Yue et al. developed an algorithm (HKAE) to perform time series forecasting. The algorithm exploits concepts derived from dynamical systems theory and Koopman theory, and it uses an autoencoder architecture to realise the link between the two. They decided to apply the algorithm to laboratory earthquakes data, and in particular to shear stress time series.

I think that the idea and the results are interesting. Nonetheless, there are several points that need to be better explained and/or further developed.

I have two major concerns about the manuscript.

1) The first problem that I see comes from your interpretations. You state that your algorithm outperforms the existing ones (e.g., you tested LSTM, TCN and MLP). But in many occasions this is not true. Your Figures 8 and 10 show that, especially for the immediate next future, LSTM performs better than HKAE. I would recommend you to not oversell the HKAE algorithm.

2) The second problem comes from the pre-processing of the data. Reading the code, I noted that there is a pre-processing step to smooth the data. In the manuscript you do not mention any filtering or smoothing step. The smoothing function that you use is taken from the statsmodels package, and it takes the closest data to perform a local linear regression. The closest data in a 1-dim time series can come from both the past and the future. This means that when you smooth the data you are introducing information from the future. This is a problem if you want to evaluate forecasting performances. You need to clarify how many data from the future are used to smooth the data.

Other two, less critical but still important, problems are the following.

One concerns the reproducibility. In order to reproduce the results, it is important that you add a README.txt file in your repository. Furthermore, you should add a requirements.txt file with the details of the packages that you used. It is a good practice to do it so that people can create a local virtual environment and reproduce your results. Some comments in your code are not in English, and you should translate them.

Finally, it is not easy to follow the reasoning and the various steps mainly because of the overall poor English structure. I was trying to write down the correction myself, but after line 70 I gave up because there were too many corrections to suggest. In the Acknowledgments section you mention that the manuscript was polished with GPT-4. Sometimes the feeling is that entire paragraphs were written automatically, without a proper logical connection with the next part. I highly recommend you ask a native English speaker to review and edit the manuscript.

For detailed comments, please see the attached file.

This manuscript explores a novel model that integrates dynamic systems theory with the nonlinear fitting capabilities of deep learning. The HKAE model utilizes the Koopman operator and an autoencoder framework to reconstruct the dynamic behavior of laboratory slip systems, with a specific emphasis on shear stress variations. Through a synthesis of theoretical analysis and experimental validation, the HKAE model showcases exceptional performance in predicting complex nonlinear systems. This study underscores the HKAE model’s advantages in handling intricate nonlinear dynamic systems and proposes promising directions for future research and applications. The following issues could enhance the manuscript’s quality.

1. In Lines 70-71, the statement “our model envisions the future prediction as the continuous evolution of laboratory fault slip systems” lacks clarity and requires a detailed explanation.
2. In Section 2.1, simple simulation data was utilized. Did the model explore more intricate numerical simulations, and what were the specific outcomes? More details about the simulation should be given. Where is the laboratory data coming from? At least the references should be provided.
3. In Section 2.2, please elucidate the advantages of Koopman theory and delay embedding theory in managing the intricate dynamics of laboratory fault slip systems.
4. Line 97, “we’re” should be “we are”. No contraction is allowed in formal English writing. Line 340, as well, among others.
5. In Section 2.3, how do the functions of these three model modules impact the model’s performance?
6. Line 305, “to shows” should be “to show”. The authors should carefully check the whole paper regarding typos.
7. In Figure 9, RMSE and R2 should be included.
8. In Line 339, the model exhibits subpar performance in Exp. 4581. Is the model particularly sensitive to specific types of data?
9. In Section 4, what is the interpretability of the HKAE model? Are there specialized methods or techniques for elucidating the model’s predictions?
10. In Section 5, the HKAE model has demonstrated robust predictive capabilities. What are the potential directions for future research?
11. The fonts in some figures are extremely large, while in others are too small to read. I highly recommend that the authors re-draw most figures to enhance their quality.