The development of lead-free KNN-based piezoceramics has become one of the most rapidly growing trends of materials science in recent decades due to their high piezoelectric response and environmental friendliness [1, 2]. The search for new functional materials has been conventionally based on an extremely difficult empirical approach [3]. Despite the multiple discoveries that approach has led to, it detracts from significant limitations, most clearly expressed in the current epoch of growing complexity of multicomponent material compositions and the requirement to minimize the time of their implementation [3–5]. Since the path from a fundamental understanding to a commercial product can take decades [6], accelerated search for material compositions and optimization methods acquire special importance.

One of the most promising tools to solve the above task is machine learning (ML). The implementation of ML methods gives new opportunities in materials science for accelerated search for compositions with preset characteristics and retrieval of latent regularities in the “composition – structure – properties” system [7]. Machine learning models can analyse large datasets, identify nonlinear relationships between the composition and properties of materials, and predict, with a high precision, the parameters of new, yet not synthesised compounds [7]. Such methods are already successfully used for the prediction of extended properties, including piezoelectric coefficients [8], Curie temperatures [9], and piezoelectric permittivity [10]. Of special interest are lead-free KNN-based piezoelectric materials, which are considered as an environmental-friendly alternative to noxious lead zirconate-titanate-based materials [11, 12]. Chemical modification through complex multicomponent doping is the key method to improve the piezoelectric properties [2], yet it leads to an exponential growth of the number of possible composition combinations, thus making them ideal for ML applications [13].

Despite the high predictive capability, most of the existing ML models, such as deep neural networks or complex ensemble methods, often operate by the “black box” principle, which limits their application in materials science [14]. The lack of interpretability complicates the understanding of latent mechanisms determining the properties of materials and undermines confidence to the predictions [15]. Therefore increasing attention is paid to the application of interpretable models.

One of the most powerful tools to develop interpretable models is the use of symbolic regression, e.g. the sure independence screening and sparsifying operator (SISSO) method [16]. Sure independence screening and sparsifying operator allows identifying compact analytical descriptors which directly associate the fundamental properties of elements with the target property, thus delivering a high degree of interpretability. For example, not only the piezoelectric coefficient d₃₃ of KNN-based ceramics was successfully predicted using SISSO, but also an interpretable descriptor was suggested which correlates with phase transition mechanisms underlying the high piezoelectric response of materials [8]. In another work, SISSO allowed identifying the key role of covalent and Van-der-Vaals radii of B-cations in the ABO₃ perovskite structure in the determination of the d₃₃ of KNN, with an error of ~40 pC/N [17].

Along with interpretable methods, e.g. SISSO, prediction accuracy is maximized in case of wide application of ensemble machine learning methods, including extra trees regressor (ETR), random forest (RF), gradient boosting regressor (GBR), and extreme gradient boosting (XGBoost). For example, the ETR model with an optimum set of seven features allowed predicting the d₃₃ of KNN with an error of ~49 pC/N [8]. Active learning based on ML models also showed its efficiency in the synthesis of a KNN composition with d₃₃ ~353 pC/N, which is far above the respective figures of most specimens in the initial training set [13]. Importantly, the efficiency of models depends on the correct choice of features. Studies show that such characteristics as electronegativity, ratio between the number of valence electrons and the nominal charge, and electron core radius (by Schubert) are the key ones not only for the prediction of the piezoelectric properties of KNN, but also for the classification of perovskites into ferro- and antiferroelectrics [4].

Despite the above achievements, most works are focused on the prediction of a single feature, typically, d₃₃, whereas the practical application of materials requires a set of characteristics be known, e.g. the piezoelectric coefficient, the Curie temperature T_C, and the dielectric permittivity ε.

The aim of this work is system analysis and comparing the efficiency of ensemble machine learning algorithms (RF, ETR, XGBoost, and GBR) in predicting the three key properties of lead-free KNN-based piezoelectric ceramics: the piezoelectric coefficient d₃₃, the Curie temperature T_C, and the dielectric permittivity ε. The work will also deal with comparison between the models for prediction accuracy, retrieval of physically meaning regularities using SISSO in combination with SHAP, and validation of the modeling results against experimental data borrowed from literature.

The target properties for this work were the piezoelectric coefficient (d₃₃), the dielectric permittivity at room temperature measured at a frequency of 10 kHz (ε), and the Curie temperature (T_C).

The key tool of machine learning model interpretation is to evaluate the contribution of individual features, and this was implemented in this work using the SHAP method. Figure 5 shows SHAP values for descriptors illustrating the degree of their impact on the prediction of the three target properties using the optimal models which demonstrated the highest accuracy.

For interpretable machine learning in the analysis of KNN-based ceramic materials, the key result is the identification of the descriptors which have the lowest SHAP values, indicating their predominant impact on the prediction of the target properties.

A significant impact of the descriptor f₂ = (NV_B – MN_B)² on the predicted piezoelectric coefficient d₃₃ can be accounted for by the physical nature of the parameter MN_B (nuclear magnetic moment of the B-cations in ABO₃-type perovskite structures). The parameter MN_B exerts a strong impact on the value of d₃₃, since it is an effective indicator which integrates the electron configuration and crystal structure that determine the piezoelectric response of materials [8]. According to earlier works, local ferroelectric distortions induced by dopants at the B-sites directly determine Landau’s local energy barrier, thus critically affecting polarization and switching under external electric fields [90, 91]. The parameter MN_B correlates with the electronegativity and hybridization degree of the d-orbitals of the ions at the B-sites with 2p-orbitals of the surrounding oxygen atoms, this controlling the rigidity of the chemical bonds and the mobility of the ion in the oxygen cluster [11].

The greatest impact on the prediction of dielectric permittivity comes from the descriptor f₄ = μ × MN_A+ (MN_B – r_A). The component µ is the octahedral factor determining the geometry of the oxygen octahedron BO₆. According to classical standpoints [86], the ratio of the ionic radii of B and O affects the distortion degree of the octahedrons. The expression μ × MN_A describes the relationship between the geometry and the electron configuration of the atoms the A-sites (volume and effective radii: the nuclear magnetic moment MN_A correlates with the electronegativity and polarizability of the ions in the A-sublattice, hence determining the contribution of electron polarization [8]. The expression (MN_B − r_A) shows the interaction between the electron configuration of the B-cations and the dimensional factor of the A-cations. As shown earlier [8], MN_B is important for the description and prediction of d₃₃ (this is, accordingly, related to the value of polarization), while the ionic radius r_A controls the free space left for ionic shifts, which is of special importance for phase transitions [92]. An increase in the polarizability of the B-cation (large MN_B) acts to increase the electronic and ionic polarization. Optimization of the octahedron geometry (μ → 1) decreases the energy barrier for ionic shifts. Achievement of a balance between the A-cation size and the electron configuration of the B-site favors the formation of morphotropic phase boundaries (orthorhombic- tetragonal, rhombohedral-orthorhombic, rhombohedral-tetragonal), leading to high dielectric responses [92]. Thus, the descriptor f₄ covers the multicomponent nature of dielectric response: electronic polarization (MN_A, MN_B), ionic polarization in phase transition regions (μ, lattice distortions).

Analysis of SHAP values allowed to highlight the descriptor f₂ = DCE_B × V_B + (DCE_B – DCE_A), which integrates the electron configuration and geometric parameters of atoms determining the Curie temperature (T_C) of KNN-based ceramics. The validity of using the above-listed atomic features was additionally confirmed by earlier works [93], showing that the valence of the B-cation, ionic radius, and ionic shifts exert a decisive impact on T_C prediction. The component DCE_B describes the electron configuration of the B-cation, thus determining the covalence degree of the B–O bond. The expression DCE_B × V_B describes the relationship between the electronic configuration and geometric parameters of the atoms at the B-sites, where the atomic volume V_B correlates with the unit cell size and the free space left for ionic shifts. The expression (DCE_B − DCE_A) represents the difference in the electron configurations of the B- and A-cations, caused by the distribution of electronic density and distortion of oxygen octahedrons. The latter descriptor covers the key factors determining the T_C of KNN-based ceramics: the electronic structure of the B-cation (DCE_B), lattice geometry (V_B), and balance of the electron configuration parameters of the A- and B-cations (DCE_B − DCE_A).

In this work, the prediction of d₃₃, ε, and T_C of KNN-based ferroelectric ceramics was carried out sequentially using a hybrid strategy, i.e., a combination of the Sure Independence Screening and Sparsifying Operator (SISSO) with ensemble machine learning methods (Extra Trees Regressor (ETR), Random Forest (RF), Gradient Boosting Regressor (GBR), and eXtreme Gradient Boosting (XGBoost)).

For each target property (the piezoelectric coefficient d₃₃, the dielectric permittivity ε, and the Curie temperature T_C), descriptors were identified that exert the maximum impact on the predictive capability of the machine learning models. We showed that the best results in d₃₃ prediction were achieved using the ETR model, delivering an average relative error of about 10.7 %. for the validation set which was not used for leaning. For ε, the smallest error (7.8 %) was also obtained for the ETR model, and for T_C, the best result was achieved using the RF model, the error being 6.7 %.

Thus, physically interpretable predictors were developed that allow accurate prediction of the key ferroelectric parameters of KNN-based ceramics. The results confirm the efficiency of the suggested approach combining the SISSO method with ensemble machine learning methods and open up new possibilities for the design of materials with tailored functional parameters.

The study was performed with financial support from the Russian Science Foundation (grant No. 24-79-00281).