We thank the participant in the community discussion for the contribution and for providing constructive feedback. Please find our replies in italic.

This is an interesting article. There are minor issues that I think would increase the value of it (in no particular order):

• M(t) (the flow between the two buckets in M0, which I guess is a snow melting flow) is not formally defined in the model description.

We introduce melting M in l.134. and in l.148, and we reference appendix A1 where M is defined for the original conceptual model in Equation A4. We are open for further specification w.r.t. what the formal definition might be lacking but we are sure that it exists.

• There is no explanation of why different errors where applied to different modules of the hybrid models. Ther eis a comment in the discussions regarding the known failures of models to capture discharge peaks, this performance is strongly linked to the type of error (or the noise distribution in a sotchastic modeling context) that was used to train the  model. Did the chosen erros for traing improve this? why? what was the criteria to choose the different errors?

There was no error function assigned. The models are deterministic and where trained using the available data points. We applied the same procedure as was done in the reference study by Jiang et al., 2020, in order to ensure comparability.

• I could not find a link to the software, and "be makde available in the near future" is too ambiguous. The software should be part of the publication work, citing:
  
  
  
  “An article about computational result is advertising, not scholarship. The actual scholarship is the full software environment, code and data, that produced  the  result.” -- Buckheit and Donoho

We fully agree, it was always our intention to provide the code. For this purpose, we have set up a Github repository at the time of our submission. However, we wanted to have a discussion about the approach at first in a review process, and only want to release the code of a method that is considered interesting by the scientific community. Now, after receiving encouraging feedback, we will release the code in the coming revision.

• The data-driven relation learn by M50/100 are clearly tuned to the data. Assuming the proposed mechanism is causal and universal, wouldn't it then make more sense to train these modules in the totality of the data, not per catchment? On the one hand, it is well stablished that non-causal data-driven can easily ourperform causal models (e.g. a casual structure X -> Y -> Z with noise in X larger than in Z will cause data-driven models to choose Z to as the best predcitor of Y, alas non-causal). On the other hand, it is unlikely that NN models will use relations that go beyond the scope of the data, hence the optinal relations found per catchment might be reflecting circumstantial relations, but the mechanisms proposed are suposed to be principled mechanisms, not cimcustantial.

We agree. This is where our current research is headed. We work on training the models on multiple basins. One of the points we want to investigate is exactly whether the learned relations go beyond the scope of the data. The physical structure of the model enforces a sort of regularization that we think might help in this respect.

We thank the referee for the thorough review, detailed feedback and valuable comments. Please find our replies corresponding to the enumerated points.

Specific comments

1. We appreciate the specific recommendation to broaden the coverage of related approaches like PINNs etc. We will add respective information and references to further support the mentioned claim.
2. Thank you for raising these points in the discussion. We assume that the referee refers to l. 40ff in the manuscript. We agree, that modelling success is always subject to task, data, system, etc. Yet, for predictive tasks at daily resolution, LSTMs have frequently proven to show overall best results. To address the remark, we suggest, to highlight the “predictive” aspect more strongly in the revision. We do think that Gauch et al. 2021 (“Rainfall–Runoff Prediction at Multiple Timescales with a Single Long Short-Term Memory Network”) is the appropriate reference on temporal resolution of LSTM predictions since their work introduces Multi-Timescale LSTMs for this purpose. We think the issue on using different data sources is addressed in Kratzert, F., Klotz, D., Hochreiter, S., and Nearing, G. S.: A note on leveraging synergy in multiple meteorological data sets with deep learning for rainfall–runoff modeling, Hydrology and Earth System Sciences, 25, 2685–2703, 2021. that we have also in our list of references. If we misunderstand the referees comment here, we appreciate further remarks in this respect.
3. In the M50 approach, two small, process-specific NNs are used to substitute two different processes: one for discharge Q and the other one for evapotranspiration ET. Qualitatively, temperature T should not have a direct impact on Q but indirectly on the entire water balance via ET (and of course melting M but M is not substituted by an NN in M50). We considered only precipitation P and water storage S_water to have an effect on Q and therefore these are the only input variables to NN^50_Q – this is how we show it in Figure 1 (b). Yet, the referee’s remark made us aware of the fact that in the top of Figure 1 (b), P is depicted as input variable to the other neural network NN^50_ET. This should rather be temperature T as it is also used in the mechanistic formulation of ET in Equation A3 and as it is discussed in the manuscript. We thank the referee for spotting typo and we will change the figure accordingly.
4. Thank you. For keeping the main article concise, we suggest to add such a requested more detailed figure to the appendix and reference it in the main article.
5. Part 1) We did not take measures to prevent overfitting during pre-training. We did not see a need for any sort of regularization in the pre-training phase since this step is only supposed to “roughly” inform the NN parameters such that the NN learns how to process the input variables appropriately to approximate the internal model processes. The target values in this step stem from hard-coded processes relations from the original conceptual and are themselves only approximations to what actually happens in reality. We minimized the sum of squared residuals to these target values as objective function over 1000 iterations in this pre-training phase.  Part 2) We are happy to provide more details w.r.t. the training: The loss function for the training of the whole Neural ODE models (conceptual model frame + NN(s) within) is NSE. The training is conducted using the GalacticOptim.jl package in Julia where the Neural ODE model parameters (i.e., the parameters of the NN(s) within the conceptual model frame) are optimized using ADAM(): For M100 the only model parameters in the model are the weights and biases of the NN^100. For M50, the model parameters that are optimized are the weights and biases of the NN^50_Q and NN^50_ET (the model parameters in the process M(t) are those from the conceptual model and are kept fixed.) Apart from this optimization, there is no additional fine-tuning. Overall we think that the release of the code will help also with this point (please see our reply to point 12)
6. Thank you. We also found Jiang et al., 2020 to serve as a good reference point for a comparison.
7. Thank you, this is a very important point. We agree that all readers should be able to easily read the shown figures. Since the major purpose of these figures was to show the ability of the models to match the temporal pattern of hydrographs, snow water equivalent and (model-based) water storage rather than the exact fitting of data-points, we decided that gaps in dashed and dotted lines do not hinder this insight. Yet, we will reiterate the figures and search for suitable alternatives that might serve all mentioned objectives.
8. This  important point raised by the referee relates to comment 7 above. We chose the color-coding “thermal” having in mind that it is an often recommended set of colors. Yet, now we read “The misuse of color in science communication” by Crameri et al., 2020, Nature Communications, to obtain better knowledge in this field. There, “thermal” is highlighted as one of the colorblindness-friendly options confirming our choice.
9. We think that Figure 4 clearly shows that for water storage values smaller than the mean estimated S1, the discharge values estimated by the three different models are very similar while for larger storage values, there is a large discrepancy. This was the main point that we discussed also in the corresponding text in l.287ff. We did not see additional insights in this respect in log-scale plots but we will look into whether providing log-scale plots (maybe in the appendix) might add additional value.
10. Thank you for the discussion of this aspect. We agree that extrapolation of neural networks is a challenging topic. In our application case we purposely also picked basins to show that this extrapolation is difficult and sometimes leads to counter-intuitive behavior – to highlight these issues. We still think that more data might help because, first, they help the network to refine relations for data ranges that are closer to the ranges of pure extrapolation. Second, since we consider natural systems where physical limits constrain the problem to be learned by the NN. These are enforced by the chosen model structure and act as regularization aside of the information provided by data (please see our answer w.r.t. regularization for remark 5 above). We think that this combination might help to elicit extrapolation relations that can then be evaluated in plausibility testing. An example for this would be a very high rainfall intensity where our data might never cover a centennial occurrence, but we would still be able to qualitatively judge whether the relation learned by network is plausible or not. Nonetheless, we agree, more data is not necessarily the solution to the problem alone. An extended discussion of this topic is useful and we will elaborate on it.
11. We partially agree. There can be confusion if “significance” is used with respect to results from statistical tests, but we do not think that this is the case here. Significant is a frequently used term also outside of only statistical tests and this was our intention to use it. Otherwise, we would have used “statistically significant”. Nonetheless, we will check the different instances to decide whether a substitution by “considerably” or comparable alternatives is an option.
12. We fully agree and we will make the corresponding code publicly available. This point was also raised in the community discussion, please see our answer there: “it was always our intention to provide the code. For this purpose, we have set up a Github repository at the time of our submission. However, we wanted to have a discussion about the approach at first in a review process, and only want to release the code of a method that is considered interesting by the scientific community. Now, after receiving encouraging feedback, we will release the code in the coming revision.”

Technical corrections

1. Thank you for spotting this. As we correctly showed in Equation 3, the text should of course say “observed average” – we will correct this.
2. We agree that directly showing and using the equation for NSE is very common in hydrology. Yet, we purposely decided to focus on CoE_alpha and to show that NSE (like the also used mNSE) is only a special case of CoE_alpha. This is not often highlighted in hydrology but we think it is an useful relation to know. We think that lines 203-213 explain this sufficiently and hope that the referee can follow our intention here.
3. Thank you, we will correct the typo.

We thank the referee for constructive remarks and we appreciate the future recommendation to advance our research. Providing a modelling approach that allows to extrapolate to different climatic conditions is one of the motivations to use NeuralODEs. We have collected all comments with corresponding lines from the pdf file that was attached and we have answered them below:

l.3 Replace the highlighted with "not yet fully understood"

- We agree, we change “given” to “understood”.

l.9 Please mention the exact number (609?)

- We will replace “several hundred” by the exact number: 569

l.12 There is no need to divide the "Introduction" section through sub-sections. Hence you may remove 1.1, 1.2 sub sections.

- We introduction brings together several big topics that are important for the remainder of the article. Our intention to use sub-sections was to help structuring this. Referee 1 commented to like this setup. We see this as confirmation of our assumption that this structure helps to keep a red line throughout section 1. Therefore, our preference is to keep the sub-sections and we hope that the referee can follow our argument here.

l.16 (PUB)

- We think the referee refers to separating the abbreviation PUB from the references. We seek to avoid two parenthesis following one another. Yet, to provide a better distinction, we suggest that a “;” rather than only a “,” might be more suitable.

l.18 You can start this sentence with "In this paper"

- Thank you for the suggestion. Since we did not refer to “this paper” anywhere else and in order to keep this style of writing, we suggest to start the sentence with “Here,…”

l.21 It would be better if you can support this sentence through literature

- We will add supporting literature.

l.22 «Although, the»

- We think there is no comma necessary here.

l.22-23 “Especially, when»

- After reading this sentence again, we suggest to drop “Especially” entirely, since it appears only to be a filling word here.

l.31 I suggest the authors to brief some of the internals whicj Nearing et al., & ... have explained (May be in 1 or 2 sentences)  

- Thank you, we will add some details here, e.g. the referenced papers demonstrate linking hidden states in LSTM models to physical variables like soil moisture content.

l.41-46 Thank you for looking into the different words that are made from word combinations. We looked into the individual cases and learned the following:

• Timescales

- following https://www.collinsdictionary.com/dictionary/english/timescale : “timescale” and “time scale” both exist, but refer to different contexts. In our case, timescales seems to be appropriate

• Continuous-time

- Agreed, we will change it to “continuous time”. Also, following this, we will delete the hyphen in “hidden-states”

• Step-wise

- Agreed, we will change it to “stepwise”

• Real-world

- following https://grammarhow.com/real-world-or-real-world/ : we keep real-world with a hyphen because it is used as adjective to a noun

• Time-continuous

- we will replace this by “continuous in time”

l.61 Provide some examples of certain bucket-type models? (HBV-light?)

- we will add examples like the mentioned HBV or GR4J

l.74 Spell out the acronymn «FUSE»

- Agreed, we will add “Framework for Understanding Structural Errors”

l.129 Material and Methods

-  We see that “Material and Methods” is often the default header for this section. Since we do not specify any material as it is used, e.g., in a lab, we prefer keeping “Methods” only.

l.168 Spell out the abbreviation “CAMELS”

- Agreed, we will add “Catchment Attributes and Meteorology for Large-sample Studies”

l.188 Was there a specific reason to use NSE. What is the advantage of using NSE over Kling Gupta Efficiency?

- We purposely only used the metrics that were also employed in our reference study by Jiang et al. (2020), i.e. NSE, FHV and mNSE. Since NSE is a commonly used training and testing metric in hydrology, we did not see a need to switch to KGE. Yet, we know that KGE with its three different parts is generally a good option for hydrologic model rating.

L.215 This has been mistakenly written here (“FHV”)

- It shall be in the parenthesis, but at the beginning, not the end. This is a TeX-based error, thank you for spotting it!

l.258-259Give a basic description about these two catchments.

- We will add a short description providing some context information about the two catchments.