactly does this difference manifest itself, and please explain clearly.
OR1.4. Thank you for the comment. We agree that statement in line 263 is not clear enough. What we intended to convey is that the PI-RNN model outperforms the NLSE model by a factor of 2,000 in terms of the computational time required to predict the intensity profile at the distance of 7 meters. This is the distance at which we compare the PI-RNN and the NLSE predictions in figures 5 and 6.
In generating the PI-RNN prediction, we employ an autoregression approach. This involves using previous PI-RNN predictions as inputs to generate subsequent predictions, allowing us to reconstruct the intensity profile's evolution along the fiber from input to output. We utilize a spatial resolution of Δz=46 mm along the fiber, resulting in 150 steps to cover the distance of 7 meters. Additionally, we employ a temporal (spectral) resolution of 500 points, capturing the main features of the pulse shape and spectrum modulations.
For the NLSE prediction, we employ a numerical method based on the split-step Fourier algorithm. The resolution along the fiber is Δz=0.7 mm, requiring 10,000 steps to reach the distance of 7 meters. We also utilize a temporal (spectral) resolution of 16,384 points.
The significant reduction in computational time with the PI-RNN model is attributed to a decrease in the number of numerical operations required for each step along the fiber and a reduction in the total number of steps needed to obtain a solution. Consequently, the PI-RNN model proves to be markedly faster than the NLSE model, all while maintaining accuracy and prediction quality.
The revised text is highlighted on page 9 of the manuscript.
R1Q5. Conclusion section need to be rewritten
OR1.5. We have revised the conclusion to enhance its clarity.
Reviewer 2 Report
Comments and Suggestions for AuthorsIn this manuscript, a physically-informed recurrent neural network (PI-RNN) is applied to predict the nonlinear evolution of pulse intensity in an active optical fiber. By considering dynamic gain profiles and Raman scattering, the PI-RNN accurately predicts the intensity evolution and exhibits a relative speedup of 2000 times compared to standard numerical methods. The improved PI-RNN architecture demonstrates superior performance in terms of speed and interpolation, and it adapts well to different propagation regimes. Using the PI-RNN for predicting pulse intensity evolution is quite interesting work, and has the potential to make some contributions to the field.
However, I have a few questions and suggestions:
1. Regarding the training dataset: it is essential to address if any filtering or degradation was applied to it, because the results obtained from standard numerical methods do not always align well with experimental results. If the training dataset is not properly filtered, the PI-RNN model might primarily focus on fitting the nonlinear Schrödinger equation (NLSE) rather than accurately representing the pulse evolution in experiments. Therefore, this manuscript may be more suitable for publication in the journals about machine learning, just like references [8] and [9] cited in your manuscript, which are from the journal “Nature Machine Intelligence”.
2. The utilization of experimental data as part of the training dataset seems promising. Can you discuss how this might improve the accuracy of pulse evolution prediction and enable a more comprehensive comparison with NLSE calculations? This comparison could provide valuable insights into the refinement of the NLSE itself and further our understanding of the underlying physics.
3. It would be beneficial to clearly outline the specific advantages of the PI-RNN over standard numerical methods. While the computational speed improvement offered by the PI-RNN is noteworthy, it would be insightful to address other tasks or applications in which the PI-RNN outperforms standard numerical methods. If the PI-RNN presented in this manuscript only offers advantages in terms of computational speed, there may be certain limitations to its practical application.
I believe addressing these points will strengthen your manuscript.
Author Response
We are glad to see the positive evaluation of our work. We would like to thank the referees for time spent on reading the manuscript and for their valuable comments and suggestions. We are happy to clarify points raised by the referees and improve manuscript accordingly. Please find below a detailed point-by-point response to all comments. Corresponding changes in the manuscript are marked with the red color.
R2Q1. Regarding the training dataset: it is essential to address if any filtering or degradation was applied to it, because the results obtained from standard numerical methods do not always align well with experimental results. If the training dataset is not properly filtered, the PI-RNN model might primarily focus on fitting the nonlinear Schrödinger equation (NLSE) rather than accurately representing the pulse evolution in experiments. Therefore, this manuscript may be more suitable for publication in the journals about machine learning, just like references [8] and [9] cited in your manuscript, which are from the journal “Nature Machine Intelligence”.
OR2.1. Thank you for the comment. Our neural network has been trained only using synthetic undisturbed data, generated by numerical simulation of the NLSE equation. We did not apply any filtering to the data, as we wanted to preserve the complex and fine modulation dynamics of the pulse evolution. However, we did reduce the resolution of the data (using interpolation) along the temporal (spectral) and spatial coordinates, from 16,384 to 500 points and from 10,000 to 150 points respectively. This dimensionality reduction was chosen based on the dataset size and the computational resources available, and it can be varied for different datasets.
We discovered that our neural network exhibits robustness to external disturbances, even when not included in the training sample. The PI-RNN can accurately capture the pulse propagation regime up to OSNR values of 20 dB, as illustrated in the figure below. This implies that our neural network can generalize effectively to noisy data, correcting for the effects of the noise. This feature enhances the practical significance of our study for future applications, enabling it to handle realistic experimental data that may contain noise or other imperfections.
The corresponding changes in the text were made on pages 4 and 9.
R2Q2. The utilization of experimental data as part of the training dataset seems promising. Can you discuss how this might improve the accuracy of pulse evolution prediction and enable a more comprehensive comparison with NLSE calculations? This comparison could provide valuable insights into the refinement of the NLSE itself and further our understanding of the underlying physics.
OR2.2. Incorporating experimental data into the training of our neural network holds the potential to enhance the accuracy of pulse evolution predictions. Experimental data can capture real-world conditions such as temperature fluctuations, mechanical disturbances, noise, and other imperfections, which may not be fully accounted for in numerical models. By learning from both numerical simulations and experimental data, our neural network can adapt its predictions to better reflect the complexities of reality, accommodating uncertainties and variations present in experiments.
Furthermore, the use of experimental data in training allows for a more comprehensive comparison with NLSE calculations, revealing potential discrepancies and limitations in the NLSE model. Assumptions and approximations in the NLSE model, such as the slowly varying envelope approximation, neglect of higher-order nonlinearities and dispersion effects, or idealization of the gain profile, may be scrutinized against real-world experimental data.
Therefore, we think that using experimental data as part of the training dataset is an interesting direction for future research, and we plan to explore this direction in our future work.
R2Q3. It would be beneficial to clearly outline the specific advantages of the PI-RNN over standard numerical methods. While the computational speed improvement offered by the PI-RNN is noteworthy, it would be insightful to address other tasks or applications in which the PI-RNN outperforms standard numerical methods. If the PI-RNN presented in this manuscript only offers advantages in terms of computational speed, there may be certain limitations to its practical application.
OR2.3. We can cite the following advantages of neural networks compared to standard numerical methods:
1. Numerical simulation of the NLSE is based on approximations and discretizations that may introduce errors and artifacts. Moreover, real experimental problems are not fully parameterizable, as there are always additional factors affecting the system, such as temperature fluctuations, mechanical disturbances, noise, etc. These factors may not be captured by the numerical model, leading to discrepancies between the simulation and the experiment. The possibility to additionally train a model pretrained on synthetic data on experimental data can make the PI-RNN to exceed this limitation.
2. Numerical simulation of the reverse NLSE problem is more difficult than the forward problem. Proposed PI-RNN offers easy and fast reversed problem solution by reversing the training process.
3. Numerical simulation of the pulse propagation along the fiber requires a lot of computational resources and time, especially for long fibers, high power pulses, or complex gain profiles. Our PI-RNN, once trained in the required range of parameters, can provide the pulse evolution along the fiber in a matter of seconds, without the need to perform any calculations.
We addressed the most important points in the revised Introduction.
Author Response File: 
Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper addresses the prediction of the nonlinear evolution of ultrashort pulses along a fiber amplifier, utilizing a Physically Informed Long Short-Term Memory Recurrent Neural Network. LSTMs, known for their efficacy in capturing long-range dependencies and learning patterns over extended sequences, are employed in this study. To provide data to the neural network, a well-founded theoretical model is developed. The paper exhibits a well-organized structure, and the results are presented with clarity. The subject matter holds significant relevance and is of great interest within the optical community. As a comment, it would be beneficial if the authors included a discussion on how they determined the optimal values for the input/output layers in the ratio of 10:1.
Nevertheless, I recommend the publication of this paper in its current form.
Author Response
We would like to thank the referee for the positive evaluation of our work. The input length of 10 sequential pulses was chosen in accordance with the sampling step and the required length of the input pulse sequence for initiating evolution prediction. It was found to depend on propagation regimes included in the sample and the required duration for the starting sequence to enable successful distinction of the regimes by the PI-RNN. The pulse sequence of length 1 (cold start problem) was chosen to be the simplest possible case as long as this length is enough to initiate the autoregressive predictions. We did not carry out fine optimization of these parameters, having tested only a few ratios and chosen what was convenient for us.
Round 2
Reviewer 1 Report
Comments and Suggestions for Authorsthe author has made necessary revision.
Comments on the Quality of English LanguageEnglish is OK
Reviewer 2 Report
Comments and Suggestions for AuthorsAccept in present form
