Abstract
The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional superconductors, including the retardation of the interaction and the Coulomb pseudopotential, to predict the critical temperature T_{c}. McMillan, Allen, and Dynes derived approximate closedform expressions for the critical temperature within this theory, which depends on the electron–phonon spectral function α^{2}F(ω). Here we show that modern machinelearning techniques can substantially improve these formulae, accounting for more general shapes of the α^{2}F function. Using symbolic regression and the SISSO framework, together with a database of artificially generated α^{2}F functions and numerical solutions of the Eliashberg equations, we derive a formula for T_{c} that performs as well as Allen–Dynes for lowT_{c} superconductors and substantially better for higherT_{c} ones. This corrects the systematic underestimation of T_{c} while reproducing the physical constraints originally outlined by Allen and Dynes. This equation should replace the Allen–Dynes formula for the prediction of highertemperature superconductors.
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Introduction
Although the theory of electron–phonon superconductivity due to Bardeen–Cooper–Schrieffer, Gor’kov, Eliashberg, Migdal, and others is wellestablished, it has not historically aided in the discovery of new superconductors. The materials space to search for new superconductors is vast, and it is, therefore, desirable to find a practical way to use theory as a guide. Recent computational developments may allow a new approach to superconducting materials discovery based on abinitio and materialsgenome type methods^{1,2,3}.
One approach to this problem, pioneered by McMillan^{4} and Allen and Dynes^{5}, is to search for a formula for T_{c} based on materialsspecific parameters derived from the Eliashberg equations of superconductivity. These parameters, mostly moments of the electron–phonon spectral function α^{2}F(ω), can be determined by experiment or, more recently, calculated within ab initio approaches. In principle, this allows one also to deduce how to optimize T_{c} if one can optimize one or more of these parameters.
The Allen–Dynes equation has played a crucial role in debates on how to achieve hightemperature superconductivity by both theorists, who use it to predict T_{c}, and by experimentalists, who extract λ from measured T_{c} and ω_{D}. Nevertheless, it is important to recall that the Allen–Dynes equation has been derived from Eliashberg theory within an approximation where the momentum dependence of the Eliashberg function is neglected. It is based on 217 Eliashberg solutions of three types of α^{2}F(ω) shapes (those obtained from tunneling data on Hg and Pb, and those obtained for a single Einstein mode).
There have been several important advances in providing more detailed solutions to the Eliashberg equations since the work of Allen and Dynes. Combescot solved the Eliashberg equations on the weak coupling side and obtained an expression for T_{c} that depends on \(\omega_{\mathrm{log}}\) and a shapedependent integral^{6}. Recently, Marsiglio et al. solved the Eliashberg equations numerically at small λ^{7,8}, exhibiting some deviations of the theory from the BCS result in this limit, in particular the correction \(\frac{1}{\sqrt{e}}\) to the BCS T_{c}^{9,10,11}. And of course the full equations can be solved numerically for any coupling, including the momentum dependence of α^{2}F if desired^{12,13}. The validity of the theory as λ increases is a subtle question, and has been the subject of a number of recent studies^{14,15,16,17,18,19}.
In this paper, we solve the Eliashberg equations using different types of electron–phonon spectral functions, including multimodal Einsteinlike spectra and a set of α^{2}F obtained from firstprinciples calculations. We find that, while the Allen–Dynes formula accurately predicts the Eliashberg T_{c} for λ values near 1.6 (the coupling constant for Hg and Pb), it nevertheless deviates from the Eliashberg T_{c} when λ is significantly larger or smaller than 1.6 and when the shape of α^{2}F(ω) differs from the simple unimodal Einstein model. This deficiency highlights the need to improve on Allen–Dynes to investigate the highpressure, hightemperature hydrides of great current interest^{20}.
In a previous paper, we used an analytical machine learning approach to try to improve on the Allen–Dynes formula, testing and training on tiny databases from the Allen–Dynes table of 29 superconducting materials^{21}. This proof of principle work showed that the SISSO framework, properly constrained by physical law, could substantially improve the performance of the Allen–Dynes equation with a smaller number of parameters. Clearly, it is necessary to apply this approach to a more extensive and diverse database.
Here, we proceed more systematically and show how we can “teach the machine Eliashberg theory” by generating large databases of α^{2}F functions from both real materials and single and multimodal artificial ones and learning the results of T_{c} from solutions to the Eliashberg equations. We additionally include in our study α^{2}F functions for superhydrides, extending training and testing to the higher λ range. We show that the Allen–Dynes equation fails in this region particularly badly, since it was designed to fit materials with the ratio of the Allen–Dynes parameters \({\bar{\omega }}_{2}/{\omega }_{{{\mathrm{log}}}\,}\simeq 1\), which is strongly violated in some of the higherT_{c} materials. Here λ is the integral \(2\int\nolimits_{0}^{\infty }{\alpha }^{2}F(\omega )/\omega \ {\rm {d}}\omega\), the frequencies \({\bar{\omega }}_{n}\) are the nth root of the nth moment of the normalized distribution g(ω) = 2/(λω) α^{2}F(ω), and \({\omega }_{{{\mathrm{log}}}\,}\equiv \exp \left [ \int_0^\infty \ln \omega \, g(\omega) \, d\omega \right ]\).
We begin by introducing the McMillan and Allen–Dynes equations, against which we will compare our results. McMillan^{4}, in an attempt to improve on the BCS weakcoupling T_{c}, incorporated elements of Eliashberg theory^{22} into a phenomenological expression, relating T_{c} to physical parameters that could in principle be extracted from tunneling data^{23},
where μ^{*} is the Coulomb pseudopotential and ω_{D} is the Debye frequency. Note that the McMillan formula predicts a saturation of T_{c} in the strongcoupling limit, λ → ∞, for fixed ω_{D}.
Allen and Dynes^{5} showed that the true Eliashberg T_{c} did not obey such a bound in this limit but rather grew as \(\sqrt{\lambda }\). They proposed an alternate approximate fit to Eliashberg theory based on data on a few lowT_{c} superconductors known in 1975
where f_{1} and f_{2} are factors depending on \(\lambda ,{\mu }^{* },{\omega }_{{{\mathrm{log}}}\,}\), and \({\bar{\omega }}_{2}\).
Results and discussion
Figure 1 outlines our methods and computational workflow. We begin by collecting α^{2}F(ω) spectral functions from ab initio calculations and augmenting the dataset with artificial spectral functions based on generated Gaussian functions. The Coulomb pseudopotential μ^{*} is sampled as a free parameter and used, alongside the spectral functions, as an input to the Eliashberg equations. Eliashberg theory yields the superconducting gap function Δ, from which we extract \(T_{\mathrm{c}}^{\mathrm{E}}\). At the same time, we extract the quantities λ, \({\omega }_{{{\mathrm{log}}}\,}\), and \(\bar{\omega}_2\) from α^{2}F. Next, we use machine learning techniques to learn the relationship between the four model inputs, or features, and the critical temperature from Eliashberg theory \({T}_{\mathrm{c}}^{\mathrm{E}}\). Finally, we compare the predictive models for T_{c} and discuss the featureT_{c} relationships.
Computational details
We compile a set of 2874 electron–phonon spectral functions α^{2}F(ω), summarized in Table 1. Of these, 13 are conventional phonon mediated superconductors, where we calculate α^{2}F using the electron–phonon Wannier package (EPW)^{12,24} of the Quantum Espresso (QE) code^{25,26}. An additional 42 (29 classic and 13 hydride superconductors) are obtained from the computational superconductivity literature. We augment the dataset by generating 2819 artificial multimodal α^{2}F(ω) functions and calculating the corresponding T_{c}’s with the EPW code. The superconducting transition temperatures are estimated by using both the Allen–Dynes equation and by solving the isotropic Eliashberg equations. The raw data are available upon request.
The artificially generated α^{2}F(ω) consist of three Gaussian peaks with randomly selected peak location and height
where G(ω) is a normalized Gaussian with width of 1/8 of the peak frequency ω_{i}.
The total λ is then equal to the sum of the λ_{i}, which simplifies sampling of the space of spectral functions. The artificial trimodal α^{2}F spectral functions resemble those of many realistic materials, see Fig. 2 for the example of LaAl_{2}. The Allen–Dynes and Eliashberg T_{c} for the hydrides are obtained from published work (see refs. in Table 1).
To ensure efficient sampling of the input spaces, we select values of λ and μ^{*} with pseudorandom Sobol sequences. As shown in Fig. 3, our uniform sampling scheme results in a set of artificially generated α^{2}F corresponding to an approximately uniform distribution of T_{c}. Next, we removed artificial entries with T_{c} > 400 K to better reflect the distribution of realistic materials. While the histogram of μ^{*} remains approximately uniform after this truncation, the histograms of λ, \({\omega }_{{{\mathrm{log}}}\,}\), and \({\bar{\omega }}_{2}\) become skewed towards lower values.
Data
In the Allen–Dynes formula, the “arbitrarily chosen” shapedependent factor f_{2} is based on the numerical solutions using the spectral functions of Hg, Pb, and the Einstein model^{5}. Because the number of α^{2}F(ω) shapes is small, it is expected that the Allen–Dynes T_{c} (\({T}_{\,{{\mbox{c}}}}^{{{\mbox{AD}}}\,}\)) would have significant errors in some instances. Figure 4 illustrates such deviations for bimodal Gaussian spectral functions. So far, we discussed the α^{2}F(ω) shapes in an abstract sense because there is no single parameter that uniquely determines their shape. Allen and Dynes proposed using the ratio \({\omega }_{\mathrm{log}} /{\bar{\omega }}_{2}\) as an indicator of the shape of α^{2}F(ω). In Fig. 4, the ratio \({{T}_{{{\mathrm{c}}}}^{{{\mathrm{AD}}}}}/{{T}_{{{\mathrm{c}}}}^{{{\mathrm{E}}}}}\) is plotted against \({\omega}_{{{\mathrm{log}}}} / {\bar{\omega}}_{2}\) for λ = 0.6, 1, 2, 3 and 4. The results demonstrate that there can be significant differences between the Allen–Dynes \({T}_{{{\mathrm{c}}}}^{{{\mathrm{AD}}}}\) and Eliashberg \({T}_{{{\mathrm{c}}}}^{{{\mathrm{E}}}}\) even for some simple cases. The root mean square error in the Allen–Dynes paper is 5.6%. When the ratio \({\omega}_{{{\mathrm{log}}}} / {\bar{\omega}}_{2}\) is 1, the shape of α^{2}F is that of the unimodal Einstein model and the Allen–Dynes T_{c} accurately predict the Eliashberg T_{c} regardless of the coupling strength. When the ratio \({\omega}_{{{\mathrm{log}}}} / {\bar{\omega}}_{2}\) decreases, i.e. the shape of α^{2}F has more structure; whether the Allen–Dynes formula can then still reasonably predict the Eliashberg T_{c} depends on the electron–phonon coupling strength.
In this work, we train and test machinelearning models using the datasets listed in Table 1. Two sizes are reported for each nonGaussian dataset, indicating the number of unique materials compared to the total number of datapoints. We sample μ^{*} between [0.1, 0.16] which covers a wide range of possible μ^{*} values^{5,27}. The calculated, artificial Gaussian, and literaturederived α^{2}F datasets are used for training all machine learning models. We left the hydride materials out of the training in order to validate the extrapolative capacity of each model.
Correction factors for T _{c} from symbolic regression
As in our previous symbolic regression effort^{21}, we use the SISSO framework to generate millions of candidate expressions by recursively combining the input variables with mathematical operators such as addition and exponentiation. We performed symbolic regression twice, sequentially, to obtain two dimensionless prefactors of the McMillan exponential, yielding a machine learned critical temperature
We name the two learned prefactors a posteriori based on their functional forms and the mechanisms by which they reduce the error in predicting T_{c}. The first factor
is obtained from the fit to the ratio \({T}_{{{\mathrm{c}}}}^{{{\mathrm{E}}}}/{T}_{{{\mathrm{c}}}}^{{{\mathrm{McMillan}}}}\) and eliminates the systematic underprediction of T_{c} at higher temperatures. Like the Allen–Dynes prefactor f_{2}, f_{ω} includes the ratio \({\omega }_{{{\mathrm{log}}}}/{\bar{\omega }}_{2}\), modifying the prediction based on the shape of α^{2}F(ω). Moreover, f_{ω} also scales with \(\sqrt{\lambda }\), like the Allen–Dynes prefactor f_{1}. This is in agreement with the correct largeλ behavior of Eliashberg theory, unlike our earlier work^{21} and the modified T_{c} equation with linear correction proposed recently by Shipley et al. ^{28}. The manifestation of both behaviors in f_{ω} gives credence to our symbolic regression approach because it incorporates the primary effects of the Allen–Dynes equation with fewer parameters. Applying the correction \({T}_{{{\mathrm{c}}}}={f}_{\omega }{T}_{{{\mathrm{c}}}}^{{{\mathrm{McMillan}}}\,}\) achieves a percent RMSE of 15.2% across the materials (nonGaussian model) data, compared to 48.6% when using the Allen–Dynes equation.
The second correction factor
is obtained from the fit to the ratio \({T}_{\,{{\mathrm{c}}}}^{{{\mathrm{E}}}}/({f}_{\omega }{T}_{{{\mathrm{c}}}}^{{{\mathrm{McMillan}}}})\), effectively correcting the residual error from the fit of f_{ω} and thus cannot be used independently. Applying the correction \({T}_{{{\mathrm{c}}}}={f}_{\omega }{f}_{\mu }{T}_{{{\mathrm{c}}}}^{{{\mathrm{McMillan}}}}\) achieves a percent RMSE of 15.1% across the materials datasets, compared to 15.2% when using f_{ω} alone. The influence of f_{μ} is more apparent when examining clusters of points corresponding to resampled μ^{*} values for a single material, where the systematic error in \({T}_{{{\mathrm{c}}}}^{{{\mathrm{ML}}}}/{T}_{{{\mathrm{c}}}}^{{{\mathrm{E}}}}\) is reduced.
Note that f_{μ} → 1 in both of the limits λ → 0 and λ → ∞, and in fact does not vary by more than ~10% from 1 over the data set.
Figure 5 shows that, apart from the lowT_{c} nonhydride materials for which the difference is smaller than 0.1 K, the corrections f_{ω} and f_{μ} dramatically improve predictions compared to using the Allen–Dynes equation. Since we excluded the hydrides from the training, these results successfully validate our datadriven symbolic regression approach by demonstrating the extrapolative capacity of the learned equations.
To further quantify the similarity between the existing Allen–Dynes prefactors and the machinelearned prefactors, we employ two statistical measures, the Spearman and distance correlation. The Spearman correlation is a measure of monotonicity in the relationship between rankings of two variables. Like the Pearson correlation coefficient for linear correlation, the Spearman correlation varies between −1 and +1, where extrema imply high correlation and zero implies no correlation. Unlike the Pearson correlation, the Spearman correlation does not assume normally distributed datasets. By construction, all four prefactors tend to unity for many materials, resulting in asymmetric distributions that are unsuitable for analysis with parametric measures like the Pearson correlation.
In addition to the Spearman correlation, we compute the distance correlation, another nonparametric measure of the dependence between two variables. The distance correlation is defined as the ratio of the distance covariance and the product of the distance standard deviations, where distance covariance is the weighted Euclidean distance between the joint characteristic function of the two variables and the product of their marginal characteristic functions. Unlike the Pearson and Spearman correlation coefficients, the distance correlation varies between 0 and 1, where 0 indicates that the variables are independent, measuring both linear and nonlinear association.
Table 2 shows a strong relationship between f_{1}, f_{2}, and f_{ω} according to both Spearman and distance correlation metrics, with values close to one. This numerical analysis reinforces the conclusion that f_{ω} reproduces characteristics of both f_{1} and f_{2}, as illustrated earlier in the comparison of functional forms. On the other hand, both Spearman correlation and distance correlation measures indicate slightly weaker relationships between f_{μ} and the other three prefactors. The relative independence of f_{μ} compared to f_{ω}, f_{1}, and f_{2} stems from the sequential nature of the fitting process.
Comparing predictive models for T _{c}
To compare existing equations for T_{c} with the corrections identified in this work, we benchmarked the RMSE across nonhydride materials, artificial Gaussians, and hydrides as tabulated in Table 3. Additionally, we compute the %RMSE by normalizing each RMSE by the mean value across the corresponding dataset. To assess the behavior of each model with increasing λ, we plot \({T}_{{{\mathrm{c}}}}/{\omega }_{{{\mathrm{log}}}\,}\) for each model in Fig. 6.
As expected, the Allen–Dynes equation improves on the McMillan equation across all three groups. On the other hand, the equation identified by Xie et al. ^{21} in an earlier symbolic regression work performs slightly worse on the lowT_{c} nonhydride dataset but achieves lower RMSE across the artificial Gaussian and hydride materials despite being trained on a small set of 29 lowT_{c} materials.
Applying the new f_{ω} prefactor to the McMillan equation reduces %RMSE in nonhydride materials from 14.4% to 8.4%, in artificial Gaussian models from 45.1% to 9.2%, and in hydrides from 36.6% to 5.8%. Moreover, applying both f_{ω} and f_{μ} results in a further, modest improvement to the RMSE. In Fig. 6, our machinelearned correction (blue) is nearly equal to the Allen–Dynes equation (gray) for values of λ up to 1 but rapidly increases at larger λ. Both bounds, for higher and lower values of \({\omega }_{{{\mathrm{log}}}\,}/{\bar{\omega }}_{2}\), exceed the bounded region of the Allen–Dynes equation, indicating that at least part of the new model’s success is due to an improvement in capturing the behavior of T_{c} with increasing λ.
We additionally fit a random forest (RF) model and an artificial neural network (ANN) model using the same training data to compare against our symbolic regression method. Hyperparameters for RF and ANN models were selected using 10fold leaveclusterout crossvalidation and the same clusters identified for symbolic regression. On the other hand, the model error was estimated using nested crossvalidation, where the inner loop was performed using a conventional 5fold crossvalidation scheme. Production models used in Fig. 6 were fit with the selected hyperparameters using the entire training set.
The RF is an ensemble model comprised of decision trees, each fit to random subsets of the data and queried to yield an independent prediction. Each decision tree uses a flowchartlike series of decisions (branches) to yield predictions (leaves) and is optimized by varying decision thresholds. While individual decision trees are prone to overfitting, a RF produces robust predictions by averaging the predictions of its members. The optimized RF model, consisting of 100 decision trees with a maximum depth of eight splits per tree, achieved the lowest RMSE across all three models, with 4.7% RMSE in the testing set of hydride materials. This success may be attributed to both the flexibility of the method and the relative complexity compared to other methods. With up to 128 nodes per tree, the RF evaluates tens of thousands of binary decisions per prediction. On the other hand, as illustrated in Fig. 6, the resulting output (green) is discontinuous. Furthermore, the RF does not have the ability to extrapolate outside of regions of the input spaces included in the training data, resulting in constantvalue outputs. This deficiency is evident in both upper and lowerbound curves above λ = 3.8, where the RF correction results in a simple rescaling of the McMillan curve.
The ANN models in this work are feedforward neural networks, also known as multilayer perceptrons, designed to learn highly nonlinear function approximators to map multiple inputs to a target output. The feedforward architecture involves an input layer consisting of one neuron per input, one or more hidden layers, and an output layer consisting of one neuron per target. The value at each noninput neuron is a weighted, linear summation of the values in the preceding layer followed by a nonlinear activation function. The optimized ANN includes three hidden layers with forty neurons each, totaling 3521 trainable parameters of multiplicative weights and additive biases. Despite the increased model complexity, the ANN performs similarly to the symbolic regression model, with slightly lower training RMSE and slightly higher testing RMSE. With increasing λ, the ANN model yields similar values of T_{c} as indicated by the overlap between the shaded regions of the symbolic regression model (blue) and the ANN (yellow).
For low to moderate values of λ, such as those originally studied by Allen and Dynes, all models behave similarly and the dimensionless corrections (f_{1}, f_{2}, f_{ω}, f_{μ}, ANN, RF) are close to unity. However, as λ increases, the ANN, RF, and symbolic regression corrections deviate significantly from the Allen–Dynes equation as well as the previous symbolic regression equation^{21}. The corrections introduced in this work successfully correct the systematic underprediction of T_{c}, with the symbolic regression solution offering simplicity and accuracy. Moreover, the monotonicity constraint in the symbolic regression search guarantees invertibility, allowing experimentalists to extract λ from measured T_{c} and the electron–phonon spectral function. This characteristic is not guaranteed for the RF and ANN models.
Summary
The present work demonstrates the application of symbolic regression to a curated dataset of α^{2}F(ω) spectral functions, yielding an improved analytical correction to the McMillan equation for the critical temperature of a superconductor. We showed that the wellknown Allen–Dynes equation, an early improvement based on fitting to a very limited set of spectral functions, exhibits systematic error when predicting the Eliashberg critical temperature of highT_{c} hydrides, a flaw due to the original training set being based on lowT_{c} superconductors. The equation we obtain here by symbolic regression has the same form as the original Allen–Dynes equation, with exactly the same McMillan exponential factor, but has two prefactors that behave very differently than those employed by Allen–Dynes. They ensure that superconductors with spectral functions, α^{2}F(ω), of unusual shapes, such that \({\bar{\omega }}_{2}/{\omega }_{{{\mathrm{log}}}\,}\) is significantly different from 1, are adequately described; this subset of conventional superconductors includes the new hydride highpressure superconductors. In addition, the machinelearned equation can be simplified by dropping one of the prefactors with negligible loss of accuracy. Since the machinelearned expression of Eqs. (6)–(8) extends the accuracy of the Allen and Dynes expression to hightemperature superconductors while maintaining the utility and simplicity of the original formula, we suggest that this equation should replace the Allen–Dynes formula for predictions of critical temperatures and estimations of λ from experimental data, particularly for highertemperature superconductors.
Using a dataset of ab initio calculations alongside artificially generated spectral functions, we mitigated the smalldata problem associated with previous symbolicregression efforts. The dimensionless correction factor identified by symbolic regression reproduces the expected physical behavior with increasing λ and achieves lower prediction errors than the Allen–Dynes corrections, despite having similar model complexity. Finally, we compared our equation to models generated with two other machinelearning techniques, which achieve modest improvements in error at the cost of far greater complexity and lack of invertibility. While the present work successfully learns the isotropic Eliashberg T_{c}, future extensions may incorporate additional data from fullyanisotropic Eliashberg calculations and experimental measurements. On the other hand, separate extensions may involve approximating α^{2}Frelated quantities from lessexpensive calculation of density functional theorybased descriptors like the electronic density of states.
Methods
Calculating electron–phonon spectral functions with density functional theory
We calculate the electron–phonon spectral functions α^{2}F(ω) for 13 compounds with densityfunctional theory using the QE code^{25,26} and the EPW^{12,24}. We use the optimized normconserving pseudopotential^{29,30} and the PBE version of the generalized gradient exchangecorrelation functional^{31}. We sample the Brillouin zone for the electron orbitals using a 24 × 24 × 24kpoint mesh and for the phonons using a 6 × 6 × 6 qpoint mesh. To obtain the critical temperatures T_{c} of the compounds, we solve the isotropic Eliashberg equations with the EPW code.
Symbolic regression
We performed symbolic regression using the sure independence screening (SIS) and sparsifying operator (SISSO) framework^{32,33}, generating millions of candidate expressions. Based on memory constraints, the subspace of expressions was limited to those generated within four iterations. This limitation precludes the appearance of expressions of the complexity of the Allen–Dynes equation, motivating our search for a dimensionless correction to the McMillan equation rather than directly learning models for T_{c}.
The initial quantities for generating expressions were the three dimensionless quantities λ, μ^{*}, and the ratio \({\omega }_{{{\mathrm{log}}}\,}/{\bar{\omega }}_{2}\). Candidates were generated using the set of operators \(\{+,,\times ,\exp ,{{\mathrm{log}}}\,,\sqrt{},\root 3 \of {}{,}^{1}{,}^{2}{,}^{3}\}\). During the SIS step, these expressions were ranked based on their correlation to the ratio \({T}_{{{\mathrm{c}}}}^{{{\mathrm{E}}}}/{T}_{{{\mathrm{c}}}}^{{{\mathrm{McMillan}}}}\) rather than \({T}_{{{\mathrm{c}}}}^{{{\mathrm{E}}}}\) to identify dimensionless, multiplicative corrections to \({T}_{{{\mathrm{c}}}}^{{{\mathrm{McMillan}}}}\).
To facilitate generalizability, we employ leaveclusterout crossvalidation during the generation of expressions using kmeansclustering with k = 10 on the combined set of 179 nonhydride and 2819 artificialGaussian entries. For each round of crossvalidation, we generate candidate equations using a different subset of nine clusters and used the remaining cluster to evaluate performance using the rootmeansquare error metric. As such, each training sample was left out of training and used for testing during one round. The top 10,000 models, ranked by rootmeansquare error (RMSE) across the training set, were returned from each round. Models that did not appear in all ten rounds, corresponding to those with poor performance in one or more clusters, were eliminated. Following the same principle, we ranked the remaining equations by the average RMSE across all ten rounds.
We note that the sparsifying operator (SO) step of the SISSO framework offers increased model complexity, as we explored in our previous work, but is limited in functional form to linear combinations of expressions generated from the preceding step. The linear combination of expressions from the initial subspace, by extension, also excludes equations as complex as the Allen–Dynes correction. Therefore, we did not consider linear combinations of expressions, meaning the SO simply selected the firstranked expression from the SIS step in each run.
Data availability
The database of critical temperatures, descriptors derived from the computed spectral functions α^{2}F(ω), and symbolic regression logs are freely available at https://MaterialsWeb.org and https://www.materialscloud.org^{34}.
Code availability
The symbolic regression workflow software we developed is freely available on Github (https://github.com/henniggroup/).
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Acknowledgements
We are grateful to L. Boeri, P. Allen, and W. Pickett for valuable discussions. We thank D. Semenok for providing data related to Actinium hydrides. The work presented here was performed under the auspice of Basic Energy Sciences, United States Department of Energy, contract number DESC0020385. Partial funding was also provided by the University of Florida Informatics Institute.
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All authors contributed extensively to the work presented in this paper. S.R.X., Y.Q., A.C.H., L.F., J.L., J.K., G.R.S., J.J.H., P.J.H., and R.G.H. conceived the overall methodology of data assembly, augmentation, Eliashberg calculations, and symbolic regression. Y.Q., B.D., J.M.D., I.S., and U.S.H. performed the literature search and collected the spectral function data for superconductors. S.R.X. and Y.Q. implemented the algorithm and performed the calculations and analysis. S.R.X., Y.Q., P.J.H., and R.G.H. contributed to the writing of the manuscript.
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Xie, S.R., Quan, Y., Hire, A.C. et al. Machine learning of superconducting critical temperature from Eliashberg theory. npj Comput Mater 8, 14 (2022). https://doi.org/10.1038/s41524021006667
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DOI: https://doi.org/10.1038/s41524021006667
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