This tutorial shows how to find descriptive parameters (short formulas) that predict whether alloyed materials are topological or trivial insulators, using the example of tetradymites. It is based on the algorithm sure independence screening and sparsifying operator (SISSO), that enables to search for optimal descriptor by scanning huge feature spaces.
This tutorial shows how to find descriptive parameters (short formulas) that predict whether alloyed materials are topological or trivial insulators, using the example of tetradymites. It is based on the algorithm sure independence screening and sparsifying operator (SISSO), that enables to search for optimal descriptor by scanning huge feature spaces.
<divstyle="padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;">R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler, L. M. Ghiringhelli: <spanstyle="font-style: italic;">SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates</span>, Phys. Rev. Materials 2, 083802 (2018) <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">[PDF]</a>.</div>
<divstyle="padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;">R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler, L. M. Ghiringhelli: <spanstyle="font-style: italic;">SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates</span>, Phys. Rev. Materials 2, 083802 (2018) <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">[PDF]</a>.</div>
With the default settings, the method reproduces the same results from:
With the default settings, the method reproduces the same results from:
<divstyle="padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;">G. Cao, R. Ouyang, L. M. Ghiringhelli, M. Scheffler, H. Liu, C. Carbogno, and Z. Zhang: <spanstyle="font-style: italic;">Artificial intelligence for high-throughput discovery of topological insulators: The example of alloyed tetradymites</span>, Phys. Rev. Materials 4, 034204 (2020) <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.4.034204">[PDF]</a>,</div>
<divstyle="padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;">G. Cao, R. Ouyang, L. M. Ghiringhelli, M. Scheffler, H. Liu, C. Carbogno, and Z. Zhang: <spanstyle="font-style: italic;">Artificial intelligence for high-throughput discovery of topological insulators: The example of alloyed tetradymites</span>, Phys. Rev. Materials 4, 034204 (2020) <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.4.034204">[PDF]</a>,</div>
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<summary>
<summary>
<divstyle="padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;"><b>Explanation of the method (click to expand/collapse)</b></div></summary>
<divstyle="padding: 1ex; margin-top: 1ex; margin-bottom: 1ex; border-style: dotted; border-width: 1pt; border-color: blue; border-radius: 3px;"><b>Explanation of the method (click to expand/collapse)</b></div></summary>
We present a tool for predicting whether alloyed tetradymite are topological or trivial insulators, by using a set of descriptive parameters (a descriptor) based on free-atom data of the atomic species constituting the $AB-LMN$ materials, where $A,B \in \{\textrm{As, Sb, Bi} \}$ and $L,M,N \in \{\textrm{S, Se, Te} \}$. We apply a recently developed method: sure independence screening and sparsifying operator (SISSO), that allows to find an optimal descriptor in a huge feature space containing billions of features. In this tutorial an $\ell_0$-optimization is used as the sparsifying operator.
We present a tool for predicting whether alloyed tetradymite are topological or trivial insulators, by using a set of descriptive parameters (a descriptor) based on free-atom data of the atomic species constituting the $AB-LMN$ materials, where $A,B \in \{\textrm{As, Sb, Bi} \}$ and $L,M,N \in \{\textrm{S, Se, Te} \}$. We apply a recently developed method: sure independence screening and sparsifying operator (SISSO), that allows to find an optimal descriptor in a huge feature space containing billions of features. In this tutorial an $\ell_0$-optimization is used as the sparsifying operator.
R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler, L. M. Ghiringhelli: <spanstyle="font-style: italic;">SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates</span>, Phys. Rev. Materials 2, 083802 (2018) <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">[PDF]</a>. <br></div>
R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler, L. M. Ghiringhelli: <spanstyle="font-style: italic;">SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates</span>, Phys. Rev. Materials 2, 083802 (2018) <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">[PDF]</a>. <br></div>
In this tutorial, we focus on the classification flavor of SISSO($\ell_0$).
In this tutorial, we focus on the classification flavor of SISSO($\ell_0$).
In the space of descriptors, each category’s domain (here, topological vs trivial insulator) is approximated as
In the space of descriptors, each category’s domain (here, topological vs trivial insulator) is approximated as
the region of space within the convex hull of the corresponding training data. SISSO finds the low-dimensional descriptor yielding the minimum overlap between these convex regions. In practice, the algorithm is iterative. At the first iteration, in the SIS step, it selects the $k$ features which yield the smallest overlap when convex regions (segments encompassing all the data in one category) over the training data are constructed. In the first iteration the feature giving the smalles overlap is already the 1D model. At each subsequent iteration $i$, in the SIS step. $k$ new features that do the same for those training points which were in the overlap regions at the previous steps (i.e., the residuals). Then, in the SO step, all $i$-tuples of features selected combining in all possible ways the features selected in the SIS steps are ranked by the size of the overlap. The $i$-tuple with the smallest overlap is the $i$D model.
the region of space within the convex hull of the corresponding training data. SISSO finds the low-dimensional descriptor yielding the minimum overlap between these convex regions. In practice, the algorithm is iterative. At the first iteration, in the SIS step, it selects the $k$ features which yield the smallest overlap when convex regions (segments encompassing all the data in one category) over the training data are constructed. In the first iteration the feature giving the smalles overlap is already the 1D model. At each subsequent iteration $i$, in the SIS step. $k$ new features that do the same for those training points which were in the overlap regions at the previous steps (i.e., the residuals). Then, in the SO step, all $i$-tuples of features selected combining in all possible ways the features selected in the SIS steps are ranked by the size of the overlap. The $i$-tuple with the smallest overlap is the $i$D model.
In order to better identify a predictive model to classify unseen data point, at each dimension (iteration) a soft-margin support-vector machine <ahref="https://link.springer.com/article/10.1007%252FBF00994018"target="_blank">[C. Cortes & V. Vapnik, Machine learning 20, 273 (1995)]</a> is trained to define the separating hyperplanes. The resulting model is identified by the coefficents and intercept of the hyperplanes.
In order to better identify a predictive model to classify unseen data point, at each dimension (iteration) a soft-margin support-vector machine <ahref="https://link.springer.com/article/10.1007%252FBF00994018"target="_blank">[C. Cortes & V. Vapnik, Machine learning 20, 273 (1995)]</a> is trained to define the separating hyperplanes. The resulting model is identified by the coefficents and intercept of the hyperplanes.
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The idea demonstrated in this tutorial is to start from simple physical quantities ("primary features", here properties of the constituent free atoms such as Pauling electronegativity), to generate millions (or billions) of candidate formulas by applying arithmetic operations combining primary features. These candidate formulas constitute the so-called "feature space". Then, SISSO is used to select only a few of these formulas that explain the data.
The idea demonstrated in this tutorial is to start from simple physical quantities ("primary features", here properties of the constituent free atoms such as Pauling electronegativity), to generate millions (or billions) of candidate formulas by applying arithmetic operations combining primary features. These candidate formulas constitute the so-called "feature space". Then, SISSO is used to select only a few of these formulas that explain the data.
By clicking directly on "Run" below, i.e., with the default selection, you can reproduce the 2D map as published in <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">PRM 2020</a>. You can also select primary features and allowed operations (by clicking the check-boxes), as well as the SISSO rung (i.e., the number of iterations in the construction of the feature space), the number of features that are selected at each iteration of the SIS step, and the max number of dimensions of the model. The materials considered here have up to 5 different atomic species in the unit cell, with the prototype formula $AB-LMN$, where the cations $A,B \in \{\textrm{As, Sb, Bi} \}$ and the anions $L,M,N \in \{\textrm{S, Se, Te} \}$. We have therefore grouped the features to be selected into those for cations and anions. This means that by selecting, e.g., a cation feature, such feature is added to the primary feature set for both $A$ and $B$ elements, but either is treated singularly in the feature construction and SISSO optimization. After the features' and other settings' selection, press "Run". \
By clicking directly on "Run" below, i.e., with the default selection, you can reproduce the 2D map as published in <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">PRM 2020</a>. You can also select primary features and allowed operations (by clicking the check-boxes), as well as the SISSO rung (i.e., the number of iterations in the construction of the feature space), the number of features that are selected at each iteration of the SIS step, and the max number of dimensions of the model. The materials considered here have up to 5 different atomic species in the unit cell, with the prototype formula $AB-LMN$, where the cations $A,B \in \{\textrm{As, Sb, Bi} \}$ and the anions $L,M,N \in \{\textrm{S, Se, Te} \}$. We have therefore grouped the features to be selected into those for cations and anions. This means that by selecting, e.g., a cation feature, such feature is added to the primary feature set for both $A$ and $B$ elements, but either is treated singularly in the feature construction and SISSO optimization. After the features' and other settings' selection, press "Run". \
After the results are shown for all models from one dimensional to the max chosen dimension, you can press "Plot interactive map" to reveal a map of tetradymites' topological vs trivial insulators, for the highest-dimensional model. If the highest-dimensional model is 2D, the support-vector-machine separation line between the two phases is shown. For higher dimensional models, the 3rd and 4th dimensions can be visualized via the size or the color of the data-point markers. Intuitive drop-down menus allow to assign axes, markers, and colors, to the descriptor components of choice.
After the results are shown for all models from one dimensional to the max chosen dimension, you can press "Plot interactive map" to reveal a map of tetradymites' topological vs trivial insulators, for the highest-dimensional model. If the highest-dimensional model is 2D, the support-vector-machine separation line between the two phases is shown. For higher dimensional models, the 3rd and 4th dimensions can be visualized via the size or the color of the data-point markers. Intuitive drop-down menus allow to assign axes, markers, and colors, to the descriptor components of choice.
With the selection of "PRM2020" (or default selection) as SISSO rung, a special feature space is uploaded, which contains much fewer features than in the production calculation used in <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">PRM 2020</a>. This allows to reobtain in the notebook the same result in a reasonsable time. Still, the provided feature space contains thousands of the top ranked features and SISSO finds the best nD model.
With the selection of "PRM2020" (or default selection) as SISSO rung, a special feature space is uploaded, which contains much fewer features than in the production calculation used in <ahref="https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.2.083802"target="_blank">PRM 2020</a>. This allows to reobtain in the notebook the same result in a reasonsable time. Still, the provided feature space contains thousands of the top ranked features and SISSO finds the best nD model.
print("\nThe number of selected features per SIS iteration is bigger than the number of features available. Please reduce the number of selected features per SIS iteration (number of features generated / max number of dimensions) or increase the number of selected features and operations.")
print("\nThe number of selected features per SIS iteration is bigger than the number of features available. Please reduce the number of selected features per SIS iteration (number of features generated / max number of dimensions) or increase the number of selected features and operations.")
except:
except:
print('The present selection does not lead to the creation of any derived features in the highest selected rung, please select at least one binary or power operator, or reduce the maximum rung')
print('The present selection does not lead to the creation of any derived features in the highest selected rung, please select at least one binary or power operator, or reduce the maximum rung')
