Introduction

Artificial-intelligence (AI) techniques have the potential to significantly accelerate the search for novel, functional materials, especially for applications where different physical mechanisms compete with each other non-linearly, e.g., quantum materials1, and where the cost of characterizing the materials makes a large-scale search intractable, e.g., thermoelectrics2. Due to this inherent complexity, only limited amounts of data are currently available for such applications, which in turn severely limits the applicability and reliability of AI techniques3. Using thermal transport as an example, we propose a route to overcome this hurdle by presenting an AI framework that is applicable to scarce datasets and that provides heuristics able to steer further data creation into regions of interest in materials space.

Heat transport, as measured by the temperature-dependent thermal conductivity, κL, is a ubiquitous property of materials and plays a vital role for numerous scientific and industrial applications including energy conversion4, catalysis5, thermal management6, and combustion7. Finding new crystalline materials with either an exceptionally low or high thermal conductivity is a prerequisite for improving these and other technologies or making them commercially viable at all. Accordingly, finding new thermal insulators and understanding where in materials space to search for such compounds is an important open challenge in this field. From a theory perspective, thermal transport depends on a complex interplay of different mechanisms, especially in thermal insulators, for which strongly anharmonic, higher-order effects can be at play8. Despite significant progress in the computational assessment of κL in solids9,10, these ab initio approaches are too costly for a large-scale exploration of material space. For this reason, computational high-throughput approaches have so far covered only a small subset of materials11,12,13. Experimentally, an even smaller number of materials have had their thermal conductivities measured, and <150 thermal insulators identified14,15.

Recently, increased research efforts have been devoted to leveraging AI frameworks to extend our knowledge in this field. In particular, various regression techniques have been proven to successfully interpolate between the existing data and approximate κL using only simpler properties11,14,16,17; however, using these techniques to extrapolate into new areas of materials space is a known challenge. More importantly, the explainbility of these models is limited by their inherent complexity. Physically motivated, semi-empirical models, e.g., the Slack model18, perform slightly better in this regard because they encapsulate information about the actuating mechanism. Recent efforts have used AI to extend the capabilities of these models2,16,19,20 to increase their accuracy in estimating κL. However, the applicability of such models is still limited by the physical assumptions entering the original expressions2,19. A general model that removes these assumptions and achieves the quantitative accuracy of AI approaches, while retaining the qualitative interpretability of analytical models, is however, still lacking.

In this work, we tackle this challenge by using a symbolic regression technique to quantitatively learn κL, using easily calculated materials properties. While symbolic regression methods are typically more expensive to train than other kernel based methods, such as Kernel-Ridge Regression (KRR) and Gaussian Process Regression (GPR), their prediction errors are typically equivalent to other methods and their natural feature reduction and resulting analytical expressions make them a useful method for explainable AI, as further illustrated below21. Furthermore, the added cost of training does not affect the evaluation time of the given models, meaning the extra time only has to be spent at the beginning. The inherent uncertainty estimate in methods like GPR, allows for a prediction of where the resulting models are expected to perform worse; however, we also propose a method to get an ensemble uncertainty estimate for symbolic regression that can be applied more generally to these types of models. We further exploit the feature reduction of SISSO and expand upon its interpretability by using a global sensitivity analysis method to distill out the key material properties that are most important for modelling κL and to find the conditions necessary for obtaining an ultra-low thermal conductivity. From here, we use this analysis to learn the conditions needed to screen materials in each step of a hierarchical, high-throughput workflow to discover new thermal insulators. Using this workflow we can then establish qualitative design principles that lend themselves to general application across material space and use them to find 80 materials with an ultra-low κL.

Results

Symbolic regression models for thermal conductivity

We use the sure-independence screening and sparsifying operator (SISSO) method as implemented in the SISSO++ code22. This method has been used to successfully describe multiple applications including the stability of materials23, catalysis24, and glass transition temperatures25. To find the best low-dimensional models for a specific target property, in our case the room temperature, lattice thermal conductivity, \({\kappa }_{{{{\rm{L}}}}}\left({{300\,{\rm{K}}}}\right)\), SISSO first builds an exhaustive set of analytical, non-linear functions, i.e., trillions of candidate descriptors, from a set of mathematical operators and primary features, the set of user-provided properties that will be used to model the target property. Here we are focusing on room temperature data only because that is what is the most abundant in the literature and relevant for potential applications; however, some temperature dependence will be inherently included via the temperature dependence of our anharmonicity factor σA. For this application the primary features are both the structural and dynamical properties for seventy-five materials with experimentally measured \({\kappa }_{{{{\rm{L}}}}}\left({{300\,{\rm{K}}}}\right)\)17,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43 (see Section IV D and Supplementary Note 1 for more details). By using the experimentally measured values for κL we avoid the issues related to the inconsistent reliability of different approaches to calculating κL for different material classes44,45, and hopefully create a universal model for it. For many of the materials of interest here the standard Boltzmann Transport approach will be unreliable44,45, but the fully anharmonic ab initio Green-Kubo approach is unnecessarily expensive to use for all materials45. Combining theoretical and experimental data in this way allows one to avoid both the cost or unreliability of calculating, κL and the challenges of experimentally synthesizing and characterizing candidate materials. As long as all samples are consistent across each feature, AI and ML based models will adapt the computational features to the experimental target.

Figure 1b illustrates the main goal of the work: to learn which primary features are important for modeling κL and what thresholds of those indicate where thermal insulators are present. As a result the figure also represents the workflow used to calculate κL and generate the primary features for the model. All of the data generated in this workflow will be calculated using ab initio methods, with each step representing an increasing cost of calculation, as shown in Fig. 1a. The total cost of calculating these primary features is several orders of magnitude smaller than explicitly calculating κL, either with the Boltzmann Transport Equation or aiGK. While using only compositional and structural features would further reduce the cost of generating them, it comes at the expense of decreasing the reliability and explainability of the models. A goal of this work is to learn the screening conditions needed to remove materials at each step of the workflow in Fig. 1b and only perform the intensive κL calculations on the most promising materials. Because of this, we feel that using the features generated from this workflow is the most logical set to use. Importantly, as described in Section IV D we use a consistent and accurate formalism for calculating all features in this workflow, and therefore expect a quantitative agreement between these features and their experimental counterparts. Even if this framework were restricted to explore only high-symmetry materials, the overall cost of the calculations in a supercell would be reduced by a factor of one hundred as shown by the non-green bars in Fig. 1a. In the more general case we would be able to screen closer to 1000 more materials using this procedure over the brute-force workflows of calculating κL for all materials. With the learned conditions one could then create a prescreening procedure by learning models for each of the relevant structural or harmonic properties using only compositional inputs, and use those to estimate κL46; however, that is outside of the scope of this work.

Fig. 1: The motivation for the work is reducing the number of calculations needed to approximate the thermal conductivity of a material.
figure 1

a The number of force evaluations needed to complete each step of a κL calculation for four representative materials: (1) Geometry relaxation (green first bar), (2) Harmonic model generation with Phonopy (yellow, second bar), (3) Evaluating κL via Phono3py (lavender third bar) or MD (purple fourth bar), and (4) estimating κL with the σA (light blue). The relaxation step typically acts on the primitive cells (~10 atoms) while all other are done on supercells with ~200 or more atoms. The number of force evaluations for Phono3py assumes all displacements are needed to calculate the third order force constants for version 2.5.1 (b) The proposed hierarchical workflow that can screen out materials before the final calculations.

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In practice, we model the \(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) instead of \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\) itself to better handle the wide range of possible thermal conductivities. The parity plot in Fig. 2(a) illustrates the performance of the identified SISSO model when the entire dataset is used (see Section IV A for more details). The resulting expression is characterized by d1 and d2

$$\begin{array}{ll}\log \left({\kappa }^{{{{\rm{SISSO}}}}}\left({{{300\,\rm{K}}}}\right)\right)\,=\,{a}_{0}+{a}_{1}{d}_{1}+{a}_{2}{d}_{2}\\ \qquad \qquad \qquad\quad \,\,{d}_{1}\,=\,\frac{{\left({m}_{{{{\rm{avg}}}}}+200.3{{{\rm{Da}}}}\right)}^{2}}{\sqrt{\mu }{\left({V}_{{{{\rm{m}}}}}+218.9{\mathring{\rm A} }^{3}\right)}^{3}{\Theta }_{{{{\rm{D,\infty }}}}}{\sigma }^{{{{\rm{A}}}}}}\\ \qquad \qquad \qquad\quad \,\,{d}_{2}\,=\,{\sigma }^{{{{\rm{A}}}}}\frac{{V}_{{{{\rm{m}}}}}\rho }{{m}_{{{{\rm{avg}}}}}}+{{{{\rm{e}}}}}^{\frac{-{\omega }_{\Gamma ,\max }}{27.11{{{\rm{THz}}}}}}+{{{{\rm{e}}}}}^{{\sigma }^{{{{\rm{A}}}}}}\end{array}$$
(1)

where a0 = 6.327, a1 = − 8.219 × 104, and a2 = − 1.704 are constants found by least-square regression and all variables are defined in Table 1. We find that this model has a training root-mean squared error (RMSE) of 0.14, with an R2 of 0.98 for \(\log \left({\kappa }^{{{{\rm{SISSO}}}}}\left({{300\,{\rm{K}}}}\right)\right)\). To better understand how these error terms translate to \({\kappa }_{{{{\rm{L}}}}}\left({{300\,{\rm{K}}}}\right)\), we also use the average factor difference (AFD)

$${{{\rm{AFD}}}}=1{0}^{x}$$
(2a)
$$x=\frac{1}{n}\mathop{\sum }\limits_{i}^{n}\left\vert \log \left({\kappa }_{{{{\rm{L}}}}}\right)-\log \left({\kappa }_{{{{\rm{L}}}}}^{{{{\rm{pred}}}}}\right)\right\vert ,$$
(2b)

where n is the number of training samples. Here, we find an AFD of 1.30 that is on par if not smaller than models previously found by other methods (e.g., 1.36 ± 0.03 for a Gaussian Process Regression model17 and 1.48 for a semi-empirical Debye-Callaway Model2). However, differences in the training sets and cross-validation scheme prevent a fair comparison of these studies for the prediction error. To see a complete representation of the training error for all models refer to Supplementary Note 2.

Fig. 2: Error evaluation for the presented models.
figure 2

a Comparison of the predicted \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\) against the measured \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\) for the model trained against all data. The gray shaded region corresponds to the 95% confidence interval. b Violin plots of the mean prediction error of all samples for the SISSO, KRR, and GPR models using all features (red, left) and a reduced set including only σA, ΘD,∞, and Vm (blue, right) and the Slack model. Gray lines are the median, white circles are the mean of the distributions, the boxes represent the quartiles, and the whiskers are the minimum and 95% absolute error. For all calculations the parameterization depth and dimension are determined by cross-validation on each training set. The red stars and blue hexagons are the outliers for the box plots. c A map of the two-dimensional SISSO model, where the features on the x − and y − axes correspond to the two features selected by SISSO. The labeled points represent the convex-hull of the scatter plot and related points.

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Table 1 List of the primary features used in this calculation.
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To get a better estimate of the prediction error, we use a nested cross-validation scheme further defined in Section IV E. As expected, the prediction error is slightly higher than the training error with an RMSE of 0.22 ± 0.02 and an AFD of 1.45 ± 0.03. As shown in Fig. 2(b), these errors are comparable to those of a KRR and GPR model trained on the same data, following the procedures listed in Sections IV B and IV C, respectively. We chose to retrain the models using the same dataset and cross-validation splits in order to single out the effect of the methodology itself, and not changes in the data set and splits. These results show that the performance of SISSO and more traditional regression methods are similar, but the advantage of the symbolic regression models is that only seven of the primary features are selected. Another advantage of the nested cross-validation scheme is that it creates an ensemble of independent models, which can also be used to approximate the uncertainty of the predictions. These results substantiates that our symbolic regression approach performs as well as interpolative methods and it outperforms the Slack model, which was originally developed for elemental cubic solids18. Interestingly, offering the features of the Slack model to SISSO does not improve the results, and even some primary features previously thought to be decisive, e.g., the Grüneisen parameter, γ. are not even selected by SISSO (see Supplementary Note 5).

A key advantage of using symbolic regression techniques over interpolative methods such as KRR and GPR is that the resulting models not only yield reliable quantitative predictions, but also allows for a qualitative inspection of the underlying mechanisms. To get a better understanding of how the thermal conductivity changes across materials space we map the model in Fig. 2c. From this map we can see that the thermal conductivity of a material is mostly controlled by d2 with d1 providing only a minor correction. While these observed trends are already helpful, the complex non-linearities in both d1 and d2 impedes the generation of qualitative design rules. Furthermore, some primary features such as Vm and σA enter both d1 and d2, with contrasting trends, e.g., σA lowers d1 but increases d2. To accelerate the exploration of materials space, one must first be able to disentangle the contradicting contributions of the involved primary features.

Extracting physical understanding by identifying the most physically relevant features via sensitivity analysis

The difficulties in interpreting the “plain” SISSO descriptors described above can be overcome by performing a sensitivity analysis or a feature importance study to identify the most relevant primary features that build d1 and d2. For this purpose, we employ both the Sobol indices, i.e., the main effect index Si and the total effect index \({S}_{i}^{{{{\rm{T}}}}}\)47, and the Shapley Additive Explanations (SHAP)48 metric for the model predictions. To calculate the Sobol indices we use an algorithm that includes correlative effects first described by Kucherenko et al.49, and later implemented in UQLAB50,51. The main advantage of this approach is its ability to include correlative effects between the inputs, which if ignored can largely bias or even falsify the sensitivity analysis results52. Qualitatively, Si quantifies how much the variance of \(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) correlates with the variance of a primary feature, \({\hat{x}}_{i}\), and \({S}_{i}^{{{{\rm{T}}}}}\) quantifies how much the variance of \(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) correlates with \({\hat{x}}_{i}\) including all interactions between \({\hat{x}}_{i}\) and the other primary features. For example, Sobol indices of 0.0 indicate that \(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) is fully independent of \({\hat{x}}_{i}\), whereas a value of 1.0 indicates that \(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) can be completely represented by changes in \({\hat{x}}_{i}\)51. Moreover, \({S}_{i}^{{{{\rm{T}}}}} < {S}_{i}\) implies that correlative effects are significant, with an \({S}_{i}^{{{{\rm{T}}}}}=0\) indicating that a primary feature is perfectly correlated to the other inputs51.

The SHAP values constitute a local measure of how each feature influences a given prediction in the data set. This metric is based on the Shapley values used in game theory for assigning payouts to players in a game based on their contribution toward the total reward48. In the context of machine learning models each input to the model represents the players and the difference between individual predictions from the global mean prediction of a dataset represents the payouts53. The SHAP values then perfectly distribute the difference from the mean prediction to each feature for each sample, with negative values indicating that the feature is responsible for reducing the prediction from the mean and a positive value is responsible for increasing it.53. A similar metric is the Local Interpretable Model-agnostic Explanations (LIME) values54. LIME first defines a local neighborhood for each data point, and then uses a similar algorithm to SHAP to compare each prediction against their corresponding local area. Because of the computational complexity of calculating SHAP values makes their exact calculation intractable with a large number of features, these values can be approximated by the Kernel SHAP method48. Originally the Kernel SHAP method assumed feature independence48, but was recently advanced to include feature dependence via sampling over a multivariate distribution represented by a set of marginal distributions and a Gaussian Copula53. However, there are some cases for small data sets with highly correlated features where the SHAP values are qualitatively different from the true Shapley values55.

Figure 3 compares the different sensitivity metrics including and excluding feature dependence. To get the global values of the SHAP and LIME indexes we take the mean absolute value for each feature across all 75 materials, but other metrics have been proposed in the literature and it is not clear which one is best56,57,58. However the local information contained in metrics such as SHAP and LIME is an advantage they have over global metrics such as the Sobol indexes as it allows for the identification of regions in the material space that do not follow the global trends. Comparing the plots in Fig. 3a and b illustrates the importance of not treating the input primary features as independent, as all four sensitivity analysis metrics are qualitatively wrong under that assumption. This is likely a result of sampling over physically unreachable parts of the feature space, e.g., areas with a high density, low mass, and high molar volume, and suggests that caution should be used when applying these techniques to highly correlated datasets. The impact of this is demonstrated in Supplementary Fig. 3, where we explicitly simplify the model to remove some of the dependencies. All three indexes that include correlative effects show that σA, Vm, ΘD,∞, and ωΓ,max predominately control the variance of \({\kappa }^{{{{\rm{SISSO}}}}}\left(300\,{{{\rm{K}}}}\right)\). The main difference between Si and the kernel SHAP metrics is the relative importance of ΘD,∞ and ωΓ,max when compared against Vm and σA. The difference between these results could be from the the Sobol indexes globally sampling the region of ΘD,∞ > 1300 K instead of relying on the two materials in that regime or Si over-estimating its importance because the higher correlation between ΘD,∞ and the other inputs. In fact, the low values of \({S}_{i}^{{{{\rm{T}}}}}\) also imply that there are significant correlative effects in place between these inputs, and no single feature can be singled out as primarily responsible for changes in \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\). For instance, the similarity between the importance of ωΓ,max and ΘD,∞ is because they are strongly correlated to each other, and only one of them needs to be considered (see the Supplementary Fig. 2). The importance of these features is further substantiated in Fig. 2b, where we compare the performance of the models calculated using the full dataset and one that only includes σA, Vm, and ΘD,∞. For all tested models, we see only a slight deterioration in performance with a predictive AFD of 1.87, 1.77, and 1.77 for the SISSO, KRR, and GPR models, respectively, compared to 1.45 for the models trained with all features. This result highlights that the trends and the underlying mechanisms describing the dependence of \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\) in materials space are fully captured by those features alone.

Fig. 3: The feature importance metrics for the models.
figure 3

Si (first bar, dark blue), \({S}_{i}^{T}\) (second bar, light blue), mean absolute SHAP index (third bar, brown), and LIME index (fourth bar, yellow) for each feature in the model by treating the inputs as (a) dependent feature and (b) independent features. The Sobol indices are plotted on the left y-axis and the SHAP and LIME indexes are plotted on the right y-axis.

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Even more importantly, our model captures the interplay between these features across materials, as demonstrated in the maps in Fig. 4. These maps showcase the strong correlation between \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\) and σA, Vm, and ΘD,∞, and that materials with high anharmonicity, low-energy vibrational modes, and a large molar volume will be good thermal insulators. Figure 4 shows the expected value of \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\), \({E}_{\hat{{{{\mathcal{X}}}}}}\left(\left.{\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\right\vert \hat{{{{\mathcal{X}}}}}\right)\), for different sets of input features, \(\hat{{{{\mathcal{X}}}}}\), shown on the axes of each plot. We then overlay the maps with the actual values of each input for all materials in the training set to evaluate the trends across different groups of materials. Figure 4c confirms that σA is already a good indicator for finding thermal insulators, with most of the materials having \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\) within one standard deviation of the expected value. For the more harmonic materials with σA < 0.2, the vanishing degree of anharmonicity is, alone, not always sufficient for quantitative predictions. In this limit, a combination of σA and Vm can produce correct predictions for the otherwise underestimated white triangles with a σA < 0.2, as seen in Fig. 4a. In order to fully describe the low thermal conductivity of the remaining highlighted materials both ΘD,∞ and Vm are needed as can be seen in Fig. 4a, b, d and e. Generally, this reflects that the three properties σA, ΘD,∞, and Vm are the target properties to optimize to obtain ultra-low thermal conductivities.

Fig. 4: The expected value of \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\) relative to select primary features.
figure 4

The expected value of \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\), \({E}_{\hat{{{{\mathcal{X}}}}}}\left(\left.{\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\right\vert \hat{{{{\mathcal{X}}}}}\right)\), where \(\hat{{{{\mathcal{X}}}}}\) is (a) \(\left\{{\sigma }^{{{{\rm{A}}}}},{V}_{{{{\rm{m}}}}}\right\}\), (b) \(\left\{{\Theta }_{{{{\rm{D,\infty }}}}},{V}_{{{{\rm{m}}}}}\right\}\), (c) \(\left\{{\sigma }^{{{{\rm{A}}}}}\right\}\), (d) \(\left\{{\Theta }_{{{{\rm{D,\infty }}}}}\right\}\), and (e) \(\left\{{V}_{{{{\rm{m}}}}}\right\}\). \({E}_{\hat{{{{\mathcal{X}}}}}}\left(\left.{\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\right\vert \hat{{{{\mathcal{X}}}}}\right)\) is calculated by sampling over the multivariate distributions used for the sensitivity analysis, and binning the input data until there are at least 10,000 samples in each bin. The red line in (c–e) corresponds to \({E}_{\hat{{{{\mathcal{X}}}}}}\left(\left.{\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\right\vert \hat{{{{\mathcal{X}}}}}\right)\) and the pink shaded region is one standard deviation on either side of the line. The gray shaded regions represent where a thermal conductivity of 10 Wm−1 K−1 or lower is within one standard deviation of the expected value. On all maps all materials in the training set are displayed. The green circles correspond to rock-salts, the blue diamonds are zincblende, the light blue pentagons are wurtzites, and black triangles are all other materials. All points with a \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\) less than one standard deviation below the expected value based on σA are highlighted in white. The points in (c–e) correspond to the actual values of \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\) for each material. Additionally we include four materials outside of the training set (yellow stars) whose thermal conductivities we calculate using ab initio molecular dynamics.

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These results can also be rationalized within our current understanding of thermal transport and showcase which physical mechanisms determine κL in material space. Qualitatively, it is well known that good thermal conductors typically exhibit a high degree of symmetry with a smaller number of atoms, e.g., diamond and silicon, whereas thermal insulators, e.g., glass-like materials, are often characterized by an absence of crystal symmetries and larger primitive cells. In our case, this trend is quantitatively captured via Vm, which reflects that larger unit cells have smaller thermal conductivities. Furthermore, it is well known that phonon group velocities determine how fast energy is transported through the crystal in the harmonic picture59, and that it is limited by scattering events arising due to anharmonicity. In our model, these processes are captured by ΘD,∞, which describes the degree of dispersion in the phonon band structure, and the anharmonicity measure, σA respectively. In this context, it is important to note that, in spite of the fact that these qualitative mechanisms were long known, there had hitherto been no agreement on which material property would quantitatively capture these mechanisms best across material space. For instance, both the γ, the lattice thermal expansion coefficient, and now σA, have been used to describe the anharmonicity of a material. However, when both γ and σA are included as primary features, only σA is chosen (see Supplementary Note 5 for more details). This result indicates that the σA measure is the more sensitive choice for modeling the strength of anharmonic effects. While γ also depends on anharmonic effects, they are also influenced by the bulk modulus, the density, and the specific heat of a material.

Validating the predictions with ab initio Green-Kubo calculations

To confirm that the discovered models produce physically meaningful predictions, we validate the estimated thermal conductivity of four materials using the ab initio Green-Kubo method (aiGK)10,45. This approach has recently been demonstrated to be highly accurate when compared to experiments45, using similar DFT settings for what is done in this work. In particular aiGK is highly accurate in the low thermal conductivity regime that we are studying here. For details of how we calculate κL see the methodology in Section IV J. For this purpose, we chose BrBaCl, LiScS2, CaF2, and GaLiO2, since these materials represent a broad region of the relevant feature space that also test the boundary regions of the heuristics found by the sensitivity analysis and mapping, as demonstrated by the yellow stars in Fig. 4. Figure 5 shows the convergence of the thermal conductivity of the selected materials, as calculated from three aiMD trajectories. All of the calculated thermal conductivities fall within the 95% confidence interval of the model, with the predictions for both CaF2 and ClBaBr being especially accurate. The better performance of the model for these materials is expected, as they are more similar to the training data than the hexagonal Caswellsilverite like materials. In addition, quantum nuclear effects play a more important role in LiScS2 and GaLiO2 than CaF2 and ClBaBr, which can also explain why those predictions are worse than CaF2 and ClBaBr. Overall these results demonstrate the predictive power of the discussed model.

Fig. 5: Validation of the predictions of the model.
figure 5

The convergence of the calculated thermal conductivity of (a) CaF2, (b) ClBaBr, (c) GaLiO2 (d) LiScS2. All aiGK calculations were done using the average of three 75 ps (ClBaBr and GaLiO2) or 100 ps (CaF2 and LiScS2) molecular dynamics trajectories. The dashed lines are the values of the thermal conductivities predicted by Eq. (1) and the shaded region is the 95% confidence interval of the prediction based on the RMSE obtained in Fig. 2b.

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Discovering improved thermal insulators

Using the information gained from the sensitivity analysis and statistical maps of the model, we are now able to design a hierarchical and efficient high-throughput screening protocol split into three stages: structure optimization, harmonic model generation, and anharmonicity quantification. We demonstrate this procedure by identifying possible thermal insulators within a set of 732 materials, within those compounds available in the materials project60 that feature the same crystallographic prototypes61,62 as the ones used for training. Once the geometry is optimized we remove all materials with Vm < 35.5 Å(60 materials) and all (almost) metallic materials (GGA bandgap < 0.2 eV), and are left with 302 candidate compounds. We then generate the converged harmonic model for the remaining materials and screen out all materials with ΘD,∞ > 547 K or have an unreliable harmonic model, e.g., materials with imaginary harmonic modes, leaving 148 candidates. Finally we evaluate the anharmonicity, σA, for the remaining materials (see Section IV D) and exclude all materials with σA < 0.206, and obtain 110 candidate thermal insulators. To avoid unnecessary calculations, we first estimate σA via \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}}\) and then refine it via aiMD when \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}} > 0.4\)8. For these candidate materials, we evaluate \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\) using Eq. (1). Of the 110 materials that passed all checks, 96 are predicted to have have a \({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\) below 10 Wm−1K−1, illustrating the success of this method.

Finally, let us emphasize that the proposed strategy is not limited to the discovery of thermal insulators, but can be equally used to find, e.g., good thermal conductors. This is demonstrated in Fig. 6, in which we predict the thermal conductivity of all non-metallic and stable materials using the SISSO and KRR models. Generally, both the SISSO and KRR models agree with each other with only 28 of the 227 materials having a disagreement larger than a factor of two and one (LiHF2) with a disagreement larger than a factor of 5, further illustrating the reliability of these predictions. We expect that the large deviation for LiHF2 is a result of the large σA value for that material (0.54), which is significantly larger than the maximum in the training data. We can see from the outset histograms of both the SISSO and KRR models that the hierarchical procedure successfully finds the good thermal insulators, with only 26 of the 122 materials with a \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\le 10\) Wm−1K−1 and 10 of the 80 materials with a \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\le 5\) Wm−1K−1 not passing all tests. Of these eight only the thermal insulating behavior of CuLiF2 and Sr2HN can not be described by the values of the other two tests that passed. Conversely, materials that do not pass the test show high conductivities. When one of the tests fail the average estimated value of \(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) increases to 1.38 ± 0.490 (24.0 Wm−1K−1), with a range of 0.95 Wm−1K−1 to 741.3 Wm−1K−1. In particular, screening the materials by their molar volumes alone is a good marker for finding strong thermal conductors as all of the 15 materials with \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\ge 100\) Wm−1K−1 have a Vm ≤ 45 Å3.

Fig. 6: A scatter plot of the prediction of both the SISSO and KRR generated models for an additional 227 materials from the same classes as the training set.
figure 6

σA is estimated via \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}}\) for all materials with a \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}}\le 0.4\) in this screening. The dataset is split up into four subsets based on if the Vm test failed (top, green), ΘD,∞ test failed (second from top, yellow), σA test failed (third from top, blue), or none of the tests failed (bottom, purple). The outlets correspond to the histogram of all predictions using the same break down. The darker shaded region represents where both predictions are within a factor of 2 of each other and the lighter shaded region where both predictions are within a factor of 5 of each other.

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Discussion

We have developed an AI framework to facilitate and accelerate material space exploration, and demonstrate its capabilities for the urgent problem of finding thermal insulators. By combining symbolic regression and sensitivity analysis, we are able to obtain accurate predictions for a given property using relatively easy to calculate materials properties, while retaining strong physical interpretability. Most importantly, this analysis enables us to create hierarchical, high-throughput frameworks, which we used to screen over a set of more than 700 materials and find a group of ~100 possible thermal insulators. Notably, almost all of the good thermal conductors in the set of candidate materials are discarded within the first iteration of the screening, in which we only discriminate by molar volume, i.e., with an absolutely negligible computational cost compared to full calculations of κL. Accordingly, we expect this approach to be extremely useful in a wide range of materials problems beyond thermal transport, especially whenever (i) few reliable data are available, (ii) additional data are hard to produce, and/or (iii) multiple physical mechanisms compete non-trivially, limiting the reliability of simplified models.

Although the proposed approach is already reliable for small dataset sizes, it obviously becomes more so when applied to larger ones. Here, the identified heuristics can substantially help steer data creation toward more interesting parts of material space. Along these lines, it is possible to iteratively refine both the SISSO model and the rules from the sensitivity analysis during material space exploration while the dataset grows. Furthermore, one can also apply the proposed procedure to the most influential primary features in a recursive fashion, learning new expressions for the computationally expensive features, e.g., σA, using simpler properties. In turn, this will further accelerate material discovery, but also allow for gaining further physical insights. Most importantly, this method is not limited to just the thermal conductivity of a material, and can be applied to any target property. Further extending this framework to include information about where the underlying electronic structure calculations are expected to fail, also provides a means of accelerating materials discovery more generally63.

Methods

SISSO

We use SISSO to discover analytical expressions for \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\)64. SISSO finds low-dimensional, analytic expressions for a target property, P, by first generating an exhaustive set of candidate features, \(\hat{{{\Phi }}}\), for a given set of primary features, \({\hat{{{\Phi }}}}_{0}\), and operators \({\hat{{{{\mathcal{H}}}}}}_{m}\), and then performing an ℓ0-regularization over a subset of those features to find the n-dimensional subset of features, whose linear combination results in the most descriptive model. \(\hat{{{\Phi }}}\) is recursively built in rungs, \({\hat{{{{\mathcal{F}}}}}}_{r}\), from \({\hat{{{\Phi }}}}_{0}\) and \({\hat{{{{\mathcal{H}}}}}}_{m}\), by applying all elements, \({\hat{{{{\rm{h}}}}}}^{m}\), of \({\hat{{{{\mathcal{H}}}}}}^{m}\) on all elements \({\hat{f}}_{i}\) and \({\hat{f}}_{j}\) of \({\hat{{{{\mathcal{F}}}}}}_{r-1}\)

$${\hat{{{{\mathcal{F}}}}}}_{r}\equiv {\hat{{{\mbox{h}}}}}^{m}\left[{\hat{f}}_{i},{\hat{f}}_{j}\right],\forall \ {\hat{{{\mbox{h}}}}}^{m}\in {\hat{{{{\mathcal{H}}}}}}^{m}\ \,{{\mbox{and}}}\,\ \forall \ {\hat{f}}_{i},{\hat{f}}_{j}\in {\hat{{{{\mathcal{F}}}}}}_{r-1}.$$

\({\hat{{{\Phi }}}}_{r}\) is then the union of \({\hat{{{\Phi }}}}_{r-1}\) and \({\hat{{{{\mathcal{F}}}}}}_{r}\). Once \(\hat{{{\Phi }}}\) is generated, the nSIS features most correlated to P are stored in \({\hat{{{{\mathcal{S}}}}}}_{1}\), and the best one-dimensional models are trivially extracted from the top elements of \({\hat{{{{\mathcal{S}}}}}}_{1}\). Then the nSIS features most correlated to any of the residuals, \({{{{\boldsymbol{\Delta }}}}}_{1}^{i}\), of the \({n}_{{{{\rm{res}}}}}\) best one-dimensional descriptors are stored in \({\hat{{{{\mathcal{S}}}}}}_{2}\). We define this projection as

$$s=\max \left({s}_{0},{s}_{1},...,{s}_{i},...,{s{}_{{{{\rm{n}}}}}}_{{{{\rm{res}}}}}\right)$$
(3)
$${s}_{i}={R}^{2}\left(\hat{\phi },{{{{\boldsymbol{\Delta }}}}}_{1}^{i}\right),$$
(4)

where \(\hat{\phi }\in \hat{{{\Phi }}}\), and R is the Pearson correlation function. We call this approach the multiple residual approach, which was first introduced by the authors65 and later fully described in Ref. 66. From here, the best two-dimensional models are found by performing an ℓ0-regularized optimization over \({\hat{{{{\mathcal{S}}}}}}_{1}\cup {\hat{{{{\mathcal{S}}}}}}_{2}\)67. This process is iteratively repeated until the best n-dimensional descriptor is found64.

For this application \({\hat{{{{\mathcal{H}}}}}}_{m}\) contains: A + B, A − B, A*B, \(\frac{A}{B}\), \(\left\vert A-B\right\vert\), \(\left\vert A\right\vert\), \({\left(A\right)}^{-1}\), \({\left(A\right)}^{2}\), \({\left(A\right)}^{3}\), \(\sqrt{A}\), \(\root 3 \of {A}\), \(\exp \left(A\right)\), \(\exp \left(-1.0* A\right)\), and \(\ln \left(A\right)\). In addition to ensure the units of the primary features do not affect the final results, we additionally include the following operators: \({\left(A+\beta \right)}^{-1}\), \({\left(A+\beta \right)}^{2}\), \({\left(A+\beta \right)}^{3}\), \(\sqrt{\alpha A+\beta }\), \(\root 3 \of {A+\beta }\), \(\exp \left(\alpha A\right)\), \(\exp \left(-1.0* \alpha A\right)\), and \(\ln \left(\alpha A+\beta \right)\), where α and β are scaling and bias constants used to adjust the input data on the fly. We find the optimal α and β terms using non-linear optimization for each of these operators22,66,68. To ensure that the parameterization does not result in mathematically invalid equations for data points outside of the training set, the range of each candidate feature is derived from the range of the primary features, and the upper and lower bounds for the features are set appropriately. When generating new expressions these ranges are then used as a domain for the operations, and any expression that would lead to invalid results are excluded66. The range of the primary features are set to be physically relevant for the systems we are studying and are listed in Table 1. Hereafter, we call the use of these operators parametric SISSO. For more information please refer to66.

All hyperparameters were set following the cross-validation procedures described in Section IV E.

Kernel-Ridge regression

To generate the kernel-ridge regression models we used the utilities provided by scikit-learn69, using a radial basis function kernel with optimized regularization term and kernel length scale. The hyperparameters were selected using with a 141 by 141 point logarithmic grid search with possible parameters ranging from 10−7 to 100. Before performing the analysis each input feature, xi is standardized

$${{{\bf{{x}}}_{{{{\rm{i}}}}}^{{{{\rm{Stand}}}}}}}=\frac{{{{{\bf{x}}}}}_{{{{\rm{i}}}}}-{\mu }_{{{{\rm{i}}}}}}{{\sigma }_{{{{\rm{i}}}}}}$$
(5)

where \({{{\bf{{x}}}_{{{{\rm{i}}}}}^{{{{\rm{stand}}}}}}}\) is the standardized input feature, μi is the mean of the input feature for the training data, and σi is the standard deviation of the input feature for the training data.

Gaussian process regression

To generate the Gaussian Process Regression Models we used the utilities provided by scikit-learn69, using a radial basis function kernel with an optimized regularization term and kernel length scale. The hyperparameters were selected using with a 141 by 141 point logarithmic grid search with possible parameters ranging from 10−7 to 100. Before performing the analysis each input feature, xi is standardized

$${{{\bf{{x}}}_{{{{\rm{i}}}}}^{{{{\rm{Stand}}}}}}}=\frac{{{{{\bf{x}}}}}_{{{{\rm{i}}}}}-{\mu }_{{{{\rm{i}}}}}}{{\sigma }_{{{{\rm{i}}}}}}$$
(6)

where \({{{\bf{{x}}}_{{{{\rm{i}}}}}^{{{{\rm{stand}}}}}}}\) is the standardized input feature, μi is the mean of the input feature for the training data, and σi is the standard deviation of the input feature for the training data. All uncertainty values were taken from the results of the GPR predictions, and in the case of the nested cross-validation the uncertainty was propagated using

$${\kappa }_{{{{\rm{GPR}}}}}^{{{{\rm{pred}}}}}=\frac{1}{3}\mathop{\sum }\limits_{i=1}^{3}{\kappa }_{{{{\rm{GPR}}}},i}^{{{{\rm{pred}}}}}$$
(7)
$${\sigma }_{{{{\rm{GPR}}}}}^{{{{\rm{pred}}}}}=\frac{1}{3}\sqrt{\mathop{\sum }\limits_{i=1}^{3}{\left({\sigma }_{{{{\rm{GPR}}}},i}^{{{{\rm{pred}}}}}\right)}^{2}},$$
(8)

where \({\kappa }_{{{{\rm{GPR}}}},i}^{{{{\rm{pred}}}}}\) and \({\kappa }_{{{{\rm{GPR}}}},i}^{{{{\rm{pred}}}}}\) are the respective prediction and uncertainty of the ith GPR model for a given data point and \({\kappa }_{{{{\rm{GPR}}}}}^{{{{\rm{pred}}}}}\) and \({\kappa }_{{{{\rm{GPR}}}}}^{{{{\rm{pred}}}}}\) are the respective mean prediction and uncertainty for a prediction.

Creating the dataset

In this study we focus on only room-temperature data for κL, since values for other temperatures are even scarcer. However, we note that an explicit temperature dependence can be straightforwardly included using multi-task SISSO70, and it is at least partially included via, the anharmonicity factor, σA8 (see below for more details). For \({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\), we have compiled a list of seventy-five materials from the literature (see Supplementary Table 1 for complete list with references), whose thermal conductivity has been experimentally measured. This list was curated from an initial set of over 100 materials, from which we removed all samples that are either thermodynamically unstable or are electrical conductors. This list of materials covers a diverse set of fourteen different binary and ternary crystal structure prototypes61,62,71.

With respect to the primary features, \({\hat{{{\Phi }}}}_{0}\), compound specific properties are provided for each material. All primary features can be roughly categorized in two classes: Structural parameters that describe the equilibrium structure and dynamical parameters that characterize the nuclear motion. For the latter case, both harmonic and anharmonic properties have been taken into account. As shown in Supplementary Note 5, additional features, such as the parameters entering the Slack model, i.e., γ, Θa, and Va, can be included. However, these features do not benefit the model and when included only Va, and not γ or Θa are selected. For a complete list of all primary features, and their definitions refer to Table 1.

The structural parameters relate to either the mass of the atoms (μ, \({m}_{\min }\), \({m}_{\max }\), mavg), the lattice parameters of the primitive cell (Vm, \({L}_{\min }\), \({L}_{\max }\), Lavg), the density of the materials (ρ), or the number of atoms in the primitive cell (nat). For all systems a generalization of the reduced mass, μ, is used so it can be extended to non-binary systems,

$$\mu ={\left(\mathop{\sum }\limits_{i}^{{n}_{{{{\rm{emp}}}}}}\frac{1}{{m}_{i}}\right)}^{-1},$$
(9)

where nemp is the number of atoms in the empirical formula and mi is the mass of atom, i. Similarly, the molar volume, Vm, is calculated by

$${V}_{{{{\rm{m}}}}}=\frac{{V}_{{{{\rm{prim}}}}}}{Z},$$
(10)

where \({V}_{{{{\rm{prim}}}}}\) is the volume of the primitive cell and \(Z=\frac{{n}_{{{{\rm{at}}}}}}{{n}_{{{{\rm{emp}}}}}}\). Finally, ρ is calculated by dividing the total mass of the empirical cell by Vm

$$\rho =\mathop{\sum }\limits_{i}^{{n}_{{{{\rm{emp}}}}}}\frac{{m}_{i}}{{V}_{{{{\rm{m}}}}}}.$$
(11)

All of the harmonic properties used in these models are calculated from a converged harmonic model generated using phonopy72. For each material, the phonon density of states of successively larger supercells are compared using a Tanimoto similarity measure

$$S=\frac{{g}_{{{{\rm{p,L}}}}}\left(\omega \right)\cdot {g}_{{{{\rm{p,S}}}}}\left(\omega \right)}{\parallel {g}_{{{{\rm{p,L}}}}}\left(\omega \right){\parallel }^{2}+\parallel {g}_{{{{\rm{p,S}}}}}\left(\omega \right){\parallel }^{2}-{g}_{{{{\rm{p,L}}}}}\left(\omega \right)\cdot {g}_{{{{\rm{p,S}}}}}\left(\omega \right)},$$
(12)

where S is the similarity score, \({g}_{{{{\rm{p,L}}}}}\left(\omega \right)\) is the phonon density of states of the larger supercell, \({g}_{{{{\rm{p,S}}}}}\left(\omega \right)\) is the phonon density of states of the smaller supercell, \(A\left(\omega \right)\cdot B\left(\omega \right)=\int\nolimits_{0}^{\infty }A\left(\omega \right)B\left(\omega \right)d\omega\), and \(\parallel A\left(\omega \right){\parallel }^{2}=\int\nolimits_{0}^{\infty }{A}^{2}\left(\omega \right)d\omega\). If S > 0.80, then the harmonic model is considered converged. From here CV is calculated from phonopy as a weighted sum over the mode dependent heat capacities. Both approximations to the Debye temperature are calculated from the moments of the phonon density of states

$$\langle {\varepsilon }^{n}\rangle =\frac{\int\,d\varepsilon \,{g}_{{{{\rm{p}}}}}(\varepsilon )\,{\varepsilon }^{n}}{\int\,d\varepsilon {g}_{{{{\rm{p}}}}}\left(\varepsilon \right)}$$
(13)
$${\Theta }_{{{{\rm{P}}}}}=\frac{1}{{k}_{B}}\langle \varepsilon \rangle$$
(14)
$${\Theta }_{{{{\rm{D,\infty }}}}}=\frac{1}{{k}_{B}}\sqrt{\frac{5}{3}\langle {\varepsilon }^{2}\rangle },$$
(15)

where \({g}_{p}\left(\varepsilon \right)\) is the phonon density of states at energy ε73. Finally vs is approximated from the Debye frequency, ωD, by20

$${v}_{{{{\rm{s}}}}}={\left(\frac{{V}_{{{{\rm{a}}}}}}{6{\pi }^{2}}\right)}^{1/3}{\omega }_{{{{\rm{D}}}}},$$
(16)

where ωD is approximated as

$${\omega }_{{{{\rm{D}}}}}=\root 3 \of {\frac{9{n}_{{{{\rm{at}}}}}}{a}}$$
(17)

and a is found by fitting \({g}_{p}\left(\omega \right)\) in the range \(\left[0,\frac{{\omega }_{\Gamma ,max}}{8}\right]\) to

$${g}_{p,D}\left(\omega \right)=a{\omega }^{2}.$$
(18)

To measure the anharmonicity of the materials we use σA as defined in8

$${\sigma }^{{{{\rm{A}}}}}(T)=\sqrt{\frac{\mathop{\sum}\limits_{I,\alpha }{\left\langle {\left({F}_{I,\alpha }-{F}_{I,\alpha }^{{{{\rm{ha}}}}}\right)}^{2}\right\rangle }_{(T)}}{\mathop{\sum}\limits_{I,\alpha }{\left\langle {F}_{I,\alpha }^{2}\right\rangle }_{(T)}}}\,,$$
(19)

in which 〈⋅〉(T) denotes the thermodynamic average at a temperature T, FI,α is the α component of the force calculated from density functional theory (DFT) acting on atom I, and \({F}_{I,\alpha }^{{{{\rm{ha}}}}}\) is the same force approximated by the harmonic model8. First we calculate \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}}\), which uses an approximation to the thermodynamic ensemble average using the one-shot method proposed by Zacharias and Giustino74. In the one-shot approach the atomic positions are offset from their equilibrium positions by a vector ΔR,

$$\Delta {R}_{I}^{\alpha }=\frac{1}{\sqrt{{M}_{I}}}\mathop{\sum}\limits_{s}{\zeta }_{s}\left\langle {A}_{s}\right\rangle {e}_{sI}^{\alpha },$$
(20)

where I is the atom number, α is the component, es are the harmonic eigenvectors, \(\left\langle {A}_{s}\right\rangle =\sqrt{2{k}_{B}T}/{\omega }_{s}\) is the mean mode amplitude in the classical limit75, and ζs = (−1)s−174. These displacements correspond to the turning-points of the oscillation estimated from the harmonic force constants, and is a good approximation to σA in the harmonic limit. Because of this, if \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}} < 0.2\) we accept that value as the true σA. Otherwise we calculate σA using aiMD in the canonical ensemble at 300 K for 10 ps, using the Langevin thermostat. When performing the high-throughput screening the threshold for when to use aiMD is increased to 0.4 because that is the point where \({\sigma }_{{{{\rm{OS}}}}}^{{{{\rm{A}}}}}\) becomes qualitatively unreliable8.

All electronic structure calculations are done using FHI-aims76. All geometries are optimized with symmetry-preserving, parametric constraints until all forces are converged to a numerical precision better than 10−3 eV/Å77. The constraints are generated using the AFlow XtalFinder Tool71. All calculations use the PBEsol functional to calculate the exchange-correlation energy and an SCF convergence criteria of 10−6 eV/Å and 5 × 10−4 eV/Å for the density and forces, respectively. Relativistic effects are included in terms of the scalar atomic ZORA approach and all other settings are taken to be the default values in FHI-aims. For all calculations we use the light basis sets and numerical settings in FHI-aims. These settings were shown to ensure a convergence in lattice constants of ± 0.1 Å and a relative accuracy in phonon frequencies of 3%8.

All primary features are calculated using the workflows defined in FHI-vibes78.

Error evaluation

To estimate the prediction error for all models we perform a nested cross-validation, where the data are initially separated into different training and test sets using a ten-fold split. Two hyperparameters (maximum dimension and parameterization depth) are then optimized using a fivefold cross-validation on each of the training sets, and the overall performance of the model is evaluated on the corresponding test set. The size of the SIS subspace, number of residuals, and rung were all set to 2000, 10, and 3, respectively, because they did not have a large impact on the final results. We then repeat the procedure three times and average over each iteration to get a reliable estimate of the prediction error for each sample79.

Calculating the inputs to the Slack model

The individual components for the Slack model were the same as the ones used for the main models, with the exception of γ, Va and Θa. For Θa, we first calculate the Debye temperature, ΘD

$${\Theta }_{{{{\rm{D}}}}}=\frac{\hslash {\omega }_{{{{\rm{D}}}}}}{{k}_{B}}$$
(21)

where ωD is the same Debye frequency used for calculating vs (see Section IV D), kB is the Boltzmann constant, and ℏ is Planck’s constant. From here we calculate Θa using

$${\Theta }_{{{{\rm{a}}}}}=\frac{{\Theta }_{{{{\rm{D}}}}}}{\root 3 \of {{n}_{{{{\rm{at}}}}}}}.$$
(22)

We use the phonopy definition of ΘD instead of ΘD,∞ because it is better aligned to the original definition of Θa. However, it is not used in the SISSO training because the initial fitting procedure to find ωD does not produce a unique value for ΘD and it is already partially included via vs. To calculate the thermodynamic Grüneisen parameter we use the utilities provided by phonopy72. The atomic volume was calculated by taking the volume of the primitive cell and dividing it by the total number of atoms.

Calculating the Sobol indexes

Formally, the Sobol indices are defined as

$${S}_{i}=\frac{{{{{\rm{Var}}}}}_{{\hat{x}}_{i}}\left({E}_{{\widetilde{{{{\mathcal{X}}}}}}_{i}}\left(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)| {\hat{x}}_{i}\right)\right)}{{{{\rm{Var}}}}\left(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\right)}$$
(23)
$${S}_{i}^{{{{\rm{T}}}}}=1-\frac{{{{{\rm{Var}}}}}_{{\widetilde{{{{\mathcal{X}}}}}}_{i}}\left({E}_{{\hat{x}}_{i}}\left(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)| {\widetilde{{{{\mathcal{X}}}}}}_{i}\right)\right)}{{{{\rm{Var}}}}\left(\log \left({\kappa }_{{{{\rm{L}}}}}\left({{{\rm{300\,K}}}}\right)\right)\right)}$$
(24)

where \({\hat{x}}_{i}\in \hat{{{{\mathcal{X}}}}}\) is one of the inputs to the model, \({{{{\rm{Var}}}}}_{a}\left(B\right)\) is the variance of B with respect to a, \({E}_{a}\left(B\right)\) is the mean of B after sampling over a, and \({\widetilde{{{{\mathcal{X}}}}}}_{i}\) is the set of all variables excluding \({\hat{x}}_{i}\).

Normally, it is assumed that all elements of \(\hat{{{{\mathcal{X}}}}}\) are independent of each other, and this assumption is preserved when calculating Si and \({S}_{i}^{T}\) in Fig. 3b. As a result of this, the variance of \(\log \left({\kappa }^{{{{\rm{SISSO}}}}}\left({{{\rm{300\,K}}}}\right)\right)\) and the required expectation values would be calculated from sampling over an nv-dimensional hypercube covering the full input range, ignoring the correlation between the input variables. However, in order to properly model the correlative effects between elements of \(\hat{{{{\mathcal{X}}}}}\), Kucherenko et al. modify this sampling approach49,51. The first step of the updated algorithm is to fit the input data to a set of marginal univariate distributions coupled together via a copula49,51. The algorithm then samples over an nv-dimensional unit-hypercube and transforms these samples into the correct variable space using a transform defined by the fitted distributions and copulas (see Supplementary Note 3 for more details). It was later demonstrated that when using the approach proposed by Kucherenko and coworkers to calculate the Sobol indices, Si includes effects from the dependence of \({\hat{x}}_{i}\) on those in \({\widetilde{{{{\mathcal{X}}}}}}_{i}\), while \({S}_{i}^{{{{\rm{T}}}}}\) is independent of these effects80. We use this updated algorithm to calculate Si and \({S}_{i}^{{{{\rm{T}}}}}\) in Fig. 3a. In both cases we use the implementation in UQLab50 to calculate Si and \({S}_{i}^{T}\).

Calculating the SHAP indexes

The SHAP values are calculated by treating the features as independent variables using the original method proposed by Lundberg and Lee48, as implemented in the python package SHAP, and as dependent variables using shapr by Aas, et al.53. The SHAP values are an extension of the Shapley values from cooperative game theory, that distributes the contribution, \(v\left({{{\mathcal{S}}}}\right)\), of each player or subset of players, \({{{\mathcal{S}}}}\subseteq {{{\mathcal{M}}}}=\left\{1,\cdot ,M\right\}\), where \({{{\mathcal{M}}}}\) is the set of all players48,53. The Shapley value, \({\phi }_{j}\left(v\right)={\phi }_{j}\), can then be calculated by taking a weighted mean over the contribution function differences for all \({{{\mathcal{S}}}}\) not containing the player, j,

$${\phi }_{j}=\mathop{\sum}\limits_{{{{\mathcal{S}}}}\subseteq {{{\mathcal{M}}}}\setminus \left\{j\right\}}\frac{\left\vert {{{\mathcal{S}}}}\right\vert !\left(M-\left\vert {{{\mathcal{S}}}}\right\vert -1\right)!}{M!}\left(v\left({{{\mathcal{S}}}}\cup \left\{j\right\}\right)-v\left({{{\mathcal{S}}}}\right)\right),j=1,\cdots \,,M,$$
(25)

where \(\left\vert {{{\mathcal{S}}}}\right\vert\) is the number of members in \({{{\mathcal{S}}}}\)53. For a machine learning problem with a training set \({\left\{{y}^{i},{{{{\boldsymbol{x}}}}}^{i}\right\}}_{i = 1,\cdots ,{n}_{{{{\rm{train}}}}}}\), where yi is the property value and xi are the target property value and input feature values for the ith data point in the training set with ntrain data points48,53, we can explain the prediction of the model, \(f\left({{{{\boldsymbol{x}}}}}^{* }\right)\) for a particular point, x*, with

$$f\left({{{{\boldsymbol{x}}}}}^{* }\right)={\phi }_{0}+\mathop{\sum }\limits_{j=1}^{M}{\phi }_{j}^{* },$$
(26)

where ϕ0 is the mean prediction and \({\phi }_{j}^{* }\) is the Shapley value for the jth feature for a prediction x = x*. Essentially the Shapley value for the model describes the difference between a prediction, \({y}^{* }=f\left({{{{\boldsymbol{x}}}}}^{* }\right)\), and the mean of all predictions48,53. The contribution function is then defined as

$$v\left({{{\mathcal{S}}}}\right)=E\left[\left.f\left({{{\boldsymbol{x}}}}\right)\right\vert {{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}^{* }\right],$$
(27)

which is the expectation value of the model conditional on \({{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}^{* }\)48,53. The expectation value can be calculated as

$$\begin{array}{ll}E\left[\left.f\left({{{\boldsymbol{x}}}}\right)\right\vert {{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}^{* }\right]&=E\left[\left.f\left({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}},{{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}\right)\right\vert {{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}^{* }\right]\\ &=\int\,f\left({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}},{{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}\right)p\left(\left.{{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}\right\vert {{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}* \right)d{{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}},\end{array}$$
(28)

where \({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}\) is the subset of all features not included in \({{{\mathcal{S}}}}\) and \(p\left(\left.{{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}\right\vert {{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}* \right)\) is the conditional probability distribution of \({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}\) given \({{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}*\)48,53. In the case where the features are treated independently, \(p\left(\left.{{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}\right\vert {{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}={{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}* \right)\) is replaced by \(p\left({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}\right)\) and \(v\left({{{\mathcal{S}}}}\right)\) can be approximated by Monte Carlo integration

$$v\left({{{\mathcal{S}}}}\right)=\frac{1}{K}\mathop{\sum }\limits_{k=1}^{K}f\left({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}^{k},{{{{\boldsymbol{x}}}}}_{{{{\mathcal{S}}}}}^{* }\right),$$
(29)

where \({{{{\boldsymbol{x}}}}}_{\widetilde{{{{\mathcal{S}}}}}}^{k}\) are samples from the training data, and K is the number of samples taken48,53. To include feature dependence the marginal distributions of the training data are converted into a Gaussian copula and that is used to generate samples for the Monte Carlo integration53.

Because the number of subsets that need to be explored grows as 2M for the number of features, calculating the exact Shapley values for a large number of inputs becomes intractable. To remove this constraint the problem can be approximated as the optimal solution of a weighted least squares problem, which can be described as Kernel SHAP, which is described in refs. 48,53.

Calculating the LIME indexes

For the LIME values we use the LIME package in python54. The values were calculated using the standard tabular explainer using all features in the model and the mean absolute value of each prediction for each feature was used to asses the global feature importance. The methodology assumes the features are independent and for algorithmic details see ref. 54

Calculating the thermal conductivity

To calculate κL, we use the ab initio Green-Kubo (aiGK) method10,81. The aiGK method calculates the αβ component of the thermal conductivity tensor, καβ, of a material for a given volume V, pressure p, and temperature T with

$${\kappa }^{\alpha \beta }\left(T,p\right)=\frac{V}{{k}_{B}{T}^{2}}\mathop{\lim }\limits_{\tau \to \infty }\int\nolimits_{0}^{\tau }{\langle G{\left[{{{\bf{J}}}}\right]}^{\alpha \beta }\left({\tau }^{{\prime} }\right)\rangle }_{\left(T,p\right)}d{\tau }^{{\prime} }$$
(30)

where kB is Boltzmann’s constant, \({\langle \cdot \rangle }_{\left(T,p\right)}\) denotes an ensemble average, \({{{\bf{J}}}}\left(t\right)\) is the heat flux, and \(G\left[{{{\bf{J}}}}\right]\) is the time-(auto)correlation functions

$$G{\left[{{{\bf{J}}}}\right]}^{\alpha \beta }=\mathop{\lim }\limits_{{t}_{0}\to \infty }\frac{1}{{t}_{0}}\int\nolimits_{0}^{{t}_{0}-\tau }{J}^{\alpha }\left(t\right){J}^{\beta }\left(t+\tau \right)dt.$$
(31)

The heat flux of each material is calculated from aiMD trajectories using the following definition

$${{{\bf{J}}}}\left(t\right)=\mathop{\sum}\limits_{I}{{{{\boldsymbol{\sigma }}}}}_{I}{\dot{{{{\bf{R}}}}}}_{I},$$
(32)

where RI is the position of the ith-atom and σI is the contribution of the ith atom to the stress tensor, σ = ∑IσI10. From here κL is calculated as

$${\kappa }_{{{{\rm{L}}}}}=\frac{1}{3}{{{\rm{Tr}}}}\left[{{{\boldsymbol{\kappa }}}}\right]$$
(33)

All calculations were done using both FHI-vibes78 and FHI-aims with the same settings as the previous calculations8 (see Section IV D for more details). The molecular dynamics calculations were done using a 5 fs time step in the NVE ensemble, with the initial structures taken from a 10 ps NVT trajectory. Three MD calculations were done for each material and the κL was taken to be the average of all three runs.