From the journal Digital Discovery Peer review history

29-Apr-2022

Dear Dr Flores:

Manuscript ID: DD-ART-04-2022-000027
TITLE: Learning the laws of lithium-ion transport in electrolytes using symbolic regression

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

Reviewer 1

The authors have attempted to elucidate the model for ionic conductivity in Li-based liquid electrolytes using symbolic regression and high throughput experimentation. Interestingly, the authors have selected 3 predictors for ionic conductivity, namely, the temperature, salt concentration and cyclic to linear carbonates ratio. It was also noted that all the experiments were performed under a moisture free environment. Subsequently, the authors derived a general model for ionic conductivity. This is a significant work in many ways. Firstly, a framework of ion conduction mechanism discovery is discussed in this work. Secondly, a demonstration of high throughput experimentation for characterising the electrolyte has been shown and can be followed for subsequent efforts in this field.
Specifically, the authors have highlighted an expression that generalises the conductivity as a function (c, T and r). Can the authors comment if this expression holds true for both high and low salt concentrations given that the solvation structures formed is different at different salt concentrations; there could be a coordination preference difference between high and low salt concentration for linear and cyclic carbonates which could affect the Li+ diffusion mechanism and potentially the conductivity. (see Phys. Chem. Chem. Phys., 2016, 18, 164-175)

Reviewer 2

Flores et al. present a study on rediscovery of the physical laws governing the transport of lithium ions in liquid electrolytes using symbolic regression.
Overall the paper is very well written and presents a nice examples of applying symbolic regression to a seemingly trivial but actually rather complex problem: the conductivity of an electrolyte containing several solvents.
I find in this paper a thorough discussion of the underlying models and a good approach to avoid overfitting.

Maybe this is a discussion for another manuscript but what I gather from the discussion and the figures is that the model seems to perform a lot better at higher temperatures. From my own groups research I found polynomial models to predict much better (well in agreement with Figure 3) but in general suffering from the same problem of overfitting as discussed here unless heavily regularised. Could the authors maybe add a sentence or two discussing the possibility of two physico-chemical regimes? I could imagine there being something like a phase boundary towards low temperatures when the viscosity starts to play a role?

An additional comparison for the discussion could be made if one did a simple comparison of a Pade or Taylor approximation or alike of the DHO model and the found expression would these approximations differ a lot and where?

Editorial note: There was a minor figure hick-up in the generated pdf. Figure 2 was for the most part grayed out but the preprint version I found did not have that issue. I was able to gather the missing info.

In general this manuscript is a very concise and well written example of using symbolic regression to (re)discover physico-chemical laws of ion transport in liquid electrolytes that I recommend to be accepted as is.

Reviewer 3

The research utilizes SR to unravel a solution expression that tries to balance several aspects simultaneously, such as: accuracy without overfitting, parsimonious or compactness, consistency, a square-root functional structure resembling the DHO law, conformity with the existing understanding of ionic transport. The results are well reported and discussed, though I have some concerns as mentioned below.

1) How do you choose the library of basis operators prior to running SR? Why do you select 'sqrt', 'exp', 'log' but not 'sin', 'cos'. The justification is not available in the paper. Is the choice solely based on the intuition or you have conducted trial runs with other set of basis operators ! Even the supplementary material does not contain this information.

2) The moment non-linear features comes in, the model can't be said linear anymore.
Rather than saying "a linear model of ...", perhaps you can write the same idea as follows.

Please rephrase the sentence: In our SR approach, we apply non-linear transformations on the original predictors to produce more informative candidate features. The aim is to approximate the conductivity with a combination of non-linear candidate features, as given below.

3) The paper mentions that the solution to the current SR approach is generally non-unique, however, there is no discussion with examples on this topic.

Can you please show some of the other expressions that give good accuracy of fit ? Discuss the underlying pros and cons (accuracy, parsimonious, consistency) with different expressions and justify why do you select the current solution.

4) The example notebook in the Github reports a mse of 0.13. This error is a bit high ! Can you explain why ?

5) At several places, I have found that some terminologies are misleading. For example: 'transformation' should be replaced with 'operation'. Please see the attached file for further minor issues.

Thank you.

We would like to thank the three referees for taking their time to provide helpful comments and so improve the manuscript. We made genuine efforts to address the referees’ comments and suggestions consistently by making adjustments and adding clarifications at several instances throughout the revised manuscript and supplementary information (highlighted by yellow background), and truly hope they satisfactorily address all issues raised.

Note: While revising the manuscript we noticed a typo in our discovered expression (Eqn. 3 in the manuscript): the temperature in the last term should be raised to the power of 2, not to the power of 5/2. Accordingly, we have corrected Eqn. 3, Figures 2b and 3a. We emphasize the typo originated only from the way the expression is written symbolically (from the sympy.simplify Python module), and not from any error in the functional behaviour of discovered expression itself. We have completely verified that there is no change in the predictions and all the associated accuracy results from the discovered expression.

Referee: 1

The authors have attempted to elucidate the model for ionic conductivity in Li-based liquid electrolytes using symbolic regression and high throughput experimentation. Interestingly, the authors have selected 3 predictors for ionic conductivity, namely, the temperature, salt concentration and cyclic to linear carbonates ratio. It was also noted that all the experiments were performed under a moisture free environment. Subsequently, the authors derived a general model for ionic conductivity. This is a significant work in many ways. Firstly, a framework of ion conduction mechanism discovery is discussed in this work. Secondly, a demonstration of high throughput experimentation for characterising the electrolyte has been shown and can be followed for subsequent efforts in this field.
Specifically, the authors have highlighted an expression that generalises the conductivity as a function (c, T and r). Can the authors comment if this expression holds true for both high and low salt concentrations given that the solvation structures formed is different at different salt concentrations; there could be a coordination preference difference between high and low salt concentration for linear and cyclic carbonates which could affect the Li+ diffusion mechanism and potentially the conductivity. (see Phys. Chem. Chem. Phys., 2016, 18, 164-175)

Authors reply: The referee raises a very pertinent and scientifically interesting question. When building the model and testing its limits, we operate under the assumption that a single expression can describe the full range of concentrations, temperatures and solvent ratios available within the training data. However, Figure 3b of the manuscript shows our expression fails at predicting the ionic conductivity at -30 ˚C and high salt concentrations. While from the machine learning perspective the under-predictions could originate from many factors (e.g. lack of data in the anomalous regime), it is clear from the data itself that the conductivity behaves differently in the low temperature, high salt concentration region.

In principle, given that our model has learned the predominant square-root trends on the data, we could use its predictions to detect regions where the measured conductivities behave “anomalously”. Figure R1 illustrates the deviations between the model predictions and the measured values, relative to the measurement error:

Deviations=κ_predicted-κ_measured (formatted equation available in the pdf version of the response)

We have tuned the colour code in order to highlight zones where the model predicts conductivities that are i) higher than the measured conductivity, in yellow, ii) lower than the measured conductivity, in blue, and, iii) comparable to the measurement error, in grey. The magnitude of the measurement error is ca. 0.8 mS/cm (quantile 0.95 from the measurement deviations in Figure 3a in the main manuscript). Circular markers indicate measurements available in the dataset.

(Figure R1 available in the pdf version of the response, and as Figure 9 in the Supplementary Information)

In general, Figure R1 shows that the model predicts significantly lower conductivities compared to the measurements (down to 2.5 mS/cm) in concentrated and low temperature regions, as well as relative diluted and high temperature regions. The effect is specially pronounced in PC-pure solutions (Figure R1, left). We briefly mentioned in the manuscript that the “anomalous regime” at low temperature and high salt concentration might result from the electrolyte structure becoming highly-viscous, where conduction of Li+ would become hoping-like, instead of the traditional vehicular mechanism. As for why the effect seems more pronounced on PC-pure electrolytes, we agree with the referee’s observation that the dielectric properties of the medium –influenced by the predominant solvent– might play a role on the structure and content of Lithium’s coordination environment, and thus affect its diffusion through the electrolyte. However, given the complexity of our electrolyte formulations -a ternary solvent mixture of PC, EC and EMC- it becomes difficult to formulate a sound hypothesis about how dielectric, bonding and steric effects would result in multiple conduction regimes. These hypotheses could be formulated based on, for instance, in silico modelling of the electrolyte as in the article provided by the referee. Therefore, we have limited our scope in the manuscript to mention the discrepancy at low temperatures and high concentrations, without delving into the scientifically-relevant but highly-complex interactions between solvents, anions and Li+. Figure R1, left also shows that for highly-concentrated, EC-rich electrolytes, the model over-predicts the conductivity in the high-temperature region, though the effect is less pronounced (ca. 1 mS/cm) compared to the under-prediction at low temperature (ca 2.5 mS/cm) on PC-pure electrolytes (Figure R1, right).

In summary, we do believe the data provides evidence of at least two regimes of ionic conduction: a “standard” regime of predominant square-root-based trends correctly captured by our model, and an “anomalous” regime, where conductivity follow non-square-root trends and such becomes poorly described by our discovered expression.

We note that Referee 2 had raised a similar discussion point, and so we believe the separation of conduction regimes deserves additional discussion in the manuscript. Accordingly, we had now added a summary of the responses to referee 1 and 2 in an additional section of the manuscript: “Selected model: deviations at low temperature and high salt concentration. Figure 3b also shows that the model is not expressive enough to describe the conductivities measured at 30 °C and concentrations above 1 mol/kg. Further comparisons between the predicted and measured conductivities (Supplementary Figure 8) show that the model predicts significantly lower conductivities compared to the measurements (down to 2.5 mS/cm) in concentrated and low temperature regions. Within this regime, the conductivity seems to decay exponentially with concentration (Figure 3b) instead of following the square-root trends that our expression learned from the rest of the predictors space. Expectedly, as temperatures drop and salt concentration increase, the electrolyte structure changes significantly and its viscosity grows exponentially, which overall would influence the functional dependency of conductivity. Notably, the effect is especially pronounced in PC-pure solutions (Supplementary Figure 8, top), indicating that our discovered expression missed certain properties of the solvent mixture that influence the ionic transport of concentrated electrolytes at low temperatures. While we assume that a single expression can describe the complete dataset, the observations of several regimes of conduction raises the potential need for either one expression per regime, or for an overarching, more sophisticated symbolic expression that collapses to the right functional behaviour in each regime. Increased accuracy within the viscous regime could be critical for specific applications such as low-temperature electrolyte engineering.” We have also included Figure R1 as part of the supplementary information (Supplementary Figure 8).


Referee: 2

Flores et al. present a study on rediscovery of the physical laws governing the transport of lithium ions in liquid electrolytes using symbolic regression.
Overall the paper is very well written and presents a nice examples of applying symbolic regression to a seemingly trivial but actually rather complex problem: the conductivity of an electrolyte containing several solvents.
I find in this paper a thorough discussion of the underlying models and a good approach to avoid overfitting.

Referee 2: Maybe this is a discussion for another manuscript but what I gather from the discussion and the figures is that the model seems to perform a lot better at higher temperatures. From my own groups research I found polynomial models to predict much better (well in agreement with Figure 3) but in general suffering from the same problem of overfitting as discussed here unless heavily regularised. Could the authors maybe add a sentence or two discussing the possibility of two physico-chemical regimes? I could imagine there being something like a phase boundary towards low temperatures when the viscosity starts to play a role?

Authors reply: The topic of multiple conductivity regimes is indeed a very interesting one, raised by Referee 1 as well. For more details we refer to our response to Referee 1. In summary, we believe there are at least two conductivity regimes: a “standard” one where conductivity follows square-root trends on salt concentration, and an “anomalous” regime, where conductivity follow non-square-root trends. Our discovered expression describes well the former, but performs poorly on the latter.

The deviations are particularly pronounced on PC-pure, highly concentrated electrolytes at low temperature (see Figure R1, right). While the anomalous regime is still far from the glass transition temperature of the electrolyte (ca. -70 ˚C, see e.g. J. Chem. Eng. Data 2003, 48, 519-528), we do expect that as temperatures drop and salt concentration increase, the electrolyte structure changes significantly (Electrochim. Acta 2017, 233,134–141) and its viscosity grows exponentially (J. Electrochem. Soc. 2005, 162 (3) A413-A420). Whether the regimes are separated by a clear phase boundary or not is not evident from the data and, in our opinion, would require additional measurements sampling the “anomalous” regime. Clearly, the presence of a discontinuity would impose a fundamental, mathematical limitation to modelling the complete variable space with a single expression: the regimes might need to be modelled either by one expression per regime, or by a more sophisticated expression that collapses to the right functional behaviour in each regime. On the more practical side, specific applications such as low-temperature electrolyte engineering might require a model with higher prediction accuracy, in which case a separate expression for the “anomalous” regime might be needed even if the regimes are separated by a continuous boundary.

We note that Referee 1 had raised a similar discussion point, and so we believe the separation of conduction regimes deserves additional discussion in the manuscript. Accordingly, we had now added a summary of the responses to referee 1 and 2 in an additional section of the manuscript “Selected model: deviations at low temperature and high salt concentration.” In addition, de have also included Figure R1 as part of the supplementary information (Supplementary Figure 8).

Referee 2: An additional comparison for the discussion could be made if one did a simple comparison of a Pade or Taylor approximation or alike of the DHO model and the found expression would these approximations differ a lot and where?

Authors reply: We considered additional comparisons between the discovered and the DHO expressions, using for instance asymptotic approximation via Taylor expansions on the dilute concentration limit. However, it can be verified that the Taylor expansion of the discovered expression around a salt concentration c = 0 is not defined, given that one of its derivatives evaluated at c = 0 become infinite even after a first-order approximation:

〖 ├ ∂f/∂c┤|〗_(c=0)= 〖 ├ 〖(β〗_1+〖5/2β〗_3 〖r^(1/2) c〗^(3/2)+〖1/4T^2 β〗_3 c^(-3/4))┤|〗_(c=0) and 〖 0〗^(-3/4)=inf
(formatted equation available in the pdf version of the response)

Furthermore, when evaluated instead at c = 1x10-4 M (where DHO should be valid), the resulting expression not only differs from DHO as it appears linear in c, but also predict negative conductivities i.e. physically meaningless. We believe that a Symbolic Regression (SR) expression will rarely comply with the expected behaviour at limits not seen within the training dataset. Hence, we can only guarantee the out-of-sample performance of our discovered expression within the data distribution used for training (0.2 < c[m] < 2.1, -30 < T[˚C] < 60). However, we are actively exploring ways to implement more sophisticated constraints, in order to improve the SR algorithm’s ability to find physically meaningful expressions, i.e. complying with boundary conditions and collapsing to the functional form of limiting laws. In a way, these constraints would effectively regularize the SR model, not by penalizing model complexity as in e.g. Lasso regression, but rather by guiding the SR model using domain-knowledge.

Referee 2: Editorial note: There was a minor figure hick-up in the generated pdf. Figure 2 was for the most part grayed out but the preprint version I found did not have that issue. I was able to gather the missing info.

Authors reply: We thank the referee for highlighting the potential issue with the rendering of Figure 2. Accordingly, we will supply a .png version of the Figure to the editorial team upon request.

Referee 2: In general this manuscript is a very concise and well written example of using symbolic regression to (re)discover physico-chemical laws of ion transport in liquid electrolytes that I recommend to be accepted as is.


Referee: 3

The research utilizes SR to unravel a solution expression that tries to balance several aspects simultaneously, such as: accuracy without overfitting, parsimonious or compactness, consistency, a square-root functional structure resembling the DHO law, conformity with the existing understanding of ionic transport. The results are well reported and discussed, though I have some concerns as mentioned below.

Referee 3: 1) How do you choose the library of basis operators prior to running SR? Why do you select 'sqrt', 'exp', 'log' but not 'sin', 'cos'. The justification is not available in the paper. Is the choice solely based on the intuition or you have conducted trial runs with other set of basis operators ! Even the supplementary material does not contain this information.

Authors reply: From the transformations available in the AutoFeat Library, we dismissed several transformations based solely on domain knowledge and without conducting additional trial runs. We have now included further details about our choice in the “Feature generation” section of the Experimental Methods: “From all the transformations available in the AutoFeat Library, we did not consider abs(x) since we expect the conductivity to be differentiable; likewise we do not consider neither sin(x) nor cos(x) since we do not expect the conductivity to be periodic with respect to any of the predictors”. Furthermore, we have edited the last paragraph of the section “Selected model: interpretations”: “At this point, we emphasize we have only implemented two domain-knowledge decisions – i) exclude non-differentiable and periodic operators and ii) constrain the intercept to zero – on an otherwise purely statistical approach.”

Referee 3: 2) The moment non-linear features comes in, the model can't be said linear anymore. Rather than saying "a linear model of ...", perhaps you can write the same idea as follows. Please rephrase the sentence: In our SR approach, we apply non-linear transformations on the original predictors to produce more informative candidate features. The aim is to approximate the conductivity with a combination of non-linear candidate features, as given below.

Authors reply: The sentence the referee refers to it was initially intended to draw parallelisms between our methodology and the linear model most researchers are familiar with. In essence, our methodology can be viewed as a linear model regressed on transformed features. Moreover, our methodology for feature selection technically is still linear from the machine learning perspective, since it optimizes the weights which are linear. However, we agree these terms might be confusing and so we gladly accept the suggestion by the referee and we have now updated our sentence accordingly: “In our SR approach, we apply non-linear operations to the original predictors to produce more informative candidate features.”

Referee 3: 3) The paper mentions that the solution to the current SR approach is generally non-unique, however, there is no discussion with examples on this topic. Can you please show some of the other expressions that give good accuracy of fit ? Discuss the underlying pros and cons (accuracy, parsimonious, consistency) with different expressions and justify why do you select the current solution.

Authors reply: Figure 2a demonstrates that the solutions are not unique: using different subsamples of the training set result in multiple candidate expressions. We further emphasize in the Figure caption: “Each data point represents an expression, whose colour indicates its parent transformation set”. In any case, we have now added an additional table in the supplementary information (Supplementary Table 4) with the 10 most accurate expressions (in terms of validation MSE), and referenced it in the “Evaluation of models” section of the manuscript: “While there are expressions with higher prediction accuracy, these not only have more terms but also repeat only once throughout the training sessions and thus are not consistent (Supplementary Table 4)”. Furthermore, we believe we have justified why we selected the current solution in the section “Evaluation of models” of the main manuscript. First, we discuss that the Pareto-optimality frontier is mostly populated by square-root models (Figure 2a) thus they offer the better compromise between parsimony and accuracy. Second, we show that one of these Pareto-optimal models is by far the most consistent (Figure 2b). Then we clearly state (after Eqn. 3): “We, therefore, select Eqn. 3 as it clearly stands out from the other competing models, for being not only consistent (discovered 15 times) but also parsimonious (four terms), comparatively accurate in the training set (MSE < 0.75), and generalizable, as evidenced by a good accuracy in the validation set.”

Referee 3: 4) The example notebook in the Github reports a mse of 0.13. This error is a bit high ! Can you explain why ?

Authors reply: We note that the example in the Github page uses a synthetic dataset (not the same used for the article) since it is intended as a guide for potential users and contributors. Moreover, the (synthetically generated) target variable k in the example is not scaled, but has a range from 1.54 to 51.07 and a standard deviation of 13.6; in the context of these statistics, an MSE = 0.13 can be considered very low. We have added a line of code in the example with summary statistics of the synthetic target, in order to provide interested users with additional context on the model performance metrics.

Referee 3: 5) At several places, I have found that some terminologies are misleading. For example: 'transformation' should be replaced with 'operation'. Please see the attached file for further minor issues.

Authors reply: We sincerely appreciate the file attached by reviewer 3: we have now examined carefully and made updates and corrections in the manuscript in order to improve it. About the terms “transformation” and “operation”, we note that often these terms are used interchangeably within the machine learning literature. Moreover, we had initially chosen the term “transformation” inheriting from the article reporting the AutoFeat Library (https://doi.org/10.1007/978-3-030-43823-4_10), so to keep the terminology consistent across articles. However, we agree that “operations” might be more intuitive for the chemistry side of the article audience, since it is often the term used in arithmetic’s and algebra to refer to mappings from numerical domains to numerical ranges, as in the case of exp(x), sqrt(x), log(x), etc. We have therefore updated our terminology to use operations as suggested by the referee.

Referee 3: Additional comments from the pdf attached by referee 3:
This sentence “Constraining models might be beneficial when selecting promising surrogate models.” is unclear and needs to be rewritten. What is a surrogate model and how it is linked with a constrained model. Precisely say in simple words.

Authors reply: We agree with the referee this sentence needs to be clarified. In general, a surrogate model is an analytical, simplified representation of a complex input-output process; in our case, a simplification of the thermodynamic and kinetic processes relating the predictors to the electrolyte conductivity. However, given the multiple audiences potentially interested in this manuscript, it might be counter-productive to include the concept of surrogate model in the manuscript. Accordingly, we have rephrased the first sentence of the section “Model constraints” in the following way: “Enforcing model constraints is an effective way to improve model consistency.” Furthermore we have clarified our argument in the following sentences: “Our choice of enforcing y0 = 0 has effectively filtered out expressions that became inaccurate under the constraint. We obtained as a result not only a smaller pool of consistent candidate expressions, but also the guarantee they all comply with the imposed boundary condition y0 = 0” We have also edited out all instances where the word “surrogate” appears after verifying it does not modify the readability of the edited sentences.

06-Jun-2022

Dear Dr Flores:

Manuscript ID: DD-ART-04-2022-000027.R1
TITLE: Learning the laws of lithium-ion transport in electrolytes using symbolic regression

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.

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Royal Society of Chemistry

Reviewer 1

The authors have addressed my comments accordingly; their response is commensurate with the scope of the work.

Reviewer 2

All comments were adequately answered and I believe this manuscript can be published as is.

Reviewer 3

The authors have properly addressed the concerns. They have shed light on the aspects: (i) the mathematical & physical role of different components in a solution expression, (ii) the variability of solution expressions and the selection the best out of many, (iii) constraints on modelling, and (iv) rephrasing the language wherever it is needed.

Therefore, their paper deserves an acceptance from my view. Thank you.