# A Two-Step Machine Learning Method for Predicting the Formation Energy of Ternary Compounds

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{2}Bi can be calculated as the composition-based weighted average of the formation energies of its constituent binary compounds, namely Zr

_{3}Bi and In

_{3}Bi, with formation energies of −0.184 eV/atom and +0.040 eV/atom, respectively, like so:

#### 2.1. Centered Adam

Algorithm 1 Centered Adam algorithm $\left(\right)$ |

Require: ${\beta}_{1},{\beta}_{2}\in \left(\right)open="["\; close="]">0,1$ ▷ hyperparametersRequire: $\nabla L:{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}\phantom{\rule{0.222222em}{0ex}}\mathrm{or}\phantom{\rule{0.222222em}{0ex}}L:{\mathbb{R}}^{d}\to \mathbb{R}$ ▷ loss function LRequire: ${\Theta}_{0}\in {\mathbb{R}}^{d}$ ▷ parameter of loss function All vector operations are element-wise ${m}_{0}:=\overrightarrow{0}\in {\mathbb{R}}^{d}$ ${v}_{0}:=\overrightarrow{0}\in {\mathbb{R}}^{d}$ $t:=0\in {\mathbb{N}}_{0}$ while t ≤ maximum iterations or sufficient convergence of ${\theta}_{t}$ do $t\leftarrow t+1$ ${g}_{t}\leftarrow \nabla L\left(\right)open="("\; close=")">{\Theta}_{t-1}$ ▷ calculate gradient ${m}_{t}\leftarrow {\beta}_{1}{m}_{t-1}+\left(\right)open="("\; close=")">1-{\beta}_{1}$ ▷ calculate first moment ${v}_{t}\leftarrow {\beta}_{2}{v}_{t-1}+\left(\right)open="("\; close=")">1-{\beta}_{2}$ ▷ calculate centered variance ${\widehat{m}}_{t}\leftarrow {m}_{t}/\left(\right)open="("\; close=")">1-{\beta}_{1}^{t}$ ▷ bias correction for first moment ${\widehat{v}}_{t}\leftarrow {v}_{t}/\left(\right)open="("\; close=")">1-{\beta}_{2}^{t}$ ▷ bias correction for centered variance ${\Theta}_{t}\leftarrow {\Theta}_{t-1}-{\alpha}_{t}{\widehat{m}}_{t}/\left(\right)open="("\; close=")">\sqrt{{\widehat{v}}_{t}}+\epsilon $ ▷ update parameters $\theta $ end while |

#### Bias Correction for Centered Adam

## 3. Computational Details

## 4. Results and Discussion

#### 4.1. Classification

#### 4.2. Regression

#### 4.3. MNIST Classifier

## 5. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Descriptors

#### Appendix A.1. Classification

- Heuristic formation energy: heuristically computed formation energy of the ternary compound (eV/atom);
- Average Pauling electronegativity: average of the Pauling electronegativity values of the constituent elements of the ternary compound;
- Average group on the periodic table: average of the group numbers of the elements;
- Average row on the periodic table: average of the row numbers of the elements;
- Average atomic mass: average of the atomic mass values of the constituent elements of the ternary compound (u);
- Average ionic radius: average of the ionic radius values of the constituent elements of the ternary compound (Å);
- Average electron affinity: average of the electron affinity values of the constituent elements of the ternary compound (eV);
- Average first ionization energy: average of the first ionization energy values of the constituent elements of the ternary compound (eV);
- Average van der Waals radius: average of the van der Waals radius values of the constituent elements of the ternary compound (Å);
- Ratio of the electronegativity of each element in the compound to the average value of electronegativity of the ternary compound;
- Ratio of the group number of each constituent element to the average for the ternary compound;
- Ratio of the row number of each constituent element to the average for the ternary compound;
- Ratio of the atomic mass of each constituent element to the average for the ternary compound;
- Ratio of the ionic radius of each constituent element to the average for the ternary compound;
- Ratio of the electron affinity of each constituent element to the average for the ternary compound;
- Ratio of the first ionization energy of each constituent element to the average for the ternary compound;
- Ratio of the van der Waals radius of each constituent element to the average for the ternary compound;
- Orbital fraction of valence electrons s: ratio of the number of s electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound;
- Orbital fraction of valence electrons p: ratio of the number of p electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound;
- Orbital fraction of valence electrons d: ratio of the number of d electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound;
- Orbital fraction of valence electrons f: ratio of the number of f electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound.

- Weight: the percentage of a binary compound’s formation energy that contributes to the formation energy of the ternary compound;
- Formation energy: the formation energy of the binary constituent of the ternary compound;
- Ratio of anions to cations: ratio of the number of anions to the number of cations in a binary compound;
- Ratio of the cations in a constituent binary compound to its number in the ternary compound;
- Ratio of the anions in a constituent binary compound to its number in the ternary compound;
- Average Pauling electronegativity: average of the Pauling electronegativities of the constituent elements of the binary compound;
- Average group on the periodic table: average of the group number of constituent elements;
- Average row on the periodic table: average of the row number of constituent elements;
- Average atomic mass: average of the atomic masses of the constituent elements of the binary compound;
- Average ionic radius: average of the ionic radii of the constituent elements of the binary compound;
- Average electron affinity: average of the electron affinity values of the constituent elements of the binary compound;
- Average first ionization energy: average of the first ionization energy values of the constituent elements;
- Average van der Waals radius: average of the van der Waals radius values of the constituent elements;
- Absolute difference in the Pauling electronegativity of the constituent elements;
- Absolute difference in the group number of the constituent elements;
- Absolute difference in the row number of the constituent elements;
- Absolute difference in the atomic mass of the constituent elements;
- Absolute difference in the ionic radius of the constituent elements;
- Absolute difference in the electron affinity value of the constituent elements;
- Absolute difference in the first ionization energy of the constituent elements;
- Absolute difference in the van der Waals radius of the constituent elements;
- Difference in the group number of the constituent elements;
- Difference in the row number of the constituent elements;
- Difference in the atomic mass of the constituent elements;
- Difference in the ionic radius of the constituent elements;
- Difference in the electron affinity value of the constituent elements;
- Difference in the first ionization energy of the constituent elements;
- Difference in the van der Waals radius of the constituent elements;
- Ratio of the Pauling electronegativity of the cations to that of the anions;
- Ratio of the group number of the cations to that of the anions;
- Ratio of the row number of the cations to that of the anions;
- Ratio of the atomic mass of the cations to that of the anions;
- Ratio of the ionic radius of the cations to that of the anions;
- Ratio of the electron affinity of the cations to that of the anions;
- Ratio of the first ionization energy of the cations to that of the anions;
- Ratio of the van der Waals radius of the cations to that of the anions;
- Orbital fraction of valence electrons s: ratio of the number of s electrons to the sum of the average of all electrons $\left(\right)$ of the binary compound;
- Orbital fraction of valence electrons p: ratio of the number of p electrons to the sum of the average of all electrons $\left(\right)$ of the binary compound;
- Orbital fraction of valence electrons d: ratio of the number of d electrons to the sum of the average of all electrons $\left(\right)$ of the binary compound;
- Orbital fraction of valence electrons f: ratio of the number of f electrons to the sum of the average of all electrons $\left(\right)$ of the binary compound.

#### Appendix A.2. Regression

- Average Pauling electronegativity: the average of the Pauling electronegativity values of the constituent elements of the ternary compound;
- Average group on the periodic table: average of the group numbers of the constituent elements;
- Average row on the periodic table: average of the row numbers of the constituent elements;
- Average atomic mass: average of the atomic masses of the constituent elements;
- Average ionic radius: average of the ionic radii of the constituent elements;
- Average electron affinity: average of the electron affinity values of the constituent elements;
- Average first ionization energy: average of the first ionization energies of the constituent elements;
- Average van der Waals radius: average of the van der Waals radii of the constituent elements;
- Ratio of the electronegativity of each element in the compound to the average value of the electronegativity of the ternary compound;
- Ratio of the group number of each element in the compound to the average value of the group number of the ternary compound;
- Ratio of the row number of each element in the compound to the average value of the row number of the ternary compound;
- Ratio of the atomic mass of each element in the compound to the average value of the atomic mass of the ternary compound;
- Ratio of the ionic radius of each element in the compound to the average value of the ionic radius of the ternary compound;
- Ratio of the electron affinity of each element in the compound to the average value of the electron affinity of the ternary compound;
- Ratio of the first ionization energy of each element in the compound to the average value of the first ionization energy of the ternary compound;
- Ratio of the van der Waals radius of each element in the compound to the average value of the van der Waals radius of the ternary compound;
- Orbital fraction of valence electrons s: the ratio of the number of s electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound;
- Orbital fraction of valence electrons p: the ratio of the number of p electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound;
- Orbital fraction of valence electrons d: the ratio of the number of d electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound;
- Orbital fraction of valence electrons f: the ratio of the number of f electrons to the sum of the average of all electrons $\left(\right)$ of the ternary compound.

## Appendix B. Dataset and Preprocessing

#### MNIST

## Appendix C. Computational Resources

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**Figure 1.**Diagram representing the two-step ML approach to predict the formation energies of the ternary compounds. (

**a**) The classification unit to predict if the heuristically computed formation energy of a ternary compound is accurate or not. (

**b**) The regression unit to predict the formation energy of a ternary compound.

**Figure 2.**Loss curves obtained for the network architecture of (117-117-117-1), the Adam optimizer, and various degrees of dropout.

Model Parameters | |
---|---|

Activation function $\left(\right)$ | ReLU |

Activation function $\left(\right)$ | Sigmoid |

Configuration | 117-117-1 |

Loss function | Binary cross-entropy |

Training parameters | |

Batch size | 128 |

Training cycles | 512 |

Validation split | 0.1 |

Test split | 0.1 |

Optimizer parameters | |

Optimizer | Adam |

Learning rate | 0.001 |

$\epsilon $ | $1/{10}^{-7}$ |

${\beta}_{1}$ | 0.9 |

${\beta}_{2}$ | 0.999 |

Model Parameters | |
---|---|

Activation function $\left(\right)$ | ReLU/Sigmoid |

Activation function $\left(\right)$ | Linear |

configuration | $\left(\right)$/ $\left(\right)$/ $\left(\right)$ |

Loss function | Log cosh/Mean squared error |

Training parameters | |

Batch size | 100/1000/10,000 |

Training cycles | 500 |

Validation split | 0.2 |

Optimizer parameters | |

Optimizer | Adam/cAdam |

Learning rate | 0.01/0.001 |

$\epsilon $ | ${10}^{-7}$ |

${\beta}_{1}$ | 0.9 |

${\beta}_{2}$ | 0.999 |

Model Parameters | |
---|---|

Activation function $\left(\right)$ | ReLU/Sigmoid |

Activation function $\left(\right)$ | Softmax/Sigmoid |

configuration | $\left(\right)$/ $\left(\right)$/ $\left(\right)$ |

Loss function | Categorical cross-entropy/ Mean squared error |

Training parameters | |

Batch size | 100/1000/10,000 |

Training cycles | 5/25/50 |

Validation split | 0.2 |

Optimizer parameters | |

Optimizer | Adam/cAdam |

Learning rate | 0.1/0.01/0.001 |

$\epsilon $ | 1/${10}^{-7}$ |

${\beta}_{1}$ | 0.9 |

${\beta}_{2}$ | 0.999 |

**Table 4.**Performance of bnn-c compared with conventional ML classifiers: logistic regression (LR), linear discriminant analysis (LDA), random forest (RF), k-nearest neighbors (KNN), and adaboost (AB).

Model | Accuracy | Precision | Recall | AUROC |
---|---|---|---|---|

LR | 0.79 | 0.73 | 0.57 | 0.85 |

LDA | 0.79 | 0.72 | 0.56 | 0.85 |

KNN | 0.82 | 0.72 | 0.68 | 0.86 |

RF | 0.74 | 0.74 | 0.29 | 0.72 |

AB | 0.81 | 0.74 | 0.63 | 0.87 |

bnn-c | 0.86 | 0.78 | 0.77 | 0.93 |

Layout | Accuracy (%) | Precision | Recall | AUROC |
---|---|---|---|---|

117-117-1 | 86.92 | 0.79 | 0.78 | 0.93 |

117-300-1 | 87.14 | 0.78 | 0.80 | 0.93 |

117-55-1 | 86.25 | 0.77 | 0.79 | 0.93 |

117-117-117-1 | 87.42 | 0.81 | 0.78 | 0.93 |

117-117-117-117-1 | 87.06 | 0.80 | 0.77 | 0.93 |

**Table 6.**Results obtained for the network architecture of (117-117-117-1) with various degrees of dropouts.

Dropout in % | Accuracy (%) | Precision | Recall | AUROC |
---|---|---|---|---|

50 | 87.82 | 0.86 | 0.72 | 0.93 |

30 | 88.24 | 0.82 | 0.79 | 0.94 |

20 | 88.29 | 0.84 | 0.77 | 0.94 |

10 | 88.51 | 0.83 | 0.78 | 0.94 |

**Table 7.**The best 5 of the 144 parameter combinations and results for the regression task. The units for the MAE values are eV/atom. Results are sorted by final validation loss.

Parameter | Best Values | ||||
---|---|---|---|---|---|

Final val loss (MAE) | 0.1116 | 0.11208 | 0.11416 | 0.11493 | 0.11534 |

Test loss (MAE) | 0.11389 | 0.11298 | 0.11493 | 0.11445 | 0.11539 |

Test loss | 0.012971 | 0.012764 | 0.0065903 | 0.0065354 | 0.0066426 |

Training time (in s) | 44.379 | 566.85 | 577.92 | 80.447 | 44.17 |

Neurons per layer | (50, 10) | (50, 10) | (50, 10) | (30, 30, 10) | (40, 20) |

Activation functions | Sigmoid | Sigmoid | Sigmoid | Sigmoid | Sigmoid |

Last activation function | Linear | Linear | Linear | Linear | Linear |

Loss function | MSE | MSE | Log cosh | Log cosh | Log cosh |

Number of epochs | 500 | 500 | 500 | 500 | 500 |

Batch size | 1000 | 100 | 100 | 1000 | 1000 |

Optimizer | Adam | cAdam | cAdam | cAdam | Adam |

Learning rate | 0.01 | 0.001 | 0.001 | 0.01 | 0.01 |

$\epsilon $ | ${10}^{-7}$ | ${10}^{-7}$ | ${10}^{-7}$ | ${10}^{-7}$ | ${10}^{-7}$ |

**Table 8.**The best 5 of the 2592 parameter combinations and results for NNs trained on the MNIST dataset. Results are sorted by final validation accuracy. Every network was trained five times, and the average values are listed.

Parameter | Best Values | ||||
---|---|---|---|---|---|

Final validation accuracy | 0.97 | 0.9691 | 0.96905 | 0.96885 | 0.9688 |

Test accuracy | 0.97006 | 0.96906 | 0.9681 | 0.97048 | 0.97084 |

Final validation loss | 0.01339 | 0.0066957 | 0.01449 | 0.0070507 | 0.96845 |

Training time (in s) | 46.666 | 43.602 | 86.105 | 86.874 | 23.122 |

Neurons per layer | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

Activation function $\left(\right)$ | ReLU | ReLU | ReLU | ReLU | ReLU |

Activation function $\left(\right)$ | Sigmoid | Sigmoid | Sigmoid | Softmax | Softmax |

Loss function | Cat-cross | Cat-cross | Cat-cross | Cat-cross | Cat-cross |

Training cycles | 50 | 25 | 50 | 50 | 25 |

Batch size | 100 | 100 | 100 | 100 | 100 |

Optimizer | Adam | cAdam | cAdam | cAdam | Adam |

Learning rate | 0.001 | 0.1 | 0.1 | 0.1 | 0.1 |

$\epsilon $ | ${10}^{-7}$ | 1.0 | 1.0 | 1.0 | 1.0 |

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## Share and Cite

**MDPI and ACS Style**

Rengaraj, V.; Jost, S.; Bethke, F.; Plessl, C.; Mirhosseini, H.; Walther, A.; Kühne, T.D.
A Two-Step Machine Learning Method for Predicting the Formation Energy of Ternary Compounds. *Computation* **2023**, *11*, 95.
https://doi.org/10.3390/computation11050095

**AMA Style**

Rengaraj V, Jost S, Bethke F, Plessl C, Mirhosseini H, Walther A, Kühne TD.
A Two-Step Machine Learning Method for Predicting the Formation Energy of Ternary Compounds. *Computation*. 2023; 11(5):95.
https://doi.org/10.3390/computation11050095

**Chicago/Turabian Style**

Rengaraj, Varadarajan, Sebastian Jost, Franz Bethke, Christian Plessl, Hossein Mirhosseini, Andrea Walther, and Thomas D. Kühne.
2023. "A Two-Step Machine Learning Method for Predicting the Formation Energy of Ternary Compounds" *Computation* 11, no. 5: 95.
https://doi.org/10.3390/computation11050095