State-of-the-Art Review on the Aspects of Martensitic Alloys Studied via Machine Learning

Abstract

Though the martensitic transformation has been a commonly investigated topic in the field of experimental and computational materials science, the understanding of this mechanism in a variety of alloys is yet far from complete. In this era of Industry 4.0, there have been ongoing trends on employing machine learning (ML) techniques for the study of the martensitic alloys, and such data-driven approaches are expected to unravel a great amount of information about the process-structure-property behaviour relationship in this class of materials. However, with the availability of a large variety of datasets and with an option to use different ML models, a bulk amount of information has already been generated with regard to martensitic alloys. The discovery and design of shape memory alloys can be accelerated if the multi-principal element functional alloys and martensitic transformation phenomenon are studied extensively using machine learning techniques. Thus, it is necessary to highlight the major categories or aspects of these alloys that have been predicted with ML. The present work performs a state-of-the-art review on the machine learning models developed for the quantification of aspects such as martensitic start temperature (Ms), materials properties, microstructure, mechanisms etc., on the alloys.

1. Introduction

Almost all metallic alloys or ceramics can exist in solid state form depending on the temperature and composition. The study of metal/alloys’ phases, their identification, various parameters of respective phases such as transformation starting and ending points is classically performed using a special type of chart called a transformation diagram [1,2]. Two types of transformation diagrams are illustrated in Figure 1, where the region or boundaries of different alloy phases are plotted in the 2D graph with the time as horizontal axis and temperature as vertical axis. The first type, as shown in Figure 1a is the time-temperature-transformations (TTT) diagram that reveals the phase transformation at isothermal dwell [2,3]. The second type, illustrated in Figure 1b is the continuously cooling transformation (CCT) diagram which describes the effect of cooling and cooling rates on phase transformation [2,4]. These transformation diagrams are an important source of information for developing data-driven models for alloys.

The phase transformation in solid–solid transformation mostly happens by the diffusion of the atom in the lattice. The diffusion transport in solid metal has been found to occur by point defects such as vacancies, interstitial, substitution, etc. that acts as the mechanism of transport [5,6]. Figure 2a illustrates an interstitial atom of different type traveling and occupying an interstitial site inside the matrix of another type atoms in the crystal structure. Similarly, in Figure 2b the diffusion of substitutional atom of different kind into a vacancy in the crystal lattice is shown at the upper part of the lattice while, a direct substitution of atoms within the crystal structure is shown at the bottom part.

Elements with small atom size such as hydrogen, carbon or oxygen are favored as the solute for the distortion or diffusion as the energy for the distortion which is elastic energy, increases squarely with the solute size making the diffusion jumps easier to perform [5].

In case of diffusionless transformation typically two types of the phenomenon are found, “lattice-distortive strain” and “shuffles”, combined or individually [9]. Lattice distortive deformation is associated with homogenous strain converting lattice shape from one to another giving rise to an elastic strain energy, along with an interfacial energy separating the phases. Figure 2c shows a cubic lattice deformed into three different shapes undergoing lattice-distortive deformation by either shear along a plane or by expansion and contration or both type of transformation in a cubic structure without the change in constituent atom in that particular structure, whereas shuffle deformation is a simple movement of atoms changing the crystal symmetry without strain, and hence only interfacial energy is produced. Figure 2d represents the shuffle displacement in strontium titanate with three different elements via a clockwise and anti-clockwise rotation of one atom (blue colored atom) around a specific another atom (green colored atom). While almost all the transformation in steel and iron alloy undergo phase change via the diffusion of atoms but a special transformation called martensitic transformation is a unique case as it undergoes diffusionless or displacive phase change by the lattice-distortive displacements dominating the kinetics of the process [9]. This gives rise to the amazing characteristics and properties in the martensitic alloys, differentiating them from other iron alloys.

In fact, phenomenological theories of martensite crystallography (PTMC) have been understood as the primary foundation for establishment of the quantitative parameters for the geometric description of martensitic transformation. These phenomenological theories of crystallography of martensitic transformation were independently developed in the 1950s by Wechsler, Lieberman and Read [10] and Bowles and Mackenzie [11]. The phenomenological theory developed by Wechsler, Lieberman and Read is referred to as WLR theory and the one proposed by Bowles and Mackenzie is termed as BM theory. Although the WLR theory and BM theory differ with each other in mathematical formulation, they have been considered as essentially equivalent theories [12,13]. Concepts and terminologies of crystallography such as invariant plane strain (IPS), habit plane indices, lattice parameters, Bain strain, lattice invariant shear (LIS), orientation relationships etc. are commonly utilized to understand the alloy or material behavior during martensitic transformation. With the notation L for LIS, R for rigid body rotation matrix, B for Bain strain, and ${ϵ}_{IPS}$ for IPS, the invariant plane strain then can be expressed as ${ϵ}_{IPS}$ = RBL [14]. As the abovementioned theories are phenomenological ones, it can be very relevant if the concepts of the PTMC theories are first integrated with data science to assess the phase transformation behavior in martensitic alloys.

As the world progresses into the new era of Industry 4.0, it is evident that the path towards this newer industrial revolution is led by state-of-the-art material design process and technologies. Machine learning assisted material synthesis is one of such tools which is at the forefront of the research and development, superseding traditional empirical methodologies with computational data driven approach. Numerous material databases [15], tools [16] and applications [17] have been developed in recent times, promoting, enhancing and speeding up the quantification and design of newer and superior materials efficiently and effectively.

Computational materials science has become one of the prime disciplines in material design and development process. With the increased computational capacity of our local machines, the machine learning tools such as artificial neural networks (ANN), support vector machine (SVM), convolutional neural network (CNN) has opened countless opportunities to material science researchers. The very nature of these tools needing a large observation of data has been a blessing to material researchers in the first place. The availability of enormous experimental, empirical and mathematical data in almost every field of material science has led to the discovery and unravelling of mysteries even within materials priorly known to us. One such example is the discovery of 244 new ternary metal nitrides [18] primarily using data science tools. This has opened a greater possibility for the development of super materials just by using the collected data from the literature and using it to predict or discover unknown properties and relations they possess.

Machine Learning (ML) techniques have been quite ubiquitous lately in almost every field of real-world engineering systems. In the renewable energy sector, their applications ranges from modeling of the solar energy systems [19] to predicting and optimizing the fuel consumption in hydrogen fuel cells [20]. Similarly, ANN has made its way to optimize the design parameters of composites for energy harvesting techniques, as in case of bistable morphing. [21]. Kosarac et al. [22] used neural networks to obtain the value of arithmetic mean roughness (Ra) in machining of aluminium alloy using a very limited data set of 27 trials which further proofs the robustness of latest neural network algorithms and their capabilities to train on small dataset as well. Another work by Wu et al. [23] in which data generated via finite element simulations were used to train a feedforward backpropagation neural network (BPNN) for estimating the residual distortion for welding sequence. Even the complex nonlinear thermo-elastic-plastic differential equations solved using finite element methods by Wu et al. could have been solved using latest algorithm called Physics Informed Neural Network (PINNS) [24]. The use of ANN by Rohman et al. [25] in predicting the dross formation in steel sheet during laser cutting at different environment showed not only a deep neural network as most accurate but could help in identifying the process parameters such as cutting environment, laser power, pulse frequency for achieving minimum dross formation.

2. Martensitic Alloys and Martensitic Transformation

The material which has shaped the course of human advancement in the whole journey of humankind is undoubtedly iron and its alloys. Steel specifically is the backbone of all the developmental challenges we have overcome in past century. The alloying of iron with carbon has given rise to different material properties in steel considering the composition, rate of cooling and other thermodynamical procedures. One such type of steel is Martensitic steel, which has gained superior fame and usage considering its properties of strength [26]. The study of special properties seen in martensitic transformation is not just curious case in academics, but the application of these materials in high scale manufacturing industry has attracted a lot of interest in detail research work on martensite, their transformation characteristics and properties. Although martensitic transformation is not just confined to steels, but the importance of the transformation is unreasonably greater and far-spread in steel [27]. As a result, microstructure and thermodynamics plays a key role in defining the intrinsic property of steel. Martensitic alloy, known for its high hardness and brittleness, has shaped its path into our day to day lives from kitchen-ware to automobile parts [28] and to the unique materials with the capability of retaining its shape even after large deformation also known as Shape Memory Alloys (SMA) used in medical industry as stents grafts, orthopedics, clinical instruments and so on [29]. NiTi-based shape memory alloys (SMA) are the important functional alloys that exhibit the phenomena of thermo-elastic martensitic transformation. Thus an adequate knowledge of martensitic transformation can be very advantageous in designing applications involving SMA. A number of amazing and remarkable properties are found in alloy undergoing martensitic transformation such as shape-memory effects, pseudoelasticity, superelasticity [9]. Products with applications involving shape-memory effect, high damping capacity, toughness in ceramics are a few examples where academic and industrial research are found to be focused on. It has to be understood well that martensitic transformation is common not only in steels but also in other alloys. As will be highlighted in Section 4, the phenomenon of martensitic transformation is nowadays getting a larger attention in the study of multicomponent shape memory alloys, including the multi-principal element shape memory alloys. Because of the bulk of studies that have already been completed by researchers in the sector of martensitic steel, it is very logical to use steel alloys as a frequent material of choice for illustrating the concepts of martensitic transformation.

Martensitic transformation is a phase transformation in solid state that occurs without any change in chemical composition [27]. Martensite start temperature, which is a critical parameter in steel design, is a prime focus of study, due to its high dependence in chemical composition. As the alloying elements are added in steel for specific application the Ms temperature varies accordingly, hence generalized prediction of the transformation temperature is still quite a challenge [30]. Besides commonly identified composition dependence, martensitic transformation temperature has also been reported to be related with energy difference between the parent phase and martensitic phases [31].

Mechanism of Martensitic Transformation

For a martensitic transformation to occur there must be lattice distortion with a shear dominant shape change where sufficiently high shear-strain energy overcomes the kinetics and morphology of the transformation [9]. The lattice-distortive transformation (Figure 2c) and shuffle transformation (Figure 2d) are the concepts enunciated to distinguish the kinetic and morphological aspects of martensitic transformation, and thus can be utilized in future models in developing the mechanistic models of martensitic transformation. Lattice-distortive transformation produces strain energy, and ideally the shuffle-deformation produces interfacial energy and no strain energy [9]. Energy considerations can be considered as the backbone of robust mechanistic models for martensitic transformation. On the other hand, the PTMC mathematically set the geometrical criteria that must be met for the system undergoing martensitic transformation to achieve its final state from a given initial state. In principle, PTMC are phenomenological theories and not mechanistic models [14]. In order to mechanistically or kinetically describe the motion of atoms during the transformation, it is necessary to combine the energy considerations with the PTMC. Machine learning algorithms related to system optimization, will be of great use in developing such hybrid models that are the combination of PTMC and thermodynamic models.

When the austenite steel is quenched to room temperature, the carbon in solid solution in the austenite cannot diffuse away due to the high transformation speed trapping the carbon within it, hence a very strong and hard phase is formed called Martensite. Martensitic transformation is different from other forms of steel such as ferrite or pearlite in a sense that diffusion is not the method of transformation for martensite [27]. In a martensitic transformation, distortion of austenite takes place and there is a development of a Body Centered Tetragonal (BCT) unit cell enclosed in two FCC unit cell as shown in Figure 3. The BCT transformation is primarily due to the lack of diffusion and high cooling rate which give rise to a lattice shear where carbon simply has no time to escape. This highly strained martensitic transformation gives rise to the high hardness and brittleness in the steel [32].

The formation of martensite begins at the temperature Ms (martensitic start temperature) and ends at the Mf (martensite finish temperature). The transformation of iron from austenite to martensite happens by a large shear and volume expansion in a diffusion-less deformation without change in the chemical composition, hence transformation of crystal structure from FCC (Face Centered Cuboid) to BCT (Body Centered Tetragonal) occurs [27] as shown in Figure 4. This transformation gives the martensite its property of strength and brittleness [33].

3. Data-Driven Approach as the Fourth Paradigm

Data-driven studies, at its core, is the use of a computational model or algorithm which can accurately predict the pattern or nature of a physical phenomenon which is hidden and quite impossible to correlate the complex relationship among different parameters inside the data just by human observation. One of the major benefits of the data-driven machine learning technique for material property prediction is that, the scope of prediction is not just limited to one specific property or class of material. The data-driven approach can equally be applied in predicting martensitic start temperature for steel as well as for other alloys including titanium or copper. After the discovery of amazing shape retaining alloys capable of maintaining or remembering their original shape even after a large deformation many researchers are attracted towards quantifying and predicting martensitic formation temperature for these alloys, where machine learning and data-guided approach has taken a significant place.

With the advent of newer technology and computational machines, the material database in current situation is all time biggest in term of information, but with an increase in data comes the challenge of data completeness, veracity, validation and standardization of experimental data and its gap between industrial and academic approach [35]. The innovative technologies such as Artificial Intelligence (AI) and machine learning (ML), are now capable of processing the biggest and largest datasets known to us and not only process them but also capable of deciphering the hidden correlations present in the data which could not simply be accessed by basic human intelligence. This new scientific era can also be referred as the fourth industrial revolution which is dictated by the data-driven approach in almost every sector of science, engineering, business, finance, politics and social life [35]. Data-driven methodologies can also be employed for inverse design of materials as well [36].

The major three scientific methods of materials study i.e., experimental, theoretical and computational has given rise to the latest, innovative and resource efficient fourth paradigm in the field of materials design and discovery called ‘big data-driven science’ which captures the essence of first three paradigms as shown in Figure 5. The data generated by the experimental, theoretical and computational/simulation study have overflowed in sector of material science that the major challenge in current time has become our ability to analyze and make a proper sense out of data rather than collecting one. However, with an unprecedented advancement in computational tools such as machine learning and artificial intelligence, now we can not only organize and store big material data in systematic order but also unveil the hidden information in them. This capability has not only made the material property prediction easier and efficient but also carved a path for the inverse materials models where we can discover a material with prior knowledge of desired property of it [37].

It is of great significance, the ability to predict microstructure based on composition and heat treatment of alloy, and to do so in a non-Edisonian approach is where the machine learning (ML) approach shines the brightest. ML technique basically makes the computer learn an intricate relationships and recognize the hidden pattern in the data. In this work, we have tried to review the works using ML tools and techniques for the prediction of Ms temperature and discuss about the shortcomings and improvement needed for the development of a precise and accurate model capable of handling large set of alloying element and concentration.

4. Machine Learning: Structure, Properties, Behavior and Application of Martensitic Alloys

The transformation of austenite to martensite happens in a non-diffusional process when there is high cooling rate which is enough to suppress the formation of microstructures as ferrite, perlite and bainite [38], which leads to the question, at what temperature does the transformation take place? To answer the question, two temperatures are taken into account, the first being the highest temperature at which martensite formation occurs called Martensitic start temperature (Ms) but for the end of transformation, there is no clear martensite finish temperature (Mf), so for convenience the point at which 95% of the transformation is completed is taken as Mf [27].

The addition of alloying elements in steel is done to achieve an enhanced physical and mechanical properties such as corrosion resistance, better machinability, improved weldability and so on. Meanwhile the addition of alloying elements enhances the desired properties in steel but they also play a greater role in changing martensitic start temperature by decreasing the saturation of carbon in austenite, as an effect Ms decreases [30]. Most of the calculations for Ms temperature involving empirical equations are found to be reasonably accurate considering lower number of alloying elements but they tend to show discrepancies in prediction when applied to a larger number of alloying elements with high alloying concentration [30,39].

As the phase transformation, cooling rate and composition have a lot of significance on mechanical property of the steel [40], it is of absolute importance to have the information about Martensitic start temperature and its relation with elemental composition. Prediction of Ms has been and is still one of the highly persuaded topics. In-spite of numerous efforts made by a lot of researchers using different mathematical tools and techniques such as thermodynamics-based modeling, multiple linear regression, artificial neural networks, Bayesian neural network, Calphad. The reason for continuous research work on developing the best performing prediction model again and again might be hidden in parameters used in the model and method itself. Machine learning tool, artificial neural network (ANN) used for prediction of MS by Vermeulen et al. [38] in 1996 is still prevalent in latest research for the same objective. Input parameters for constructing neural network by Vermeulen et al. (austenitising temperature and 17 elemental composition) are still being used in today’s modeling of neural network were composition is regarded as a major feature for prediction. M. Peet [40] suggests that the approach of use of composition as a sole criteria for prediction of Ms should be reinvestigated and newer models should consider a change in chemical driving forces due to binary and tertiary interaction as well for a realistic approach of predicting Ms.

Zhang et al. [30] utilized Gaussian process regression (GPR) model to elucidate the relationship between alloying elements and Ms temperature. A total of 1119 data observations with Ms temperature between 153 to 938 K were taken into consideration. Eighteen elements as input descriptors with five kernel functions were used to for the development of the model and the model was evaluated for prediction of Ms result based on correlation coefficient, mean absolute error and root mean square error with reported values of 99.65%, 9.979, and 6.354 respectively.

The reversible transformation between the parent and martensitic phase has led to the development of a special type of material which can retain its original shape even after a large amount of deformation: the shape memory alloys. These alloys have made their way into highly sensitive scientific and biomedical application in recent times from ‘stents’ to open up clogged arteries in human body, to the wheels of extraterrestrial rover. Since these unique materials are used in some of the most sophisticated and expensive products it goes without saying that the most important property or characteristic of these shape memory alloy is none other than transformation temperature as it dictates the limitation of their application [41].

The use of ML tools in martensitic and austenitic alloys are not just limited to prediction of martensitic start (transformation) temperature. In a work by Osman et al. [42], authors have developed ML model to predict the rupture strength in iron based martensitic and austenitic alloys. They have used three different ML algorithms namely Gaussian Process Regression, Neural Network and Gradient Boosted Decision Tree. The conclusion of the study was that for a fairly small data observations accuracy of Neural Network with input parameters of 20 elemental composition is constrained by learning and training of complex system parameters, the most accurate and robust model they got was using Gradient Boosted Decision Tree with a R${}^2$ value of 0.98 and 0.95 for martensitic and austenitic alloy respectively. They have pointed out that a possible drawback on prediction of mechanical or thermodynamical properties of the material using ML models lies in the domain knowledge and physical laws that are difficult to achieve from open-source literature. As the critical information about outsider parameters or failure parameters are hardly available in any published dataset. It is imperative to include all the cases of experimental data in publication so that for any ML algorithm trained under such data can make an assessment of the boundary or domain area where an experimental and predicted value line up. For example in case of 9–12% Cr Ferritic Martensitic Alloy reported by [42], addition of Carbon increases the strength up to a certain concentration after which the addition of Carbon is detrimental, but study about this point and beyond is rarely carried. So, the dataset does not contain information about the reverse effect of carbon addition hence ML trained in the dataset with incomplete information would result in inaccurate prediction.

Martensitic transformation is a complex process involving different microstructural changes and criteria to be fulfilled for it to occur. One of such requirements is driving force for the transformation, which can be considered as a barrier for the formation of martensite. Once the available driving force exceeds barrier, the martensitic transformation starts. The available driving force depends on the temperature and alloying constituent proportion making it a thermodynamical property where as the barrier force depends on the mechanisms of transformation itself [43].

Lath and plate are two types of microstructures observed in martensitic steel alloys [43,44,45]. In context of binary Fe–Ni alloys, the two microstructures are respectively distinguished through the use color-coded inverse pole figures (IPF) of Figure 6, wherein image (a) represents lath martensite for Fe-25Ni alloy and (b) depicts plate martensite for Fe-30Ni alloy [46]. The quantitative value of driving force for the start of formation of these microstructures is formulated with the help of information on Martensitic start temperature (Ms) lines [43,44]. The Gibbs free energy associated with these microstructure is obtained from the differences of corresponding Ms lines [44]. In other words, the martensitic microstructure evolution can be directly correlated with Ms temperature [47]. It is reported that the Gibbs free energy change is related to Martensitic start temperature, chemical driving force and entropy difference between austenite and martensite as:

$$\Delta G=\Delta G^{M_s}+\Delta S^{M_s}(T-M_s)$$

(1)

where T is the absolute temperature, $\Delta G^{M_s}$ is chemical driving force and $\Delta S^{M_s}$ is difference in entropy between austenite and martensite [48]. In other words, the martensitic microstructure evolution can be directly correlated with the $M_s$ temperature [47]. This highlights the necessity to use Ms temperature as a phase transformation based features alongside volume % of phases (microstructure related feature) in context of machine learning algorithms associated with prediction of mechanical behavior of steels [47,49] and other martensitic alloys. Hence, data-driven methodologies for estimation of Ms temperatures as described earlier in this section is in fact a starting work related to the microstructure prediction in martensitic alloys.

One of the most important applications of knowledge of martensitic transformation is in the discovery, design and development of shape memory alloys (SMA), and the use of artificial intelligence [50] can help in accelerating this process. One of the major challenges in designing SMA system is in finding a candidate that can perform excellently at elevated temperatures [51]. Ni–Ti alloy is a commercially popular SMA. However, its sub−100 ${}^{°}$C transformation temperature (TT) has restricted its application in elevated temperatures. As it has been found that ternary alloying of NiTi with an additional element such as Au, Hf, Pd, Pt or Zr can help in attaining the objective of high TTs. In order to design the composition of the Ni${}_x$Ti${}_y$Hf${}_{100-x-y}$ alloy for optimized TTs, Catal et al. [51] have trained an artificial neural network (ANN) with the alloy datasets. Through the prediction model of an ANN, they have concluded that the ternary alloy Ni${}_{49.7}$Ti${}_{26.6}$Hf${}_{23.7}$ characterized by an austenite finish temperature of 403.5 ${}^{°}$C, has an astounding cyclic stability and could be utilized for technological applications requiring reversible austenite-to-martensite phase transformation at temperatures beyond 400 ${}^{°}$C. Another significant challenge commonly observed in the state-of-the-art application of SMAs in solid-state actuation is relatively low efficiency and functional instability caused by the transformation thermal hysteresis and large temperature ranges during the martensitic phase transformation [52]. Recently, an initiatives on data-driven studies on multicomponent functional alloys and multi-principal element functional alloys [51,52,53,54,55,56] have opened the potentiality to discover novel SMAs with enhanced efficiency and larger stability. Xue et al. [54] have employed a statistical learning approach to predict the transformation temperature of multi-component alloys using three materials descriptors related to chemical bonding and atomic radii of the constituent elements. In context of multicomponent SMAs, the presence of vast search space offers a challenge in discovering the material with targeted properties and behavior. In this scenario, the application of adaptive design as demonstrated in the works [53,54] can accelerate the search for multicomponent SMA materials with the targeted properties. Through an adaptive design methodology, Xue et al. [54] have distilled out less than 50 targeted compositions from a potential space of ≈800,000 compositions of Ni${}_{50-x-y-z}$Ti${}_{50}$Cu${}_x$Fe${}_y$Pd${}_z$ multicomponent SMA, that have both low thermal hysteresis and high transformation temperature. The methodology or procedure of multi-objective optimization of transition temperature and thermal hysteresis for SMAs using adaptive design has been elaborated in Gopakumar et al. [55]. Trehern et al. [52] have utilized an artificial intelligence enabled materials discovery framework to detect SMA chemistries and identify related thermomechanical processing steps which result narrow thermal hysteresis and targeted transformation range during an applied stress. Using this framework, they have predicted that for the NiTi-based SMA system, the composition Ni${}_{32}$Ti${}_{47}$Cu${}_{21}$ (multi-principal element alloy) has the narrowest thermal hysteresis as well as transformation range, and confirmed it as the optimized composition. Liu et al. [56] have demonstrated the use of physics informed machine learning to enable composition-process-property in SMAs. The features in the dataset related to heat treatment and phase transformation, have been enabled physically consistent by mathematically treating with functions derived from established growth kinetics model and physical models.

5. Limitations of Stand-Alone Machine Learning Techniques

Machine learning or in borader sense, data driven approach in materials science has opened a new pathway for the prediction of properties and behavior of martensitic alloys. Lately as the computational power of personal computers are progressing exponentially, it has become even easier, economical and computationally efficient to undergo simulations and predictions using latest machine learning algorithms. Even though machine learning has dominated many field of artificial intelligence such as Natural Language Processing, Image Processing, Speech recognition and so on, but for interdisciplinary field of study such as computational material science, machine learning has just begun its accent. The backbone of any machine learning algorithm is the data it trains on, the amount of valid, correct and quality data in context of material science is still a challenge to collect let alone process and featurize it.

The primary concern for any machine learning model predicting material properties of martensitic alloy or phenomena associated with martensitic transformation is the accuracy of the model and process of validating the prediction from the model. The accuracy of a predictive supervised model is generally studied as a difference (loss) between the predicted value and true value [57]. The user can choose which statistical tool to use for the calculation of loss, hence a uniformity cannot be found in the accuracy of the model in general. One has to have the grasp of statistics to clearly understand what the actual model accuracy is. Along with that, as newer and newer machine learning algorithms are being developed with numerous functions to calculate accuracy, loss, optimization rules, the selection of best suitable hyperparameter has created extra hassle to get the highest accuracy model. Even after selection of the best set of hyperparameters and statistical tools, the major difficulty lies within the validation of trained model. Validation is not a simple task of cross checking the accuracy of the model with data outside the training set, but it is an actual representation of the trained model, all the parameters selected, quality of data used. Many categories of validation technique (such as train/test split, k-fold cross validation, nested-cross validation, etc.) have been used in machine learning. Validation of the model is an intricate part of any machine learned model, it is the best way of knowing any trained model’s capability to predict on an unknown or previously not seen data and to identify whether the model is over or under trained.

In case of materials science, the bigger limitation in present scenario are considered to be volume, velocity, variety, and veracity [58,59]. For the volume or observations in dataset, the size of data in context of some field of material science might be considered as considerably low from computer science perspective hence, the use of data and purpose of the model development might play a bigger role than the size of the data itself. However, in many sector of material science there are enormous amount of data in open literature, which brings us to a newer challenge of veracity in material data. The amount of incorrect, potential low quality data generated from not accurate simulations or experiments might result not only in inaccurate model but the model might not even be good enough than a random guess by the human. Third factor that might come into play specially in context of material data is the pace of data generation and publication or velocity of data production which might hinder the necessary growth in development of progressive models. As the machine learning model primarily updates its conclusion with every information update it gathers form the data, the speed at which experimental or simulation data disseminated in public domain is still a hindrance to the machine learning model developers. Finally, the variety or diversity of similar data type might be another big wall to overcome for a computational material science researcher [60]. The extra resource one needs to input for the proper and accurate featurization of numerous data types and format in materials science in general is one of the key limitation in the sector.

6. Integrated Approach for Materials Design and Modeling

Integrating machine learning with other methods [36] effectively multiplies an overall outcome of the study related with material’s structure and properties. Among the several phenomena, strain induced martensitic transformation (SIMT) has received particular interest in relation to the employment of data-driven approaches. So, the concept of integrated approach is first illustrated in data-driven studies of SIMT. Mirzadeh and Najafizadeh [61] have used an artificial neural network (ANN) to model the effects of strain and deformation temperature on volume fraction of strain-induced martensite (SIM) in context of cold-worked AISI 304 stainless steel. The model is built upon a dataset consisting of 115 observations. The associated physical model related to SIM is the Olson Cohen (O-C) equation [62]. The O-C equation [62] expressed mathematically as Equation (2) consists of two important parameters—(i) $\theta$ (parameter representing the rate of shear band formation with respect to strain ), and (ii) $\psi$ (parameter associated with the probability that a shear band intersection might generate a martensitic embryo).

$$f^{mv}=1-exp\{-\psi {[1-exp(-\theta ϵ)]}^n\}$$

(2)

In Equation (2), $f^{mv}$ is the total martensitic volume fraction, and $ϵ$ is macroscopic strain. The exponent term n has a constant value which gives the best fitting range for function of strain induced martensite (SIM). Since the work by [61] has compared results of ANN model with the fitted values of this equation, it can be considered as the integrated machine learning and physical modeling approach to study a SIMT phenomenon.

An expanded approach to the prediction of martensitic transformation induced by strain has been provided in the works by Mu et al. [63]. The authors have combined two models i.e., machine learning model alongside a physical model to not only enhance the prediction accuracy but also to increase the dataset needed for training of ML model. While an initial datasets were obtained from the literature for AISI 200-series (Mn alloyed) steels, AISI 300-series (Ni alloyed) steels as well as related alloys, the authors have then used the O-C model or O-C equation, for expanding the collected experimental dataset with information of strain and temperature dependence. The schematic diagram for the combination of models used by [63] is shown in Figure 7. The O-C model was used to fit the strain in a range for 0 to 1 for every experimental observation to get a continuous function with an interpolated and extrapolated strain values. To further increase the data points to 16,500 with respect to temperature, O-C model was again used to interpolate temperature between experimental temperatures. This was done by evaulating $\theta$ and $\psi$ parameters through modelling experimental data for specific grade steel at known temperature and then using these two parameters to find new interpolated temperature by fitting it linearly, the overall outcome of O-C model was a continuous data which was then discretized before applying to ML model by the authors. The result of ML can be considered as a good correlation with the data, and the credit for that might be the underlying use of physical model to obtain strain values. The model developed from the study uses three inputs, chemical composition, temperature and strain and it provides Stress Induced Martensite (SIM) fraction as output, which can put a different perspective on analyzing the Martensitic start temperature with SIM fraction. The combinatorial model [63] consists of aspects such as feature engineering and ensemble methods, to study the SIMT in different steel alloys.

Integrating machine learning (ML) algorithms with computational models can be a very useful concept. There can be many ways to integrate these two fields but the most intuitive approach is to hierarchically couple the data generated from computational model with the ML algorithm. Data from computational modeling (e.g., phase field simulation) are physically realistic and meaningful machine learning models can be built upon such datasets [36,64]. Phase field method has been utilized in the work by Shchyglo et al. [65] to study the martensite microstructure formation in low-carbon steel. In phase field method, the spatial-temporal evolution of the microstructures is described numerically through the solution of associated partial differential equation(s). As the images of the martensite microstructures obtained from the phase field simulations have physical information, these data can have significant potential to be employed suitable for developing explainable and interpretable machine learning in future. Another way to integrate the computational model with machine learning is by using the inference of computational simulations in the ML model to make a necessary adaptation in the latter. The adaptive design method for multicomponent shape memory alloys design and discovery in the work of Xue et al. [53,54], is tightly coupled with experiments and has taken many inferential inputs from the density functional theory (DFT) computations; and thus is an integrated approach utilized in the study of martensitic alloys.

7. Future Outlook: Towards Physically Intuitive ML Models

The past and current study of martensitic transformation specifically, transformation /start temperature involves more or less empirical methodologies of observing the experimental phenomenon and trying to figure out the causes for the nature of change in structure or property afterwards. This way of collecting and interpreting the information and data often results in inaccuracy and is insufficient to properly describe the transformation. Information collection via purely experimental methods might not only be susceptible to human errors but have high chances of missing the important features particularly non-quantifiable and measurable during experiment but responsible enough to change the course of final result. Hence, incorporating basic known laws and nature of physical transformation by properly utilizing the knowledge of physics and chemistry behind the change could be a major milestone for the study of martensitic start temperature prediction in coming days.

As illustrated in the references [56,61,63], it is now clear that when the information from physical model is integrated with the ML algorithm, then the prediction model will become physically intuitive and accurate. This fact thus highlights the emergence of physics-informed machine learning models in the design and discovery of martensitic alloys in nearby future. Applying Physics Informed Neural Network (PINN) [24] is one of the way by which one can potentially try to discover the physics behind martensitic transformation. By first establishing set of governing equations (mostly partial differential equation (PDE)) and then developing the model capable of solving those equations, it can be possible to incorporates the real physics behind the phenomenon and thus make a more physically intuitive prediction. PINNs are the latest neural network algorithms in today’s fast growing artificial intelligence and machine learning sector which uses mathematical equation governing the physics of the phenomenon and find the solution to those set of equation. These algorithms are different from traditional Finite Element Analysis (FEM) in a sense that PINNs do not need a geometrical mesh model where an element is divided into finite pieces to solve for values at every node but the governing equation is tested and iterated upon the boundary and initial condition by reducing the Mean Squared Error at every iteration using automatic differentiation technique [66]. Figure 8 illustrates the working mechanism of physics informed neural network, where a PDE is solved by minimizing two types of losses, the first where initial and boundary conditions (ICs and BCs) are forced to the governing differential equation and mean square error (MSE(u, BC, IC)) is calculated between calculated value and actual value at initial time and boundary [67]. Secondly, mean square error (MSEf) of the residual function is calculated and minimized on every iteration of neural network by backpropagation [67].

Another way of incorporating a physics-informed neural network is via a combinatory work of empirical data collected from experimental study combined with the known physics and governing laws behind the martensitic transformation to develop a model. A hybrid model can be used where experimental data can be used to enforce physical metric to the physics informed neural network along side the use of initial and boundary conditions.

Development of a digital twin using neural networks trained upon the physics based governing equations and experimental data validation could be the next step in the sector of studying real time martensitic transformation. A digital twin is a computer based virtual environment which acts as a replica of the actual hardware or physical phenomenon capable of providing process information in real time as well as capable of course correcting the path of ongoing phenomenon by controlling the process parameters in a two way data communication between the computer model and physical model [68,69]. By using digital twins in martensitic transformation one can not only predict the transformation temperature but also guide process needed for the desired transformation characteristics by controlling parameters of the system which dictates the transformation such as heat treatment time, temperature of the quenching fluid. A digital twin developed with physics informed neural network model trained with experimental data can be a powerful system for predicting and guiding the martensitic transformation by fully utilizing the intelligent information captured within the digital models which can redirect path of the transformation if found to be deviated from the optimized efficient one in real time.

8. Conclusions

Martensitic start temperature can be considered as one of the major criteria of the martensitic transformation and microstructure formation and hence the study predicting Ms temperature has been carried out since a long time now. Although numerous research works and publications are present in public domain the lack of accurate, precise and uniform prediction of Ms temperature for different type of process conditions is one of the hindrances for the fully utilization of the amazing properties of martensitic alloys. As the data driven method for prediction of mechanical, physical or chemical properties of materials have become quite popular among material science researchers, the martensitic transformation temperature prediction is of no exception where machine learning models are preforming quite better than the empirical models. Since machine learning models are quite resource efficient, high performing and show impeccable accuracy, the use of such models is now becoming quite common in prediction of Ms temperature. In the field of design and discovery of multicomponent shape memory alloys, machine learning models can be utilized to distill out the suitable composition of the alloys which can guarantee lower thermal hysteresis, targeted transformation temperature ranges, high efficiency and high cyclic stability.

Although ML algorithms have become quite strong, the actual prediction accuracy primarily depends on the data on which a model is trained upon. The availability of an accurate, verifiable, quality data in the open literature is one of the major setbacks for the ML model in present time for Ms temperature detection. Along with veracity in the data available, the complex phenomenon of martensitic transformation seems quite difficult to estimate will state of the art machine learning model as seen in the literature.

Hence, it can be concluded that for the accurate and reliable prediction of martensitic start temperature can be imagined only with integrated approach of model development. The use of parameters besides composition such as energy difference between parent and martensite phase of the alloy for the prediction of Ms temperature can be one approach to enhance the model prediction accuracy. Alongside the integrated approach of study, the use of comprehensive combinatory model which not only uses the empirical data from experimental study but also utilizes the actual physics behind the transformation by incorporating governing differential equation of the transformation can be the next research paradigm for martensitic transformation. Physics informed neural networks along side with the traditional experimental data driven machine learning model can be the next big thing in the study of martensitic transformation.