From the journal Digital Discovery Peer review history

12-Jul-2022

Dear Dr Valleti:

Manuscript ID: DD-ART-06-2022-000065
TITLE: Bayesian Optimization in Continuous Spaces via Virtual Process Embeddings

Thank you for your submission to Digital Discovery, published by the Royal Society of Chemistry. I sent your manuscript to reviewers and I have now received their reports which are copied below.

I have carefully evaluated your manuscript and the reviewers’ reports, and the reports indicate that major revisions are necessary.

Please submit a revised manuscript which addresses all of the reviewers’ comments. Further peer review of your revised manuscript may be needed. When you submit your revised manuscript please include a point by point response to the reviewers’ comments and highlight the changes you have made. Full details of the files you need to submit are listed at the end of this email.

Digital Discovery strongly encourages authors of research articles to include an ‘Author contributions’ section in their manuscript, for publication in the final article. This should appear immediately above the ‘Conflict of interest’ and ‘Acknowledgement’ sections. I strongly recommend you use CRediT (the Contributor Roles Taxonomy from CASRAI, https://casrai.org/credit/) for standardised contribution descriptions. All authors should have agreed to their individual contributions ahead of submission and these should accurately reflect contributions to the work. Please refer to our general author guidelines https://www.rsc.org/journals-books-databases/author-and-reviewer-hub/authors-information/responsibilities/ for more information.

Please submit your revised manuscript as soon as possible using this link:

*** PLEASE NOTE: This is a two-step process. After clicking on the link, you will be directed to a webpage to confirm. ***

https://mc.manuscriptcentral.com/dd?link_removed

(This link goes straight to your account, without the need to log on to the system. For your account security you should not share this link with others.)

Alternatively, you can login to your account (https://mc.manuscriptcentral.com/dd) where you will need your case-sensitive USER ID and password.

You should submit your revised manuscript as soon as possible; please note you will receive a series of automatic reminders. If your revisions will take a significant length of time, please contact me. If I do not hear from you, I may withdraw your manuscript from consideration and you will have to resubmit. Any resubmission will receive a new submission date.

The Royal Society of Chemistry requires all submitting authors to provide their ORCID iD when they submit a revised manuscript. This is quick and easy to do as part of the revised manuscript submission process.   We will publish this information with the article, and you may choose to have your ORCID record updated automatically with details of the publication.

Please also encourage your co-authors to sign up for their own ORCID account and associate it with their account on our manuscript submission system. For further information see: https://www.rsc.org/journals-books-databases/journal-authors-reviewers/processes-policies/#attribution-id

I look forward to receiving your revised manuscript.

Yours sincerely,
Dr Kedar Hippalgaonkar
Associate Editor, Digital Discovery
Royal Society of Chemistry

************

Reviewer 1

The manuscript by Valleti et al describes an interesting method for potentially accelerating process optimization using autonomous experimentation. In particular, many process variables may be adjusted as a function of time in a manner that is too complex and not physical to be encoded as an independent series of variables. Instead, the authors posit that a variational autoencoder (VAE) can be used to learn a reduced dimensionality space that describes the space of useful process trajectories in a manner that is conducive to optimization strategies such as Bayesian optimization. This is an intriguing concept and the authors explore it using a simulated ferroelectric experiment. The paper is well-written, easy to follow, and the graphics are generally quite good. I have two comments that should be addressed prior to publication.

(1) The data that is generated to train the VAE might be high dimensional in the sense that there are a lot of time points, but it is all initially represented in a very low dimensionality form. For example, the Legendre polynomials are defined by 15 scalars, of which the higher order ones are defined to be smaller. It is clear that the VAE is mostly learning latent variables that are close to proportional to the lowest order Legendre terms. This is clear in Figure 2a where A1 seems proportional to L2 while in Figure 2b, A2 seems proportional to L1. Is it actually any better to represent the data using L1 and L2 vs A1 and A2? What would the error be if the authors tried to compress the data just using A1 and A2 and evaluated the RMSE (ie add a curve to Figure 1d that shows the RMSE that comes from just representing the curves as fewer Legendre polynomials)?

(2) The crux of the paper is that it is efficient to perform optimization using the latent space. This is explored using a ferroelectric simulation. Here, the authors are selecting electric field traces which are defined by three or four parameters (four are given but the range is only given for three). While it is nice to see that optimization works in this latent space, 3-4 variables are a totally manageable number of variables in which to perform optimization. Indeed, these 3-4 variables already present a pretty low dimensionality space in which to operate. The authors should compare Bayesian optimization in the (2D) latent space against Bayesian optimization in this slightly larger parameter space to evaluate whether the compression using the VAE is actually providing any value.

Reviewer 2

This paper proposes a scheme to optimize the parameters of a system described as a set of differential equations. The idea is to generate a lot of trajectories from the system to train a VAE and then applying BO in the latent space. I think this paper contains a useful content for those working in the field related to FerroSIM, but it requires significant improvements.

1) I do not know why section headings are missing. It makes this paper hard to read. Please add them.

2) The idea about combining VAE and Bayesian optimization is not new. So, as the authors mentioned, how to select training examples for VAE would be an interesting point. Please emphasize this point and clarify your contribution in this point.

3) Most of this paper is dedicated to FerroSIM, but literature about FerroSIM and how to optimize its parameters is not really explained. I think it is beneficial to explain more about the basics and motivation of FerroSIM.

Reviewer 3

The authors proposed an interesting idea to use BO in the latent space that is generated by VAE. However, the manuscript is not very well written, so I found this manuscript may not be easy to follow. Readers for this journal would be broad, including materials scientists and chemists from different domains. I suggest the authors to consider re-organizing the contents completely. Put more emphasis of the specific challenges to optimize the ferro-electric materials, and then clearly state how the VAE + BO could assist to solve this challenge. I would also like to suggest authors to remove some of the contents) that is not closed the main theme into supplementary materials/appendix, e.g., the test of VAE’s capability to work with noisy data (and some other data in the first half). As the current state of the manuscript, I would not recommend it for publication in Digital Discovery although the presented topic is very interesting. In addition, I would like to ask the authors to pay attention to the precise use of technical language/terminologies as well as formatting consistency.

1. Comments and suggestions on the introduction of the paper (Page 2 and Page 3):
• The authors attempted to give a broad introduction by framing the common challenge in the field at the beginning. However, I found it very difficult to follow without know what a specific problem to be solved. I suggest the authors to go back and forth between specific problems and generalization instead. Or, at least, mention the specific challenge for ferroelectric early on. This will help researchers who are not in the field to understand the impact of the work.
• The authors could use the titles and sub-titles to guide readers. Without any titles or sub-titles, I found it not easy to follow, given this is a relatively long article with nice Figures.
• Based on the same rationale from point 1, the readers would need more explanations on terminology at the beginning. For example, what does the trajectories mean? Is curl of polarization the objective to optimize?
• Perhaps, a schematic drawing of the workflow (dataflow) can help the understanding, especially for those who are not familiar with VAE and BO.
2. Comments on latent space analysis (page 4 and page 5):
• Formatting consistency:
o “Equation.1” should not have “.” for consistency.
o RMSE may be used in both all caps and italic.
o Page 4: paragraph in the bottom. ‘rmse’ is used before the full definition; and also, please be consistent with whether you want to use capital RMSE or not.
o Equation 4: “where” should not be italic. Some other formatting conventions are consistent, e.g., “()” is sometimes marked as italic in the paragraph above Equation 3.
• It is unclear what the Figure 1 shows. Figure 1 caption seems to get the (b) and (c) wrongly labelled? And what is the axis label for Figure 1a? What is the color bar labels?
3. Comments on the correlation between latent space and A values (Page 7):
• In figure 2, why are A1, A2 and A4 chosen? Why skip A3? The color bar labels are missing for 2d and 2f.
• Why A values are important in the ferroelectric materials system? Would them be the variables to optimize?
4. Comments on the corrupted data test (Page 8)
• In Figure 3, could you explain why the green dots for corrupt data is not separated from the red dots for correct data?
• Why is figure 3 important in the context of BO with latent space optimization if they could not separate the corrupted data from the correct data in the latent space
5. Comments on the description of the ferroelectric problem (Page 9 – Page 12):
• In the equations 2-8 and the accompanying text. Pay attention to the format of “where”, it should not be italic and “w” in lower case. The brackets should not be italic.
• Figure 4: Should it be consistent to separate the a, b, c, d caption with a full stop (.)? Similar questions to some of the earlier figures.
• “20*20” should be “20  20”.
• In Equation 9, “exp” should not be italic. Please also check other italic conventions.
6. Comments on ferroelectric VAE training (Page 13 – 18):
• One thing I get confused is: how are the time and space domain curves or plots in Figure 4 converted into the L1 and L2 plots in Figure 5a and 5b? And then, are these the inputs and outputs into the VAE to reconstruct?
• Figure 6. Could you please re-organize the figure with a, b, c subplots? And, in the latent space plot, please enlarge the numbers, and please label the color bar. All axis labels are too small.
• In Figure 8, could you please differentiate the color of curl from the electric field? Maybe green for curl as it was used as green in Figure 7.
7. Comments on (Page 19 – 22):
• In Figure 9, the lines in 9b affect the evaluation of the sequence. Suggest to remove the lines.
• In this work 1500 data points need to be simulated for the initial data set, and only have 100 additional point to get to the prefect. I’m wondering what the contour plot of GP model looks like with the 1500 data points in Figure 9a. Would it still be necessary to get additional 100 points to know where the maximum is?
• In BO, we typically have a small initialization assuming no historical data exist. Can you reduce the initial data set and run BO for longer iterations?
• “In our case, the BO is set to explore 40% of the time while selecting the points randomly and exploit the rest 60% trying the find the maximum of the curl in the latent space.” How was it implemented sequentially? Why do the authors random sampling? Does it defeat the purpose of building a GP model? “The rest 60%” will be determined by predicted mean? How does this setting compare with an upper confidence bound with (beta parameter = 1)?
• In addition, when VAE is trained, the authors have used 10k data points as input. It means that 10k data points are already known. Since they are known, why do we not use all the data to build a regression model? I would think the authors should consider reduce the VAE training with BO as well, otherwise the framework seems not to have a practical use case.
8. Comments on reference:
• BO has been reported in materials sciences, catalysis, 3D printing, solar cells, chemical reactions, etc. Please consider citing your references more broadly from different fields and different teams.

Dear Editor,
Firstly, apologies for the delay in response. The reviewers’ comments have led us to redo a major portion of the analyses and the results of which can be found in the updated manuscript.

We are resubmitting the manuscript “Bayesian Optimization in Continuous Spaces via. Virtual Process Embeddings” by Mani Valleti et al. after exhaustively addressing the comments by the reviewers.

Regarding the manuscript and the scope of the work, the first reviewer mentions “This is an intriguing concept, and the authors explore it using a simulated ferroelectric experiment. The paper is well-written, easy to follow, and the graphics are generally quite good.” , while the second and the third reviewers commented “I think this paper contains a useful content for those working in the field related to FerroSIM” and “the presented topic is very interesting” respectively.

Both the second and the third reviewers suggested the necessity of section headings to aid the reader, and the third reviewer even suggested a schematic workflow of the entire manuscript. Both these comments are addressed, and a new figure (Fig. 1) delineating the entire workflow is now added to the manuscript.

The third reviewer’s comments on Bayesian Optimization (BO) criticized the seemingly large number of training data points as the initial seeding for the Gaussian process. To rectify this, we have modified the analysis to use a much smaller number of seed points in the updated manuscript and explained the same in response to reviewers. The comments on the corrupted dataset led us to redo the analysis and along which we identified a slight irregularity in the convergence of variational autoencoder (VAE) now the analysis has been revamped to have a more stable convergence. These changes are reflected in the response letter to the reviewers, the updated manuscript, and in the jupyter notebook that accompanies the manuscript.

There are also several minor reviewer comments regarding formatting, citations, and section headings. These are all addressed as detailed in the attached response to the reviewer.

To summarize, we believe that we have addressed the reviewer’s comments in full and we believe that the manuscript is now in a much-improved state for publication. We want to thank all the reviewers for the same.

On behalf of the authors
Mani Valleti and Sergei Kalinin.

This text has been copied from the Microsoft Word file response to reviewers and does not include any figures, images or special characters:

Referee: 1

The manuscript by Valleti et al describes an interesting method for potentially accelerating process optimization using autonomous experimentation. In particular, many process variables may be adjusted as a function of time in a manner that is too complex and not physical to be encoded as an independent series of variables. Instead, the authors posit that a variational autoencoder (VAE) can be used to learn a reduced dimensionality space that describes the space of useful process trajectories in a manner that is conducive to optimization strategies such as Bayesian optimization. This is an intriguing concept and the authors explore it using a simulated ferroelectric experiment. The paper is well-written, easy to follow, and the graphics are generally quite good. I have two comments that should be addressed prior to publication.

Response to the reviewer: We thank the reviewer for the positive comments on this work.

(1) The data that is generated to train the VAE might be high dimensional in the sense that there are a lot of time points, but it is all initially represented in a very low dimensionality form. For example, the Legendre polynomials are defined by 15 scalars, of which the higher order ones are defined to be smaller. It is clear that the VAE is mostly learning latent variables that are close to proportional to the lowest order Legendre terms. This is clear in Figure 2a where A1 seems proportional to L2 while in Figure 2b, A2 seems proportional to L1. Is it actually any better to represent the data using L1 and L2 vs A1 and A2? What would the error be if the authors tried to compress the data just using A1 and A2 and evaluated the RMSE (ie add a curve to Figure 1d that shows the RMSE that comes from just representing the curves as fewer Legendre polynomials)?

Response to the reviewer: The low dimensional smooth encoding of the functions defined by the linear combination of Legendre polynomials elucidates the capabilities of VAEs. Legendre polynomials form an orthogonal basis set in [0, 1] and we have chosen one setting where the curves are not dominated by higher order polynomials. As the reviewer rightly pointed out, this made the encoded latent space a strong function of A1 and A2. But even in that scenario, since VAE forces its latent space to be smooth, the latent space can be decoded to form functions that vary smoothly even in the higher order polynomial space. We have demonstrated this in Fig. 2d-e.

(2) The crux of the paper is that it is efficient to perform optimization using the latent space. This is explored using a ferroelectric simulation. Here, the authors are selecting electric field traces which are defined by three or four parameters (four are given but the range is only given for three). While it is nice to see that optimization works in this latent space, 3-4 variables are a totally manageable number of variables in which to perform optimization. Indeed, these 3-4 variables already present a pretty low dimensionality space in which to operate. The authors should compare Bayesian optimization in the (2D) latent space against Bayesian optimization in this slightly larger parameter space to evaluate whether the compression using the VAE is actually providing any value.

Response to Reviewer: The 3–4-dimensional space discussed in the case of FerroSIM acts as a proof of concept. As discussed in the Legendre polynomials case, the VAE can find low dimensional smooth representation of any set of curves, chosen either via much higher dimensional parametrization or suggested by domain expert. And as shown in the FerroSIM case, the BO algorithm can optimize the latent space even when it contains multiple local optima. The sinusoidal curves are selected for FerroSIM case as they make practical sense and formed a complicated latent space for curl optimization. In general, the curves do not need to be defined by a set of equations and they can be as general as the ones drawn by hand.

 
Referee: 2

This paper proposes a scheme to optimize the parameters of a system described as a set of differential equations. The idea is to generate a lot of trajectories from the system to train a VAE and then applying BO in the latent space. I think this paper contains a useful content for those working in the field related to FerroSIM, but it requires significant improvements.

Response to Reviewer: We thank the reviewer for the positive comments on the work. However, the FerroSIM model although has a lot of significance in the field of ferroelectrics, the workflow in the manuscript can still be applied to any black-box model that has high-dimensioanl time-dependent input and for which the user is trying to optimize a scalarized output of the black-box. Hence, the workflow described in this manuscript has applications beyond FerroSIM and ferroelectrics and can be applied to any process optimization problem.

1) I do not know why section headings are missing. It makes this paper hard to read. Please add them.

Response to Reviewer: The section headings are now added to the updated manuscript to guide the reader.

2) The idea about combining VAE and Bayesian optimization is not new. So, as the authors mentioned, how to select training examples for VAE would be an interesting point. Please emphasize this point and clarify your contribution in this point.

Response to Reviewer: The strength of the proposed approach is that it allows to optimize the process based on the domain specific knowledge. Here, we have generated the examples that allow for variation of linear slope and periodic component, these functional forms being suggested by the physics of ferroelectric switching (and implicitly by earlier work on ship demagnetization during WWII), since we expect that during switching cycle the system will go though point of maximum instability where the defect effect can break symmetry and give rise to curl in the most effective manner. However, we of course do not know which of the trajectories will maximize this effect, and this is where BO helped.

As applied to more general context, we propose to rely on the optimization based on the domain expertise. Here, the vast majority of the experimental fields are associated with large body of phenomenological know-how in terms of specific workflows and processing. The VAE-BO approach allows to both interpolate domain knowledge and optimize within this interpolated space.

3) Most of this paper is dedicated to FerroSIM, but literature about FerroSIM and how to optimize its parameters is not really explained. I think it is beneficial to explain more about the basics and motivation of FerroSIM.

Response to Reviewer: The manuscript shows how to perform optimization in the high dimensional space of functions. Although FerroSIM acts as a domain specific practical example with a complicated latent space, the process still applies when it is replaced by a black box which takes a function as an input and the outputs a scalar. To make sure that the idea of the optimization in high dimensional spaces is not lost in the descriptions of FerroSIM, we have consciously restricted the explanations regarding the simulations. However, we have cited references on FerroSIM where the simulation is described and added the link to the python package that runs FerroSIM for interested readers.

 
Referee: 3

The authors proposed an interesting idea to use BO in the latent space that is generated by VAE. However, the manuscript is not very well written, so I found this manuscript may not be easy to follow. Readers for this journal would be broad, including materials scientists and chemists from different domains. I suggest the authors to consider re-organizing the contents completely. Put more emphasis of the specific challenges to optimize the ferro-electric materials, and then clearly state how the VAE + BO could assist to solve this challenge. I would also like to suggest authors to remove some of the contents) that is not closed the main theme into supplementary materials/appendix, e.g., the test of VAE’s capability to work with noisy data (and some other data in the first half). As the current state of the manuscript, I would not recommend it for publication in Digital Discovery although the presented topic is very interesting. In addition, I would like to ask the authors to pay attention to the precise use of technical language/terminologies as well as formatting consistency.

Response to Reviewer: We thank the reviewer for reading the manuscript and the high estimate of its usefulness and potential impact, as well as high interest to the topic. We fully agree that this work, as any interdisciplinary work, can be presented both from the domain perspective and from more general ML perspective using the domain example for illustration. In this case, we have chosen the more general perspective as the way to demonstrate the principle and universality of the method, and use the domain example to illustrate that even in relatively simple cases this approach allows to discover the non-trivial physical responses (in the context – discover the field history which give rise to global curl in the system where there should be none based on symmetry. In other words, the algorithm amplified the fluctuation close to the phase transition point). We deeply appreciate the reviewer suggestion to restructure the manuscript, but we believe that maximum impact and generality (and matching to the journal scope) are achieved with the current structure.

1. Comments and suggestions on the introduction of the paper (Page 2 and Page 3):
• The authors attempted to give a broad introduction by framing the common challenge in the field at the beginning. However, I found it very difficult to follow without know what a specific problem to be solved. I suggest the authors to go back and forth between specific problems and generalization instead. Or, at least, mention the specific challenge for ferroelectric early on. This will help researchers who are not in the field to understand the impact of the work.
• The authors could use the titles and sub-titles to guide readers. Without any titles or sub-titles, I found it not easy to follow, given this is a relatively long article with nice Figures.

Response to Reviewer: The section headings are now added to the updated manuscript to guide the reader.

• Based on the same rationale from point 1, the readers would need more explanations on terminology at the beginning. For example, what does the trajectories mean? Is curl of polarization the objective to optimize?


• Perhaps, a schematic drawing of the workflow (dataflow) can help the understanding, especially for those who are not familiar with VAE and BO.
Response to Reviewer for the above two questions: The trajectories generally refer to the time-dependent input to the system that control in dynamics. The curl is the measure of the polarization rotation in the system, chosen here as the simplest non-trivial descriptor of materials response. The explanations along with the simplified workflow are now added at the start of manuscript as Fig. 1.


2. Comments on latent space analysis (page 4 and page 5):
• Formatting consistency:
• “Equation.1” should not have “.” for consistency.
Response to Reviewer: This issue has been addressed in the modified manuscript
• RMSE may be used in both all caps and italic.
Response to reviewer: RMSE is now used in all caps throughout the manuscript
• Page 4: paragraph in the bottom. ‘rmse’ is used before the full definition; and also, please be consistent with whether you want to use capital RMSE or not.
Response to reviewer: This is now corrected in the updated manuscript
• Equation 4: “where” should not be italic. Some other formatting conventions are consistent, e.g., “()” is sometimes marked as italic in the paragraph above Equation 3.
Response to Reviewer: The brackets and the equation 4 are now modified.
• It is unclear what the Figure 1 shows. Figure 1 caption seems to get the (b) and (c) wrongly labelled? And what is the axis label for Figure 1a? What is the color bar labels?
Response to Reviewer: Figure 1 is extensively described in the text earlier, but the descriptions weren’t adequate in the captions. The captions for figure 1 are now updated and the colorbar in figure 1c is now named to be RMSE.


3. Comments on the correlation between latent space and A values (Page 7):
• In figure 2, why are A1, A2 and A4 chosen? Why skip A3? The color bar labels are missing for 2d and 2f.
• Why A values are important in the ferroelectric materials system? Would them be the variables to optimize?

Response to Reviewer: Fig. 2a-c elucidates the encoding part of the latent space where the functions randomly generated are projected into the 2D latent space. The latent space appears to be a strong function of the coefficients of lower order Legendre polynomials in the linear expansion of the function. The dependency dies down as we go up the order in the expansion. This is due to how the coefficients in the expansion are normalized and is discussed in the manuscript. (Under section “VAEs for reducing dimensionality of arbitrary functions” in the updated manuscript”) Fig. 2a-c is supposed to communicate this. A4 is plotted to show the drastic change in the dependency from A1 to A4. However, same plots with any of the possible 15 coefficients can be plotted using the jupyter notebook that accompanies the manuscript.

Fig2d-e are plotted as RGB images as discussed in the manuscript. This method is chosen because an RGB image will be able to communicate the information about three coefficients in a single image but plotting colorbars for all three channels becomes voluminous. However, the point of this image is to communicate that the latent space is smoothly varying with multiple optima and the actual values of the coefficients do not affect this.

The Legendre polynomials exercise is a toy problem to show how the variational autoencoder works on trajectories. It does not have anything to do with the ferroelectric materials system. The input to the ferroelectric system is the family of sinusoids modulated by the exponentials which come later in the manuscript. (Under the section “Generating field trajectories and sampling in a reduced space” in the updated manuscript)



4. Comments on the corrupted data test (Page 8)
• In Figure 3, could you explain why the green dots for corrupt data is not separated from the red dots for correct data?
• Why is figure 3 important in the context of BO with latent space optimization if they could not separate the corrupted data from the correct data in the latent space

Response to Reviewer for the above 2 questions: We would like to thank the reviewer for these comments. A slight irregularity in the convergence of the VAE has been spotted while trying to answer these comments and the analysis has been updated now. The angle made by the line segment with the x-axis is one of the parameters sampled in the interval [0, ] to produce linear curves. The tangent of the sampled angle is used as the slope of the line segment and the tangent function diverges at /2 which blows up the range of linear curves which in turn led to incorrect normalization which was the source of irregularity in the convergence. In the updated version, the slope of the linear curves within a non-exploding interval and the updated results are attached below for your perusal and can also be found as Fig. 4 in the updated manuscript.

To answer the question about significance of this analysis, the following points are supposed to conveyed by this section:
• VAEs can simultaneously encode two (or more) families of curves in a smoothly varying latent space.
• The area associated with each family of curves is proportional to its variance.
• When the areas between the two families of the curves are decoded, they produce data points with features associated with both families. These data points are not part of the input dataset and can only be produced using generative models like VAEs. Sinusoidal curves overlapped on top of the linear curves are observed in the latent space below (a-c) when moving radially away from the center.




5. Comments on the description of the ferroelectric problem (Page 9 – Page 12):
• In the equations 2-8 and the accompanying text. Pay attention to the format of “where”, it should not be italic and “w” in lower case. The brackets should not be italic.
Response to Reviewer: The change has now been made to the manuscript.
• Figure 4: Should it be consistent to separate the a, b, c, d caption with a full stop (.)? Similar questions to some of the earlier figures.
Response to Reviewer: All the figure captions are now edited to be consistent.
• “20*20” should be “20  20”.
Response to Reviewer: The change has now been made to the manuscript.
• In Equation 9, “exp” should not be italic. Please also check other italic conventions.
Response to
Response to Reviewer: The change has now been made to the manuscript




6. Comments on ferroelectric VAE training (Page 13 – 18):
• One thing I get confused is: how are the time and space domain curves or plots in Figure 4 converted into the L1 and L2 plots in Figure 5a and 5b? And then, are these the inputs and outputs into the VAE to reconstruct?
Response to Reviewer: It is just the electric field trajectories in time that are encoded into the latent space (L1-L2). The latent space is then sampled and decoded to produce and electric field trajectory in time that is used as the input to a FerroSIM simulation. The Polarization vs. Field plots are obtained at the end of the simulation.

• Figure 6. Could you please re-organize the figure with a, b, c subplots? And, in the latent space plot, please enlarge the numbers, and please label the color bar. All axis labels are too small.
Response to Reviewer: The colorbar is now labeled, the image in the middle of the figure is now named (a). The numbers in the curl plot and the axis labels are now enlarged.
• In Figure 8, could you please differentiate the color of curl from the electric field? Maybe green for curl as it was used as green in Figure 7.
Response to Reviewer: Curl in Figure 8 is now plotted in green to differentiate it from the electric field.




7. Comments on (Page 19 – 22):
• In Figure 9, the lines in 9b affect the evaluation of the sequence. Suggest to remove the lines.
Response to Reviewer: The figure is now edited, and the lines are now removed.
• In this work 1500 data points need to be simulated for the initial data set, and only have 100 additional point to get to the prefect. I’m wondering what the contour plot of GP model looks like with the 1500 data points in Figure 9a. Would it still be necessary to get additional 100 points to know where the maximum is?
• In BO, we typically have a small initialization assuming no historical data exist. Can you reduce the initial data set and run BO for longer iterations?
Response to Reviewer for the above two questions: We fully agree with the reviewer that BO is largely designed for situations when only a small number of initial “measurements” is available. Here, we showed that BO can be also used to refine the location of the optima when we start from a relatively large number of randomly measured points.
However, we understand how the large number of initial points in Figure. 9 does not fully display the capabilities of BO and hence we have now modified the analysis. In the modified version, only 50 randomly sampled points are used to initialize the algorithm and an UCB acquisition function is then used to sample new point at each BO step. The results of this analysis are now in the updated Figure. 9
• “In our case, the BO is set to explore 40% of the time while selecting the points randomly and exploit the rest 60% trying the find the maximum of the curl in the latent space.” How was it implemented sequentially? Why do the authors random sampling? Does it defeat the purpose of building a GP model? “The rest 60%” will be determined by predicted mean? How does this setting compare with an upper confidence bound with (beta parameter = 1)?
Response to Reviewer: The referee is correct. In the revised manuscript, we used the UCB acquisition function to identify regions of the parameter space where the behavior of interest is maximized, starting with only 0.5% of all points. We have updated the text and the notebook accordingly.


• In addition, when VAE is trained, the authors have used 10k data points as input. It means that 10k data points are already known. Since they are known, why do we not use all the data to build a regression model? I would think the authors should consider reduce the VAE training with BO as well, otherwise the framework seems not to have a practical use case.
Response to Reviewer: The latent space comprises of 10000 points and the brute force calculation on the entire latent space were run for the discussion related to the FerroSIM (Figures 6-8). Then we show separately, without using the brute force results, that the optima of the curl in the latent space can be explored using only 550 points (50 initial points + 500 BO iterations) instead of the whole 10000 points.


8. Comments on reference:
• BO has been reported in materials sciences, catalysis, 3D printing, solar cells, chemical reactions, etc. Please consider citing your references more broadly from different fields and different teams.
Response to Reviewer: The references are now added to the section where BO algorithm is introduced.

14-Oct-2022

Dear Dr Valleti:

Manuscript ID: DD-ART-06-2022-000065.R1
TITLE: Bayesian Optimization in Continuous Spaces via Virtual Process Embeddings

Thank you for submitting your revised manuscript to Digital Discovery. I am pleased to accept your manuscript for publication in its current form. I have copied any final comments from the reviewer(s) below.

You will shortly receive a separate email from us requesting you to submit a licence to publish for your article, so that we can proceed with the preparation and publication of your manuscript.

You can highlight your article and the work of your group on the back cover of Digital Discovery. If you are interested in this opportunity please contact the editorial office for more information.

Promote your research, accelerate its impact – find out more about our article promotion services here: https://rsc.li/promoteyourresearch.

If you would like us to promote your article on our Twitter account @digital_rsc please fill out this form: https://form.jotform.com/213544038469056.

By publishing your article in Digital Discovery, you are supporting the Royal Society of Chemistry to help the chemical science community make the world a better place.

With best wishes,

Dr Kedar Hippalgaonkar
Associate Editor, Digital Discovery
Royal Society of Chemistry

Reviewer 1

The authors have addressed my comments and I support publication at this stage.

Reviewer 2

None