J. Mater. Sci. Technol. ›› 2022, Vol. 103: 113-120.DOI: 10.1016/j.jmst.2021.05.076
• Research Article • Previous Articles Next Articles
Xin Lia,b, Guangcun Shana,b,*(), C.H. Shekb
Received:
2021-02-16
Revised:
2021-05-18
Accepted:
2021-05-18
Published:
2022-03-20
Online:
2021-08-27
Contact:
Guangcun Shan
About author:
* Beihang University, School of Instrumentation Science and Opto-electronics Engineering, No.37, Xueyuan Road, Haidian, Beijing 100191, China. E-mail address: gshan2-c@my.cityu.edu.hk (G. Shan).Xin Li, Guangcun Shan, C.H. Shek. Machine learning prediction of magnetic properties of Fe-based metallic glasses considering glass forming ability[J]. J. Mater. Sci. Technol., 2022, 103: 113-120.
Feature no. | Feature name | Description |
---|---|---|
1 | Supercooled liquid region [ | ΔTx =Tx -Tg |
2 | Theoretical melting point (Tm) | ${{T}_{\text{m}}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}{{T}_{\text{m}i}}$ |
3 | Theoretical density [ | $\rho =100/\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{w}_{i}}}{{{\rho }_{i}}} \right)$ |
4 | Mean atom radius ($\bar{r}$) | $\bar{r}=\sum_{i=1}^{n} c_{i} r_{i}$ |
5 | Theoretical molar volume [ | $V=\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{c}_{i}}{{m}_{i}}}{{{\rho }_{i}}}$ |
6 | Atomic size difference [ | $\delta=\sqrt{\sum_{i=1}^{n} c_{i}\left(1-\frac{r_{i}}{\bar{r}}\right)^{2}}$ |
7 | Configurational entropy [ | $\text{ }\!\!\Delta\!\!\text{ }{{S}_{c}}=-\text{R}\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}\ln {{c}_{i}}$ |
8 | Electronegativity [ | $\chi=\sum_{i=1}^{n} c_{i} \chi_{i}$ |
9 | Valence electron concentration [ | $VEC=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}{{\left( \text{VEC} \right)}_{i}}$ |
10 | Valence electron concentration without FeCoNi [ | $VE{C}'=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}{{\left( \text{VEC} \right)}_{i}}-{{c}_{\text{Fe}}}{{\left( \text{VEC} \right)}_{\text{Fe}}}-{{c}_{\text{Co}}}{{\left( \text{VEC} \right)}_{\text{Co}}}-{{c}_{\text{Ni}}}{{\left( \text{VEC} \right)}_{\text{Ni}}}$ |
11-15 | Atomic ratio of Fe, Co, Ni, B, Si | cFe, cCo, cNi,cB, cSi |
Table 1 Description of all the features in the Fe-based MGs dataset.
Feature no. | Feature name | Description |
---|---|---|
1 | Supercooled liquid region [ | ΔTx =Tx -Tg |
2 | Theoretical melting point (Tm) | ${{T}_{\text{m}}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}{{T}_{\text{m}i}}$ |
3 | Theoretical density [ | $\rho =100/\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{w}_{i}}}{{{\rho }_{i}}} \right)$ |
4 | Mean atom radius ($\bar{r}$) | $\bar{r}=\sum_{i=1}^{n} c_{i} r_{i}$ |
5 | Theoretical molar volume [ | $V=\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{c}_{i}}{{m}_{i}}}{{{\rho }_{i}}}$ |
6 | Atomic size difference [ | $\delta=\sqrt{\sum_{i=1}^{n} c_{i}\left(1-\frac{r_{i}}{\bar{r}}\right)^{2}}$ |
7 | Configurational entropy [ | $\text{ }\!\!\Delta\!\!\text{ }{{S}_{c}}=-\text{R}\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}\ln {{c}_{i}}$ |
8 | Electronegativity [ | $\chi=\sum_{i=1}^{n} c_{i} \chi_{i}$ |
9 | Valence electron concentration [ | $VEC=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}{{\left( \text{VEC} \right)}_{i}}$ |
10 | Valence electron concentration without FeCoNi [ | $VE{C}'=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{c}_{i}}{{\left( \text{VEC} \right)}_{i}}-{{c}_{\text{Fe}}}{{\left( \text{VEC} \right)}_{\text{Fe}}}-{{c}_{\text{Co}}}{{\left( \text{VEC} \right)}_{\text{Co}}}-{{c}_{\text{Ni}}}{{\left( \text{VEC} \right)}_{\text{Ni}}}$ |
11-15 | Atomic ratio of Fe, Co, Ni, B, Si | cFe, cCo, cNi,cB, cSi |
Fig. 3. K-fold cross-validation predictive performance in training dataset evaluated by (a) R2, (b) MAPE and (c) RMSE for the ML models based on XGBoost, ANN and RF, respectively. XGBoost, ANN and RF denote ML models trained by the original 15 features, and XGBoost', ANN' and RF' denote ML models trained by the reduced 12 features. The insets of (a)-(c) plot the averaged k-fold cross-validation results of the three types of ML models with the original 15 features and the reduced 12 features, respectively. Final predictive performance calculated on test dataset for (d) XGBoost, (e) ANN and (f) RF with the reduced 12 features, respectively.
Metrics | XGBoost | XGBoost' | ANN | ANN' | RF | RF' |
---|---|---|---|---|---|---|
R2 | 0.910±0.024 | 0.910±0.021 | 0.913±0.087 | 0.908±0.097 | 0.836±0.069 | 0.845±0.066 |
MAPE (%) | 5.64±0.93 | 5.68±0.86 | 5.53±2.53 | 5.47±2.71 | 8.46±3.09 | 8.15±2.97 |
RMSE (T) | 0.093±0.015 | 0.093±0.012 | 0.079±0.039 | 0.080±0.038 | 0.121±0.037 | 0.118±0.036 |
Table 2 Evaluation metrics of the averaged 10-fold cross-validation results on the training dataset.
Metrics | XGBoost | XGBoost' | ANN | ANN' | RF | RF' |
---|---|---|---|---|---|---|
R2 | 0.910±0.024 | 0.910±0.021 | 0.913±0.087 | 0.908±0.097 | 0.836±0.069 | 0.845±0.066 |
MAPE (%) | 5.64±0.93 | 5.68±0.86 | 5.53±2.53 | 5.47±2.71 | 8.46±3.09 | 8.15±2.97 |
RMSE (T) | 0.093±0.015 | 0.093±0.012 | 0.079±0.039 | 0.080±0.038 | 0.121±0.037 | 0.118±0.036 |
Fig. 4. (a) Schematic of the ANN architecture designed to predict Bs of Fe-based MGs based on the selected features. (b) RMSE of ANN with different hyperparameters. (c) Training and validation loss during the learning process of ANN with tuned hyperparameters.
Fig. 5. Hyperparameters tuning results for XGBoost and RF evaluated by R2. Tuning process of (a) the n_estimators in XGBoost and RF, (b) the max_depth and min_child_weight in XGBoost, and (c) the max_depth and min_samples_split of RF. (d) Feature importance ranking derived from the trained XGBoost model. Importance scores were normalized by dividing the maximum score, and the maximum score was set to 100.
Feature space no. | Feature space | a R2 | b F(ΔTx) |
---|---|---|---|
1 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi,cB, cSi | 0.910 ± 0.021 | 100 |
2 | ΔTx, ρ, Tm, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi | 0.911 ± 0.015 | 91.5 |
3 | ΔTx, ρ, Tm, χ, r¯, cFe, cCo, cNi, cB, cSi | 0.873 ± 0.025 | 80.0 |
4 | ΔTx, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi | 0.902 ± 0.023 | 88.8 |
5 | ΔTx, ρ, Tm, VEC, VEC', cFe, cCo, cNi, cB, cSi | 0.907 ± 0.026 | 88.6 |
6 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, δ, ΔSc, V | 0.910 ± 0.024 | 100 |
7 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi,cP, cC | 0.914 ± 0.014 | 100 |
8 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, cP, cC, cNb, cMo | 0.916 ± 0.016 | 100 |
9 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, cZr, cGa cAl, cDy | 0.911 ± 0.015 | 98.5 |
10 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, cCu,cCr, cY, cNd | 0.911 ± 0.016 | 97.9 |
Table 3 The F-scores of ΔTx given by XGBoost models trained on different feature spaces. The first one is the XGBoost model trained on the reduced 12 features.
Feature space no. | Feature space | a R2 | b F(ΔTx) |
---|---|---|---|
1 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi,cB, cSi | 0.910 ± 0.021 | 100 |
2 | ΔTx, ρ, Tm, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi | 0.911 ± 0.015 | 91.5 |
3 | ΔTx, ρ, Tm, χ, r¯, cFe, cCo, cNi, cB, cSi | 0.873 ± 0.025 | 80.0 |
4 | ΔTx, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi | 0.902 ± 0.023 | 88.8 |
5 | ΔTx, ρ, Tm, VEC, VEC', cFe, cCo, cNi, cB, cSi | 0.907 ± 0.026 | 88.6 |
6 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, δ, ΔSc, V | 0.910 ± 0.024 | 100 |
7 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi,cP, cC | 0.914 ± 0.014 | 100 |
8 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, cP, cC, cNb, cMo | 0.916 ± 0.016 | 100 |
9 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, cZr, cGa cAl, cDy | 0.911 ± 0.015 | 98.5 |
10 | ΔTx, ρ, Tm, VEC, VEC', χ, $\bar{r}$, cFe, cCo, cNi, cB, cSi, cCu,cCr, cY, cNd | 0.911 ± 0.016 | 97.9 |
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