# Data driven prediction of thermal hysteresis in NiTi-based high entropy shape memory alloys

Léo Thiercelin

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Léo Thiercelin. Data driven prediction of thermal hysteresis in NiTi-based high entropy shape memory alloys. Materials Letters, 2026, 406, pp.140002. 10.1016/j.matlet.2025.140002. hal-05441364

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# Data driven prediction of thermal hysteresis in NiTi-based high entropy shape memory alloys

Léo Thiercelin*

Arts et Métiers Institute of Technology, CNRS, Université de Lorraine, LEM3-UMR 7239, F-57000 Metz, France

## ARTICLE INFO

Dataset link: NiTi-Like SMAs transformation temperatures dataset (Reference data)

**Keywords:**

High entropy alloys

Shape memory alloys

Martensitic transformation

Thermal hysteresis

Machine learning

Extremely randomized trees

## ABSTRACT

This study provides a data-driven approach to predict thermal hysteresis in NiTi-based high-entropy shape memory alloys (HESMAs), using a dataset of NiTi-based shape memory alloys that includes the four transformation temperatures ($M_s$, $M_f$, $A_s$, $A_f$). Machine learning models based on extremely randomized trees were proposed to predict $M_s$ and the thermal hysteresis ($\Delta T = A_f - M_s$), using 11 physical material descriptors as inputs. $M_s$ prediction is dominated by mixing enthalpy, the valence electron ratio, and the difference in electronegativity. In contrast, hysteresis depends equally on 8 additional balanced features such as the melting temperature, electronegativity, atomic radius, and the mixing entropy. The predictions models are highly reliable with mean absolute errors of 31 °C for $M_s$ and 11 °C for $\Delta T$. Finally, a design strategy is proposed for (NiCuCo)(TiZrHf) HESMAs that optimises both $M_s$ and $\Delta T$ with targeted values.

## 1. Introduction

First introduced by Cantor and Yeh, High-Entropy Alloys (HEAs) are multi-component systems that maximise configurational entropy and stabilise solid-solution phases. The concept has then inspired the development of High-Entropy Shape Memory Alloys (HESMAs), which overcome the limitations of conventional shape memory alloys (SMAs) by offering improved thermal stability and functional properties across a wide temperature range [1–3]. The properties of SMAs, shape memory, superelasticity or damping are governed by the transformation temperatures ($M_s$, $M_f$, $A_s$, $A_f$) which are highly sensitive to the chemical composition and thermomechanical treatments. These factors have a significant impact on fatigue resistance and elastocaloric performance. Designing HESMAs with tailored thermal hysteresis ($\Delta T = A_f - M_s$) is therefore critical for durable high-temperature applications.

Machine learning (ML) is a powerful approach for exploring the vast compositional space of HESMAs. Previous studies, such as that of Tian et al. [4] have predicted $\Delta T$ using a limited dataset of 207 NiTi-based alloys but have not considered HESMAs or the effects of heat treatment. This study addresses these shortcomings by introducing an expanded dataset of 518 alloys from Thiercelin et al. [5], including HESMAs, and by incorporating homogenization criteria. Eleven chemical and thermodynamic descriptors commonly used for HEA design were

used as input parameters alongside Extremely Randomized Trees and rigorous cross-validation to predict Martensitic start temperature ($M_s$) and thermal hysteresis $\Delta T$. Beyond prediction, this study proposes a design strategy for NiTi-based HESMAs with optimised hysteresis and transformation temperatures.

## 2. Methodology

The NiTi-like SMA dataset, which was initially compiled by Thiercelin et al. [5] through a literature review, was expanded in this study, to include 518 alloys and their four transformation temperatures. These alloys include binary, ternary, quaternary and high-entropy alloys with the following elements: Ni, Fe, Co, Cu, Pd, Pt, Ti, Zr, Hf, Nb and Ta. The outputs of the ML models were the $M_s$ temperature and the thermal hysteresis $\Delta T$. 11 physical descriptors were proposed as inputs including the mixing entropy ($\Delta S_{mix}$), mixing enthalpy ($\Delta H_{mix}$), and averaged properties, denoted as, $\bar{\omega}$, weighted by atomic fraction: atomic radius, electronegativity, melting temperature, and valence electron number. To account for the mismatch effect among elements, the variation coefficient ($\delta^*$) is introduced, defined as the ratio of the mean value to the standard deviation of each property. Additionally, the valence electron ratio (ver), defined as the ratio between the average valence electron number and the average atomic number of the alloy was included.

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: leo.thiercelin@ensam.eu.</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Pearson's correlation matrix (absolute values) between the 11 materials descriptors and the two outputs of the ML model ($M_s$ and $\Delta T$).</figcaption></figure>

Pearson's correlation coefficients (absolute values considered) between the indicators and the two outputs given in Fig. 1, shows that $\delta\chi$, $\Delta H_{\text{mix}}$, $\bar{r}$ and $\text{ver}$ are strongly correlated with $M_s$ whereas no clear correlations in observed for $\Delta T$. Note that the dataset used for ML training was

filtered by considering a homogenization indicator for HESMAs, $|M_s - M_f|$. A value below 50 °C, suggests a homogenized and single-phase state, whereas higher values often reflect a multiphase state with precipitates. This can cause discrepancies in transformation temperatures, even for the same chemical composition [6]. The dataset was therefore filtered with alloys such as $|M_s - M_f| < 50$ °C. Two ML algorithms are based on Extremely Randomized Trees algorithms using the same hyperparameters: 15 trees and a depth of 10. To ensure representativeness, 80 % of each alloy class (binary, ternary, quaternary, and high entropy) was allocated to the training set. To improve robustness, this procedure was repeated across 100 random splits. Model performance was evaluated using the Mean Absolute Error (MAE), selected for its uniform penalization of prediction errors. MAE values were computed for each alloy class and averaged over the 100 iterations and the feature importance of each descriptor was evaluated using Gini metrics in scikit-learn.

### 3. Results and discussion

Fig. 2 summarizes the database used to study the influence of chemical elements and entropy (number of elements) on the four transformation temperatures. Two derived features were analyzed: thermal hysteresis ($A_f - M_s$) and the homogenization indicator, $|M_s - M_f|$. Figs. 2a–b shows ternary NiTi-like alloys: most have hysteresis below 100 °C, except NiTiHf (>30 at.% Hf) reaching 150 °C and $M_s$ above 400 °C, with $|M_s - M_f| > 50$ °C even a single phase state [6]. Figs. 1c–d illustrate the effect of element count: few compositions combine hysteresis >100 °C with $M_s > 200$ °C, mainly alloys with Hf and Zr. A slight

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Overview of the NiTi-like SMAs dataset. a) hysteresis temperature $A_f - M_s$ vs $M_s$ for ternary alloys; b) hysteresis temperature $A_f - M_s$ vs $|M_s - M_f|$ for ternary alloys; c) hysteresis temperature $A_f - M_s$ vs $M_s$ for quaternary and HEAs; d) hysteresis temperature $A_f - M_s$ vs $|M_s - M_f|$ for quaternary and HEAs.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Evaluation of the ML models for the Ms. and ΔT predictions: a) features importance of the Ms. prediction model b) Ms. prediction vs Ms. experimental. The dashed line represents a MAE tolerance of 50 °C; c) features importance of ΔT predictions model; d) ΔT prediction vs ΔT experimental. The dashed line represents a MAE tolerance of 20 °C.</figcaption></figure>

correlation between hysteresis and $|M_s - M_f|$ reflects multiphase states observed microscopically except for some composition. Overall, Hf strongly increases hysteresis, especially in complex alloys with Cu, Co, Pd, and Zr. Exploring this vast compositional space requires ML tools; thus, the database was used to train two models for $M_s$ and hysteresis (ΔT).

Fig. 3 shows ML model accuracy and feature importance for $M_s$ predictions (Figs. 2a–b) and hysteresis $\Delta T = A_f - M_s$ (Figs. 2c–d). For the $M_s$ prediction, $\delta\chi$, $\Delta H_{mix}$, and ver are highly the most important features, followed by $\delta r$ and $\bar{r}$ with moderate importance. Reversely, the importance of the features shows that all the included parameters contribute equally to the predictions. These tendencies correlate fairly with Pearson's analysis both for Ms. and ΔT (Fig. 1). The $M_s$ predictions are highly reliable with average MAE of 10 ± 1 °C for training and 31 ± 4 °C for validation, increasing with alloy complexity: 27 ± 6 °C for ternary, 29 ± 7 °C for quaternary, and 49 ± 16 °C for high-entropy alloys. For ΔT prediction, MAE is 4 ± 0.3 °C (training) and 11 ± 2 °C (validation), also rising with the number of elements.

The models were used to design a NiCuCoTiZrHf HESMA by optimizing the composition proposed by Peltier et al. [5]: of Ni₂₅Cu₁₇.₅Co₇.₅Ti₁₆.₇Zr₁₆.₇Hf₁₆.₇ with both high Ms. and ΔT (Ms = 198 °C, ΔT = 157 °C). Reliable predictions were achieved ($M_s$ = 162 °C and ΔT = 100 °C). This approach fixes the composition of Ni, Cu, Co while varying the compositions Ti, Zr, Hf using ternary diagrams, as shown in Fig. 4.

Predictions indicate that the relative Ti content compared to the contents of (Zr,Hf) strongly governs both Ms. and ΔT in high-entropy

shape memory alloys (HESMAs). When Ti exceeds 17 at.%, Ms. approaches 0 °C, making these alloys suitable for superelastic applications near room temperature. In contrast, Ti contents below 17 at.% lead to $M_s$ values above 200 °C, regardless of the Zr/Hf ratio, positioning them for high-temperature actuation. Hysteresis follows a similar trend: Ti-rich compositions (Ti > 17 at.%) exhibit narrow hysteresis (<40 °C), favoring cyclic stability, whereas Ti-lean alloys (Ti < 17 at.%) show broad hysteresis (>80 °C), typical of high-temperature SMAs. In addition, the high Hf proportion would highly improve both $M_s$ and ΔT which is well in line with the observations for ternary NiTiHf SMAs [6]. However, these design domains are derived from extrapolations ML models rather than direct experimental validation. While the predictions align with trends observed in simpler alloy systems, the lack of extensive experimental data for high-entropy compositions introduces uncertainty. The strong influence of Ti content on $M_s$ and hysteresis should therefore be considered indicative rather than definitive.

## 4. Conclusions

This work extends Thiercelin et al.'s model, originally limited to predicting martensitic start temperature ($M_s$), by expanding the dataset to include four transformation temperatures ($M_s$, $M_f$, $A_s$, $A_f$). This allowed improving Ms. prediction and developing a new model for hysteresis ($\Delta T = A_f - M_s$), both using Extremely Randomized Trees and 11 physical descriptors. $M_s$ prediction is mainly driven by three indicators: $\delta\chi$, $\Delta H_{mix}$, and ver, while ΔT depends on the 11 equally important indicators. Validation shows strong accuracy with MAE of
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** ML prediction of transformation temperatures for Ni₂₅Cu₁₇.₅Co₇.₅(TiZrHf)₅₀ HESMA: a) M_s temperature; b) Hysteresis temperature. Predictions are extrapolated from the homogenized designed HESMA by Peltier et al. [7], Ni₂₅Cu₁₇.₅Co₇.₅Ti₁₆.₇Zr₁₆.₇Hf₁₆.₇ with predicted values (M_s = 162 °C and ΔT = 100 °C) close to experimental values (M_s = 198 °C and ΔT = 157 °C).</figcaption></figure>

31 °C for M_s and 11 °C for ΔT, though high-entropy alloys exhibit larger errors due to sensitivity to homogenization treatments. Finally, the models were applied to design NiCuCoTiZrHf alloys, revealing two distinct domains: Ti contents below 17 at.% correspond to low-hysteresis superelastic alloys, whereas Ti contents above 17 at.% favor high-temperature SMAs with larger hysteresis. These results, extrapolated from ML models trained on less complex ternary and quaternary alloys, should be considered indicative of a potential design strategy. Experimental validation through the synthesis of new HESMAs in these domains, combined with thermodynamic calculations, will be essential to confirm these predictions.

CRediT authorship contribution statement

**Léo Thiercelin:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Léo Thiercelin reports was provided by Arts et Métiers Institute of Technology. If there are other authors, they declare that they have no
known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported by the M²ARTS group at LEM3. The
authors thank the LEM3 facilities (“Procédés”, “MicroMat,”
“MécaRhéo”) and the “Cassiopée” computing platform at ENSAM, as
well our colleague Noé Venet for assistance with data mining. Artificial
intelligence was used for alloy design and helped to correct syntax for
writing this article.

**Appendix A. Supplementary data**

Supplementary data to this article can be found online at https://doi.
org/10.1016/j.matlet.2025.140002.

Data availability

The dataset is available in open access.

NiTi-Like SMAs transformation temperatures dataset (Reference
data) (NiTi-Like SMAs transformation temperatures dataset)

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