Machine Learning-Assisted Discovery of Empirical Rule for Martensite Transition Temperature of Shape Memory Alloys

Abstract

Shape memory alloys (SMAs) derive their unique functional properties from martensitic transformations, with the martensitic transformation temperature (T_M) serving as a key design parameter. However, existing empirical rules, such as the valence electron concentration (VEC) and lattice volume (V) criteria, are typically restricted to specific alloy families and lack general applicability. In this work, we used a data-driven methodology to find a generalizable empirical formula for T_M in SMAs by combining high-throughput first-principles calculations, feature engineering, and symbol regression techniques. Key factors influencing T_M were first identified and a predictive machine learning model was subsequently trained based on these features. Furthermore, an empirical formula of T_M = $82(\overline{\rho }·\overline{MP})-700$ was derived, where $\overline{\rho }$ and $\overline{MP}$ represent the weight-average value of density and melting point, respectively. The empirical formula exhibits strong generalizability across a wide range of SMAs, such as NiMn-based, NiTi-based, TiPt-based, and AuCd-based SMAs, etc., offering practical guidance for the compositional design and optimization of shape memory alloys.

1. Introduction

Shape memory alloys (SMAs) are a class of functional materials renowned for their unique phase transformation behaviors, including the shape memory effect [1], superelasticity [2], and elastocaloric effect [3]. These functionalities originate from martensitic transformation. Among various properties, the martensitic transformation temperature (T_M) is a critical parameter in SMA design, as it directly determines the operational temperature window of the material. For example, Ni₂MnGa, one of the most extensively studied shape memory alloys, exhibits a T_M around 220 K, thereby restricting its application in ambient or high-temperature environments such as actuators or sensors. In contrast, cryogenic applications like solid-state cooling require the T_M to be substantially below room temperature. Therefore, precise control and reliable prediction of the T_M are essential for tailoring the functional properties of SMAs for diverse applications.

Several strategies have been developed to tune T_M, including modifications of chemical composition, microstructure, atomic ordering, and vacancy concentration [4,5,6,7], with compositional tuning emerging as the most effective approach. Empirical rules, such as the valence electron concentration (VEC) rule [3,8,9,10] where the T_M generally increases with higher VEC, and the lattice volume rule [11] where a reduced austenite lattice volume tends to raise the T_M, have been widely employed. Based on these principles, empirical models like T_M = 705(VEC) − 5067 for Ni-Mn-Ga alloys have been proposed [12]. Other composition-related empirical formulas include T_M = 1000 × (Ni% − 50) for Ni-Ti with 50~60 at.% [13,14], and for Ti-Nb where each 1 at.% increase in Nb content decreases the T_M by approximately 43 K [15,16]. However, these models often suffer from limited generalizability across different alloy systems due to their strong dependence on specific material families. Therefore, identifying universal key factors that govern T_M across various shape memory alloys (SMAs) remains a critical challenge. Recently, machine learning (ML) has emerged as a powerful technique to model complex, nonlinear relationships and predict material properties. Although previous studies have applied ML to forecast T_M in SMAs, they often rely on computationally expensive features derived from density functional theory (DFT) calculations or experimental inputs [17], or are restricted to specific alloy families [18,19]. Therefore, developing fast and generalizable models to predict T_M based on readily available information holds significant theoretical and practical value.

To address this challenge, we introduced a big data-driven machine learning framework based on elemental and simple substance properties—a state-of-the-art approach for resolving multifactorial materials design problems [18,20,21,22]. In this work, high-throughput ab initio calculations were employed alongside extensive data collection from the literature to expand the training dataset. Subsequently, a five-step descriptor screening process was implemented to identify the key factors governing T_M. Based on these critical features, a highly accurate random forest machine learning model was developed to predict T_M. Furthermore, a simple yet robust empirical relationship between T_M and the identified parameters was discovered, exhibiting exceptional generalization performance across a wide range of SMAs.

2. Methods

A random forest (RF) machine learning model, as implemented in the Scikit-Learn software package (version 1.3.2) [23] was employed in this study. The number of decision trees was set to 50. During model construction, the bootstrap sampling strategy [24] was adopted, and the maximum tree depth was left unconstrained to allow for full data-driven learning. To mitigate the risk of overfitting, 10-fold cross-validation was performed [25]. Model performance was evaluated using the mean absolute error (MAE), root mean square error (RMSE) and mean square error (MSE). To identify the dominant factors influencing T_M, a comprehensive feature selection strategy was employed, involving the variance screening, the Pearson correlation screening [26], the univariate screening [27,28], the recursive elimination screening [29], and the exhaustive screening [21]. Detailed descriptions of these methods can be found in Ref. [22].

High-throughput ab initio calculations were conducted using spin-polarized density functional theory within the framework of the projector augmented-wave (PAW) method [30], as implemented in the Vienna Ab initio Simulation Package (VASP, version 6.4.2) [31]. The exchange-correlation energy was treated using the Perdew-Burke-Enzerh (PBE) functional within the generalized gradient approximation (GGA) [32,33]. The plane-wave kinetic energy cutoff, total energy convergence criterion, and force convergence criterion were set to 600 eV, 10⁻⁵ eV, and 10⁻⁴ eV/Å, respectively. A 16-atom supercell was adopted for all calculations. The first Brillouin zone was sampled using a Monkhorst-Pack [34] k-point grid of 13 × 13 × 13.

3. Results and Discussion

3.1. Dataset

Based on the literature [17,18,35], a dataset comprising 82 experimental data points of the T_M in various shape memory alloys was collected. These systems include NiMn-based, NiTi-based, AgCd-based, AuCd-based, PdTi-based, PtTi-based, CoNi-based, TiNb-based, TiTa-based, ZnAuCu-based, TiAu-based, and MgSc-based alloys, effectively covering nearly all known shape memory alloy families to date (details in Appendix A 

Table A1. Chemical compositions and T_M values of alloys from experimental (Exp.) in the literature.

Compositions	TM (K)	Ref.	Compositions	TM (K)	Ref.
Ti500Ni452Cu10Fe38	248	[14]	Ti500Ni468Cu9Fe20Pd3	290	[14]
Ti500Ni444Cu20Fe36	239	[14]	Ti500Ni442Cu19Fe38Pd1	243	[14]
Ti500Ni428Cu40Fe32	231	[14]	Ti500Ni467Cu8Fe23Pd2	282	[14]
Ti500Ni436Cu30Fe34	242	[14]	Ti500Ni442Cu19Fe39	242	[14]
Ti500Ni460Cu0Fe40	234	[14]	Ti500Ni481Cu2Fe15Pd2	302	[14]
Ti500Ni445Cu15Fe30Pd10	223	[14]	Ti500Ni445Cu16Fe37Pd2	244	[14]
Ti500Ni340Cu130Pd30	299	[14]	Ti500Ni482Cu6Fe9Pd3	320	[14]
Ti500Ni340Cu100Pd60	303	[14]	Ti500Ni465Cu11Fe22Pd2	284	[14]
Ti500Ni340Cu120Pd40	311	[14]	Ti500Ni483Fe16Pd1	302	[14]
Ti500Ni420Cu50Fe30	226	[14]	Ti500Ni490Fe2Pd8	358	[14]
Ti500Ni350Cu120Pd30	321	[14]	Ti500Ni486Fe9Pd5	332	[14]
Ti500Ni400Pd100	279	[14]	Ti500Ni435Cu20Fe45	226	[14]
Ti500Ni340Cu140Pd20	319	[14]	Ti500Ni250Pd250	456	[14]
Ti500Ni340Cu160	333	[14]	Ag51Cd49	48	[13]
Ti500Ni440Cu10Fe10Pd40	239	[14]	Au505Cd495	308	[13]
Ti500Ni340Pd160	331	[14]	Au50Ti50	843	[13]
Ti500Ni364Cu120Fe16	286	[14]	Pd50Ti50	761	[13]
Ti500Ni412Cu60Fe28	220	[14]	Pt50Ti50	1291	[13]
Ti500Ni380Cu100Fe20	274	[14]	Ni50Ti50	252	[13]
Ti500Ni348Cu140Fe12	271	[14]	Co50Ni24Ga26	336	[13]
Ti500Ni396Cu80Fe24	258	[14]	Ti505Ni245Pd250	454	[13]
Ti500Ni500	364	[14]	Ti505Ni245Pt250	721	[13]
Ti500Ni440Cu20Fe40	220	[14]	Ti50Pt25Ir25	1395	[13]
Ti500Ni445Cu21Fe34	248	[14]	Ni50Mn25Ga25	278	[13]
Ti500Ni440Cu23Fe36Pd1	241	[14]	Ti75Nb22	476	[13]
Ti500Ni404Cu46Fe10Pd40	270	[14]	Ti75Nb25	381	[13]
Ti500Ni457Fe43	231	[14]	Ti75Ta25	558	[13]
Ti500Ni458Fe42	234	[14]	Mg81Sc19	83	[13]
Ti500Ni428Cu36Fe28Pd8	234	[14]	Ag51Cd49	223	[30]
Ti500Ni439Cu21Fe40	233	[14]	Au50Cd50	308	[30]
Ti500Ni445Cu17Fe37Pd1	245	[14]	Ti50Au50	843	[30]
Ti500Ni457Cu12Fe30Pd1	264	[14]	Zn45Au30Cu25	235	[30]
Ti500Ni440Cu23Fe37	234	[14]	Co50Ni24Ga26	336	[30]
Ti500Ni451Cu14Fe35	248	[14]	Ti76Nb24	338	[30]
Ti500Ni446Cu19Fe34Pd1	250	[14]	Ni50Mn25Ga25	278	[30]
Ti500Ni438Cu26Fe36	239	[14]	Ti50Ni50	252	[30]
Ti500Ni439Cu15Fe46	228	[14]	Ti50Pd50	761	[30]
Ti500Ni445Cu19Fe34Pd2	243	[14]	Ti50Pt50	1291	[30]
Ti500Ni460Cu11Fe28Pd1	261	[14]	Ti50.5Ni24.5Pd25	454	[30]
Ti500Ni438Cu20Fe41Pd1	231	[14]	Ti50.5Ni24.5Pt25	721	[30]
Ti500Ni439Cu20Fe40Pd1	232	[14]	Ti50Pt25Ir25	1395	[30]

).

To further expand the dataset, high-throughput ab initio calculations were performed on 109 kinds of L2₁ structured Ni₂MnGa_1−xZ_x (x = 0.25, 0.5, 0.75 and 1) alloys, where Z represents various common elements from the periodic table, as illustrated in Figure 1. The Bain distortion model was employed to compute the energy difference (ΔE) between the martensitic and austenitic phases (see calculation details in Ref. [22]). The T_M was then estimated using the relation $T_M={\displaystyle \frac{∆E}{n{k}_B}}$ [36,37] where n is the number of atoms in the supercell, and k_B is the Boltzmann constant which connects the macroscopic temperature and microscopic energy parameters. Figure 1 displays the calculated martensitic transformation temperatures (T_M) for Ni₂MnGa_1−xZ_x alloys, where Z spans 35 different substituting elements and x takes values of 0.25, 0.5, 0.75, and 1.00. For each element, corresponding T_M values are shown beneath the elemental symbol. A backslash symbol (“\”) indicates compositions where the austenitic phase is energetically more stable than the martensitic phase even at 0 K, corresponding to a negative ΔE. In these cases, martensitic transformation does not occur, and the alloy remains in a stable austenitic state.

To assess the applicability of known empirical rules across this heterogeneous dataset, the relationships between T_M and two classic descriptors were examined: Figure 2a,b shows the correlations of the T_M with the valence electron concentration (VEC) and lattice volume (V), respectively. Although prior studies have demonstrated that an increase in VEC or a decrease in lattice volume correlates tends to elevate the T_M within specific alloy families [3,8,9,10,11,12] these trends do not consistently extend across a broader and more diverse dataset. In our analysis, neither VEC nor V exhibited a strong or reliable correlation with T_M when considering the full dataset, thereby underscoring the limited generalizability of these traditional empirical rules. In addition, the transformation temperatures span a wide range— from approximately 100 K to 1300 K—reflecting the intrinsic diversity of thermal responses among different SMA compositions. This rich and representative dataset thus provides a robust foundation for the development of data-driven predictive models and enables cross-system generalization.

3.2. Feature Engineering

To represent the different alloys in the training set, a total of 64 constituent element-associated parameters were initially considered, given that the structural phase transition as well as the physical and chemical properties of compounds are predominantly determined by their constituent elements [38,39,40,41]. Specifically, the weight-average value and standard deviation of 31 types of element or elemental-substance-associated parameters were calculated for each alloy [39,40,42], as summarized in

Table 1. Adopted elemental, simple substance, and phase stability-associated features.

Feature Category	Feature Description	Abbreviation
Elemental properties	Atomic number	Z
	Periodic table column	C
	Atomic weight	AW
	Mendeleev number	MN
	Periodic table row	PR
	Atomic radius	AR
	Number of s valence electrons	Ns
	Number of p valence electrons	Np
	Number of d valence electrons	Nd
	Number of f valence electrons	Nf
	Number of total valence electrons	Nt
	Number of unfilled s states	Us
	Number of unfilled p states	Up
	Number of unfilled d states	Ud
	Number of unfilled f states	Uf
	Number of total unfilled states	Ut
Simple substance properties	Melting point	MP
	Boiling point	BP
	Heat capacity	HC
	Heat fusion	HF
	Pauling Electronegativity	EN
	Covalent radius	CR
	Ionic radius	IR
	Density	$\overline{\rho }$
	Magnetic moment	M
	Volume	V
	Band gap	Gap
	First ionization energy	E
	Space group number	SG
	Bulk modulus	B
	Shear modulus	G
Phase stability properties	Mixing entropy	$∆S_{mix}$
	Atomic size difference	$\delta$

.

In addition to these elemental descriptors, two phase stability-related parameters, i.e., mixing entropy $∆S_{mix}$ and atom radium difference $\delta$ [20], were included. Considering that shape memory alloys generally possess mixed metallic, covalent, and ionic chemical bonds [43,44], three different atom radius definitions, including atomic radius, covalent radius, and ionic radius, were employed to estimate $\delta$. In total, 64 descriptors were utilized to featurize the alloys within the training dataset.

3.3. Screening of Key Features

To eliminate the redundant or insignificant descriptors and to identify the key materials parameters governing T_M, a successive five-step screening procedure was employed, as illustrated in Figure 3. This process combined a series of complementary descriptor selection methods applied sequentially: variance screening (Step 1) [45], Pearson correlation screening (Step 2) [26], univariate feature selection (Step 3) [27,28], recursive feature elimination (Step 4) [29], and exhaustive screening (Step 5) (more details in Appendix B and Ref. [21]). The detailed description of these selection methods can be found in Ref. [22].

First, redundancy among descriptors was eliminated using the variance method (Step 1) [45], by removing descriptors that exhibited identical values across all data points in the training set. Second, strongly correlated descriptors were identified through Pearson correlation screening (Step 2) [26]. The Pearson correlation coefficient (r) between every pair of descriptors was computed, and for each pair with |r| > 0.75, one descriptor was removed to reduce redundancy. For instance, the atomic number (Z) and atomic weight (AW) showed a perfect correlation (r = 1); thus, one of the two, such as the atomic number in this case, was eliminated. Following this screening process, 27 descriptors were retained for further analysis.

Figure 4a–c presents the results of feature selection using univariate filtering, recursive feature elimination (RFE), and exhaustive search, respectively. Following these three successive screening steps, the number of features was reduced from the initial 66 to 16, and finally to 6. This systematic feature selection process effectively narrowed the original set of candidate descriptors down to six key features, which are summarized in

Table 2. The selected descriptors for the univariate screening, recursive elimination, and exhaustive screening methods for the dataset.

			Descriptors (Abbr.)	Descriptors (Abbr.)
Univariate screening			Standard deviation of melting point (σMP)	Weight-average value of p valence electron ($\overline{VE}$p)
			Weight-average value of valence electron ($\overline{VE}$)	Weight-average value of s valence electron ($\overline{VE}$s)
			Weight-average value magnetic moment ($\overline{Mag}$)	Weight-average value of bulk modulus ($\overline{G}$)
			Standard deviation of Mendeleev number (σMN)	Standard deviation of heat fusion (σHF)
			Standard deviation of valence electron (VEsd)	Standard deviation of s valence electron (σVEs)
	Recursive elimination screening	Exhaustive screening	Weight-average value of melting point ($\overline{MP}$)	Standard deviation of atomic radius (σAR)
			Weight-average value of ionization energy ($\overline{IE}$)	Weight-average value of shear modulus ($\overline{G}$)
			Weight-average value of heat capacity ($\overline{HC}$)	Weight-average value of density ($\overline{\rho }$)

.

The final six descriptors identified as critical for determining the martensitic transformation temperature (T_M) are: the weight-average value of ionization energy ($\overline{IE}$), weight-average value of heat capacity ($\overline{HC}$), weight-average value of shear modulus ($\overline{G}$), and standard deviation of atomic radius (σ_AR). These descriptors were subsequently used to train the machine learning models, as they represent the dominant factors influencing T_M.

Using the six final selected features summarized in Table 2, a random forest (RF) regression model was retrained to predict the martensitic transformation temperature (T_M) of the alloys. The model’s performance was evaluated through a ten-fold cross-validation procedure to ensure its generalization ability and predictive accuracy. The resulting coefficient of determination (R²) reached 0.82, demonstrating that the model exhibits excellent predictive performance.

Figure 5 presents the feature importance ranking obtained from the random forest model. By assessing the relative contribution of each feature to the model’s prediction capability, the following order of importance was determined: $\overline{MP}$(39) > $\overline{\mathrm{\rho }}$ (37) > $\overline{G}$ (10) > σ_AR (6) > $\overline{IE}$ (5) > $\overline{HC}$(3). These results highlight the dominant roles of melting point and density in governing the T_M of shape memory alloys.

3.4. Linear Relationship Between $\overline{\rho }·\overline{MP}$ and T_M

To further investigate the influence of the six key features identified by machine learning on the martensitic transformation temperature (T_M) of shape memory alloys (SMAs), detailed visual analyses were conducted.

Figure 6a illustrates the relationship between the weight-average value of density ($\overline{\rho }$), the weight-average value of melting point ($\overline{MP}$), and T_M. As shown, the feature space defined by $\overline{\rho }$ and $\overline{MP}$ can be effectively divided into two distinct regions corresponding to different T_M ranges. The green region in the lower left corner is characterized by low values of both $\overline{MP}$ and $\overline{\rho }$; this region corresponds to relatively low martensitic transformation temperatures (T_M < 300 K). The red region in the upper right corner is marked by higher values of both $\overline{MP}$ and $\overline{\rho }$; this region corresponds to higher transformation temperatures (T_M > 300 K). This separation suggests that $\overline{MP}$ and $\overline{\rho }$ act as discriminative features that jointly determine T_M in SMAs. Figure 6b provides an enlarged view of the black-boxed region from Figure 6a, further confirming the observed trend: higher values of the weight-average of melting point ($\overline{MP}$) and weight-average value of density ($\overline{\rho }$) correspond to higher martensitic transformation temperatures (T_M). This observation reinforces the conclusion that increases in $\overline{MP}$ and $\overline{\rho }$ are closely associated with elevated T_M values. Figure 6c illustrates the relationships between the weight-average value of shear modulus ($\overline{G}$), the weight-average value of ionization energy ($\overline{IE}$), and T_M. The results reveal weak correlations between these features and T_M, suggesting that $\overline{G}$ and $\overline{IE}$ have a minimal influence on the martensitic transformation temperature. Similarly, Figure 6d depicts the relationships between the standard deviation of atomic radius (σ_AR), the weight-average value of heat capacity ($\overline{HC}$), and T_M. Again, no clear trends are observed, indicating that these descriptors also exert relatively minor influence on T_M. In sum, the weight-average value of melting point ($\overline{MP}$) and weight-average value of density ($\overline{\mathrm{\rho }}$) are identified as the dominant factors affecting the martensitic transformation temperature. Consequently, these two features were selected for the construction and analysis of an empirical predictive formula.

Using $\overline{MP}$ and $\overline{\rho }$ as the two key features, a symbolic regression approach was employed to develop an empirical formula for predicting the martensitic transformation temperature (T_M) of SMAs. Symbolic regression [46] is a data-driven technique for discovering functional relationships, aiming to autonomously derive optimal analytical expressions within a predefined mathematical operator space. In this study, symbolic regression was implemented using a genetic algorithm [46]. The mathematical operator space consisted of basic arithmetic operations, including addition, multiplication, subtraction, and division. The input features were $\overline{MP}$ and $\overline{\rho }$, and the target variable was T_M. The fitness of each candidate formula was evaluated based on the correlation coefficient (r) between the predicted and actual T_M values.

After 500 generations of iterative optimization, the final empirical formula was identified:

$$T_{\mathrm{M}}=82(\overline{\rho }·\overline{MP})-700$$

where $\overline{\rho }$ and $\overline{MP}$ represent weight-average values of density and melting point, respectively, and T_M is the martensite transition temperature of SMAs. This result highlights that T_M scales approximately linearly with the combined descriptor $\overline{\rho }·\overline{MP}$.

Figure 7 illustrates the relationship between $\overline{\rho }·\overline{MP}$ and T_M across the entire dataset. A strong linear correlation is observed (r = 0.81 and MAE = 120 K), confirming that the empirical relationship generated via symbolic regression accurately captures the dependence of T_M on the key features. Additionally, the distribution of the absolute error δ(T_M) is shown in the inset of Figure 7. Notably, for the majority of datapoints (58%), δ(T_M) is less than 100 K. To further evaluate the model’s performance, both the 95% confidence interval and the 95% prediction interval were constructed in Figure 7. Most data points fall within the prediction intervals, are closely aligned with the confidence intervals, and are distributed symmetrically around the regression line. These results indicate that the model provides an accurate and reliable estimation of the overall trend in T_M across different compositions.

This simple formula offers an effective method to predict T_M across a diverse range of alloy families, including NiTi-based, TiPd-based, NiMn-based, TiAu-based, MgSc-based, AgCd-based, AuCd-based, PdTi-based, PtTi-based, CoNi-based, TiNb-based, TiTa-based, and ZnAuCu-based SMAs. This stands in contrast to traditional empirical rules such as the valence electron concentration (VEC) and lattice volume (V) rules, or other composition-related rules, which are typically applicable only to specific alloy families (see Figure 2 and Refs. [3,8,9,10,11,12]). While previous studies have used ML to forecast T_M in SMAs, many faced similar limitations to traditional empirical approaches: (1) some models are restricted to specific alloy systems [18,19], and (2) models capable of predicting across multiple systems often rely on computationally expensive features derived from density functional theory (DFT) calculations and require complex frameworks such as artificial neural networks [17]. These DFT-based models can take hours or even days to process a single data point, and their computational cost increases rapidly with prediction set size. In contrast, the empirical formula proposed in this work enables near real-time predictions of T_M for hundreds or even thousands of alloy compositions, as it relies solely on simple substance properties that are readily available. The key advantage of our formula lies in its ability to quickly and efficiently predict the martensitic transformation temperature across multiple SMA systems.

4. Conclusions

In this study, a systematic data-driven approach was developed to derive an interpretable and generalizable empirical formula for predicting the martensitic transformation temperature (T_M) of shape memory alloys (SMAs). By integrating high-throughput first-principles calculations, extensive feature selection techniques, and symbolic regression methods, we identified the key physical factors that govern T_M behavior. The final empirical model, expressed as T_M = $82(\overline{\rho }·\overline{MP})-700$, highlights the combined influence of material density and melting point on martensitic transformation characteristics. The proposed formula demonstrates strong predictive capability across diverse classes of SMAs, such as NiMn-based, NiTi-based, TiPt-based, and AuCd-based SMAs, etc. The reliability of the model was quantitatively assessed by evaluating performance metrics and the associated confidence/prediction intervals were reported to ensure accuracy of the model. Compared to previous empirical rules that were often limited to narrow alloy families, the present work offers a unified and practically applicable tool for SMA composition design, offering significant potential for high-throughput material design and optimization.