Predictions of Lattice Parameters in NiTi High-Entropy Shape-Memory Alloys Using Different Machine Learning Models

Abstract

This work applied three machine learning (ML) models—linear regression (LR), random forest (RF), and support vector regression (SVR)—to predict the lattice parameters of the monoclinic B19′ phase in two distinct training datasets: previously published ZrO₂-based shape-memory ceramics (SMCs) and NiTi-based high-entropy shape-memory alloys (HESMAs). Our findings showed that LR provided the most accurate predictions for a_c, a_m, b_m, and c_m in NiTi-based HESMAs, while RF excelled in computing β_m for both datasets. SVR disclosed the largest deviation between the predicted and actual values of lattice parameters for both training datasets. A combination approach of RF and LR models enhanced the accuracy of predicting lattice parameters of martensitic phases in various shape-memory materials for stable high-temperature applications.

1. Introduction

The most widely used NiTi-based shape-memory alloys (SMAs) are known for their good shape memory and superelastic characteristics, based on the transformation between austenite and martensite phases [1,2]. However, their low martensitic transformation temperatures (TTs) below 100 °C are the most common challenges for high-temperature applications [3]. A composition engineering strategy in adding more alloying elements of Pt, Pd, Co, Cu, Zr, and Hf is believed to significantly increase TTs in NiTi-based high-entropy SMAs (HESMAs) [4,5,6]. In addition to high TTs, small thermal hysteresis is expected for stable high-temperature applications of the NiTi-based HESMAs.

The application of machine learning (ML) techniques has garnered significant attention for their potential in predicting and designing alloy compositions with optimized structural and functional properties [7,8,9,10,11]. Xue et al. applied different ML models to predict the transformation temperatures in NiTi-based SMAs [12]. Several ML approaches such as linear regression (LR) [9,13], random forest (RF) [11,14], neural network (NN) [11,15], and support vector regression (SVR) [11,16], have been commonly employed for computational prediction of crystal structures. Li et al. employed the MLatticeABC algorithm, a random forest model, to predict the lattice constants of crystal materials [11]. Pang et al. have recently proposed a linear regression (LR) model to successfully compute the lattice parameters of monoclinic and tetragonal structures in ZrO₂-based shape-memory ceramics (SMCs) [9]. Their approach has shed light on the composition design of new martensitic materials with desirable low thermal hysteresis and high TTs. Consequently, we are highly motivated to extend their promising strategy to our research field on the NiTi-based HESMAs.

The present study aimed to identify the most effective ML model for accurately predicting the lattice parameters of a monoclinic structure. We first followed previously published research to forecast the lattice parameters of monoclinic structures from a public training dataset of ZrO₂-based SMCs [9] using two more nonlinear models of random forest (RF) and support vector regression (SVR), besides LR. We then applied these three various ML models to the training dataset of the NiTi-based HESMAs to evaluate the most effective ML approach in predicting all lattice parameters of monoclinic B19′ (a_m, b_m, c_m, and β_m) and cubic B2 (a_c) structures.

2. Materials and Methods

Three distinct ML models were applied to predict the lattice parameters in the two training datasets of ZrO₂-based SMCs and NiTi-based HESMAs. LR, a fundamentally predictive method in statistics, is suitable for datasets with a roughly linear relationship between variables, and it can be either simple or multivariate. RF and SVR are considered for datasets with nonlinear relationships. RF is an ensemble learning method composed of numerous decision trees, with the final output being an aggregation of the individual trees’ predictions. This method is highly effective in evaluating feature importance, handling high-dimensional data, managing missing values, and demonstrating robust performance in diverse and unbalanced datasets. Therefore, RF enhances model diversity through random feature selection, effectively minimizing overfitting and increasing predictive accuracy. Conversely, SVR functions by mapping data to a high-dimensional space to identify the optimal fitting line or hyperplane. Employing kernel functions, SVR allows the construction of nonlinear relationships within the original feature space. A distinctive aspect of SVR is its ε-insensitivity band, tolerating minor prediction deviations without penalty, making it ideal for datasets with intricate or nonlinear associations. SVR’s regularization ensures strong generalization capabilities, effectively preventing overfitting. Both linear and nonlinear issues can be adeptly managed by selecting appropriate kernel functions. Like RF, SVR remains potent in scenarios where feature count surpasses sample size, ensuring reliable model predictions.

The feature for each data point was modeled as follows [9]:

$$x^{ave}={\displaystyle \sum _i{f}^i{x}^i}$$

(1)

where x^ave is the weighted average of all constituent elements, fⁱ is the mole fraction of a specific element i, and xⁱ is the feature value of element i.

Our training dataset of NiTi-based HESMAs included data points of 152 cubic and 176 monoclinic structures comprising constituent compositions of Ni, Ti, Hr, Hf, Cu, Fe, and Co in the Ni_50-x-y-zTi_50-h-kHf_hZr_kCu_xFe_yCo_z, which was collected from previously published studies [4,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. A set of 14 input features were electronegativity, electron affinity, number of valence electrons, Pettifor chemical scale, ionic radius, slater atomic radius, Clementi atomic radius, size difference, Waber atomic radius, atomic number, atomic mass, temperature, mixing enthalpy, and mixing entropy. The test root mean square error (RMSE) during cross-validation of the best feature set was used to evaluate the accuracy of ML models. Figure 1 illustrates the procedure for the computational prediction of lattice constants in the NiTi-based HESMAs.

3. Results

In accompanying LR used in published research [9], we applied two more different nonlinear models of RF and SVR to the public training dataset of ZrO₂-based SMCs to explore the most appropriate model in predicting the lattice parameters of monoclinic structure. Figure 2 shows the predicted versus actual lattice parameters fitted using three distinct models. A comparison of the test RMSE among the three ML models is presented in

Table 1. A comparison of test RMSE values among three ML models in predicting the lattice parameters of the ZrO₂-based SMCs and NiTi-based HESMAs.

Materials	Phase	Predicted Lattice Parameters	ML Model	Test RMSE	Test RMSE [9]
ZrO2-based SMCs	Monoclinic (B19′)	am	Linear regression	0.0041	0.0041
			Random forest	0.0044
			Support vector regression	0.04
	Monoclinic (B19′)	bm	Linear regression	0.0053	0.0053
			Random forest	0.0036
			Support vector regression	0.0381
	Monoclinic (B19′)	cm	Linear regression	0.0044	0.0045
			Random forest	0.0054
			Support vector regression	0.0314
	Monoclinic (B19′)	βm	Linear regression	0.0646	0.066
			Random forest	0.049
			Support vector regression	0.1273
NiTi-based HESMAs	Monoclinic (B19′)	am	Linear regression	0.0777
			Random forest	0.0919
			Support vector regression	0.1252
	Monoclinic (B19′)	bm	Linear regression	0.0326
			Random forest	0.0488
			Support vector regression	0.0785
	Monoclinic (B19′)	cm	Linear regression	0.0403
			Random forest	0.0437
			Support vector regression	0.0896
	Monoclinic (B19′)	βm	Linear regression	0.9571
			Random forest	0.7355
			Support vector regression	2.0401
	Cubic (B2)	ac	Linear regression	0.0076
			Random forest	0.0081
			Support vector regression	0.0520

. Lower test RMSE values indicate higher accuracy between the predicted and actual values. The test RMSE values of a_m, b_m, c_m, and β_m modeled by LR in the present work were in good accordance with those obtained in previously published work [9], seen in Table 1, demonstrating the reliability of our predictions in modeling the lattice parameters of the monoclinic phase.

The LR in Figure 2a–d revealed a closer fit between the predicted and actual values for a_m and c_m, in agreement with R² values. The RF in Figure 2e–h showed a better match for b_m and β_m, as also seen in the test RMSE and R². It is noted that a lower accuracy of b_m and β_m than a_m and c_m by LR was found to be modeled more accurately by RF. The predicted values of a_m, b_m, and c_m could not be distinguished from their actual corresponding values, and a very high value of the test RMSE in predicting β_m was obtained by SVR, suggesting that SVR was not suited to compute the lattice parameters of monoclinic structures, shown in Figure 2i–l and Table 1.

Both LR and RF were found to be more suitable ML models for forecasting all lattice parameters of ZrO₂-based SMCs, as there was a negligible discrepancy in the test RMSE between them. However, RF demonstrated higher accuracy in predicting b_m and β_m compared to LR, as evidenced by its lower test RMSE and higher R^² values.

To verify the potential of LR and RF ML models, we employed them to forecast the lattice parameters of monoclinic structures in the NiTi-based HESMAs in Figure 3. The predicted values of a_m, b_m, c_m, and a_c exhibited the best match with the actual values when using LR, as illustrated in Figure 3a–e. β_m demonstrated the highest accuracy between the predicted and actual values using RF, as shown in Figure 3f–j and corroborated by the test RMSE values in Table 1. The remarkable accuracy of RF in predicting β_m in the NiTi-based HESMAs was also observed in the ZrO₂-based SMCs. Conversely, SVR exhibited the poorest accuracy in modeling all lattice parameters of the monoclinic phase, as shown in Figure 3k–n. Moreover, no distinguishable prediction of a_c was computed by SVR in Figure 3o, which was similar to the predicted values of a_m, b_m, and c_m by SVR in the ZrO₂-based SMCs (Figure 2i–k). A trivial discrepancy of experimental lattice parameters between data points was ascribed to the predicted values indistinguishable from the actual values in SVR.

As shown in Figure 4, the lowest test RMSE values for β_m were achieved in both datasets, indicating that RF is the most effective approach for predicting β_m. Superior prediction of lattice angles using RF model has also been reported [11]. The test RMSE values of a_m, b_m, c_m, and a_c in LR were slightly lower than those in RF, especially in the training dataset of the NiTi-based HESMAs, suggesting the superiority of the LR model in computing a_m, b_m, c_m, and a_c.

4. Conclusions

Among the three various ML approaches, the LR and RF models achieved lower test RMSE values and better prediction performance of the lattice constants for both monoclinic and cubic phases in the NiTi-based HESMAs. There was a better match between the predicted and actual values of a_m, b_m, c_m, and a_c using LR in the NiTi-based HESMAs, which was also demonstrated in the published dataset of ZrO₂-based SMCs. Meanwhile, greater accuracy of the β_m was found using the RF model in both the ZrO₂-based SMCs and NiTi-based HESMAs. The concurrent combination of LR and RF models is expected to attain the highest accuracy in predicting all the unit-cell edge lengths and the inclination angles of the crystal structures. These initial predictive results are very crucial to the compositional design of new NiTi-based HESMAs with large recoverable strain and low density for their stable and cost-effective high-temperature applications in biomedicine, actuators, and the aerospace industry.