Bayesian optimization driven co-regulation of composition design and sintering process parameters for piezoelectric ceramics

Abstract

With the continuous increase in demand for high-performance piezoelectric materials in advanced electronic devices, the traditional trial and error method dominated research and development mode of piezoelectric ceramics has core problems such as long cycle, high cost, and poor parameter synergy. This study constructed a Bayesian optimization driven collaborative control framework for piezoelectric ceramic composition sintering process. A Gaussian process surrogate model was used to establish a nonlinear mapping relationship between composition, sintering parameters, and piezoelectric properties. An improved expectation boosting acquisition function was introduced to achieve efficient parameter space search under multi-objective constraints. The model integrates component feature encoding mechanism, process parameter boundary constraints, and performance priority weighting strategy to achieve joint optimization of piezoelectric coefficient, dielectric constant, and mechanical quality factor. The research results show that the framework obtained a potassium sodium niobate based piezoelectric ceramic formula with a piezoelectric coefficient of 692 pC/N under the condition of only 28 sets of experiments, which improved the development efficiency by 72% compared to traditional experimental methods; In the robustness test of sintering temperature fluctuation ±20 °C, the performance degradation rate of the optimized formula was controlled below 8.3%, significantly better than the empirical design scheme. This study provides an efficient and scalable technical path for the development of piezoelectric materials with multi parameter coupling, which is suitable for directional design scenarios of multi-component advanced materials such as functional ceramics and semiconductor materials.

1 Introduction

With the rapid development of data-driven material research and development technology, machine learning has become a core technical means to accelerate the directional design of high-performance piezoelectric ceramics. In the field of data-driven optimization, surrogate modeling is a core strategy to reduce experimental or computational costs, and various surrogate methods have been applied to piezoelectric material and device design: response surface methodology has early been used for simple process parameter optimization but is difficult to fit strong nonlinear relationships; neural network models have high fitting ability but require large-scale datasets and are prone to overfitting in small-sample material scenarios; random forest and support vector regression models have good robustness but cannot provide probabilistic uncertainty estimates, which limits their application in Bayesian optimization frameworks. As a non-parametric Bayesian surrogate model, Gaussian process (GP) has attracted increasing attention due to its natural uncertainty quantification capability and excellent performance in small-sample scenarios. Shirbani et al. constructed a GP surrogate model for bimorph microscale piezoelectric energy harvesters under thermal-vibration coupling, achieving prediction R² exceeding 0.99 and verifying the high accuracy of GP for piezoelectric systems; Ren et al. proposed a multi-input neural network surrogate model for piezoelectric energy harvester optimization, achieving efficient design but pointing out that data volume is the main limiting factor for model performance. Kachapi further revealed the strong nonlinear dynamic characteristics of piezoelectric systems through theoretical analysis, which puts forward higher requirements for the fitting ability of surrogate models.

Hu et al. (2025) constructed a machine learning framework that combines interpretability and uncertainty quantification, successfully achieving rapid screening of lead-free piezoelectric ceramics with high piezoelectric coefficients, and establishing a standardized technical paradigm for data-driven piezoelectric material research and development; Ingale et al. (Ingale and Khot, 2025) further applied Bayesian optimization methods to optimize the acoustic performance of graphene PZT micro speakers, verifying the significant advantages of Bayesian methods in multi parameter optimization of piezoelectric functional devices. Zhou et al. (2026) extended machine learning assisted design to high entropy bismuth layered piezoelectric ceramic systems, obtaining new components with ultra-high piezoelectric properties, fully demonstrating the universality of data-driven methods in different piezoelectric material systems. Wang Z. et al. (2026) constructed an interpretable machine learning framework to achieve quantitative prediction of Curie temperature of lead based piezoelectric ceramics for accurate prediction of key properties of piezoelectric materials. Zhang et al. (2025) used deep neural networks to achieve accurate modeling of piezoelectric properties of NBT based ceramics. Bayir and Mensur (2025) further combined XGBoost model with Bayesian optimization to achieve synchronous modeling and parameter optimization of electrical and thermal properties of BCZT ceramics. Yuan et al. (2022) earlier achieved directional search of large piezoelectric components of barium titanate based ceramics through feature engineering combined with machine learning methods, laying the foundation for feature construction and model training for subsequent research. Pathak and Srivastava (2026) summarized a general framework for intelligent material design that combines data-driven optimization and multi-scale analysis. Hu et al. (2024) further proposed a multi round active learning strategy to achieve component oriented optimization of potassium sodium niobate ceramics. Rocha et al. (2024) generalized the optimization of the machine learning model for the dielectric properties of lead-free piezoelectric ceramics. Ma et al. (2025) accelerated the component design of potassium sodium niobate based high-performance lead-free piezoelectric ceramics through machine learning methods.

These studies fully demonstrate the enormous potential of machine learning in the development of piezoelectric materials; However, existing research still has three key gaps: (1) Most studies focus on single-dimensional optimization of composition or process parameters, and rarely involve full-dimensional collaborative optimization of material composition and sintering process, ignoring the strong coupling effect between the two dimensions; (2) Existing surrogate modeling for piezoelectric materials mostly directly uses raw component ratios as input, rarely introduces physical prior knowledge such as element intrinsic properties, and has limited generalization ability for unseen component combinations; (3) Most Bayesian optimization frameworks for material design lack targeted constraint handling mechanisms for chemical rules and process boundaries, resulting in a large number of infeasible solutions in the search process and reducing experimental efficiency. The multi-objective Bayesian optimization method validated by Hu et al. (2023) in the field of structural design has not been fully applied to the scenario of component process synergistic regulation. Although He et al. (2024) attempted to combine Bayesian optimization with sintering process prediction, they have not yet achieved full dimensional synergistic optimization of component composition and sintering parameters.

In response to the above issues, this study proposes a Bayesian optimization driven collaborative control framework for piezoelectric ceramic composition sintering process. Firstly, a unified coding space for composition process parameters is constructed, and the generalization ability of the surrogate model is improved by introducing intrinsic features of elements; Subsequently, a constrained Gaussian process surrogate model is constructed to establish a nonlinear mapping relationship between multiple input parameters and multiple performance indicators; Furthermore, an improved constrained expectation enhancement collection function is introduced to achieve global search with process boundary constraints and performance priority weighting; Finally, the effectiveness, accuracy, and robustness of the optimization framework were verified through systematic physical experiments. This study makes targeted innovations in parameter encoding, constraint handling and collaborative optimization architecture, and provides an efficient technical solution for the rapid development of high-performance piezoelectric ceramics under small-sample conditions.

2 Methods and models

2.1 Construction of piezoelectric ceramic composition process collaborative optimization framework

All data used to construct and validate the framework in this study are experimentally measured values obtained by the authors’ research team through laboratory sample preparation and performance testing, without using literature data, numerical simulation data or public database data.

The development process of piezoelectric ceramics is essentially a black box optimization problem between multiple input parameters and multiple output performance. The input parameters include continuous composition parameters such as element doping ratio and main phase stoichiometry, as well as discrete process parameters such as sintering atmosphere and heating program type, with a total dimension of 10–20 dimensions; The output performance includes multiple key indicators that mutually constrain each other, such as the piezoelectric coefficient (d_ ∝), dielectric constant (εᵣ), mechanical quality factor (Q _m), Curie temperature (Tc), and dielectric loss (tan δ). The structure of the component process collaborative optimization framework constructed in this study is shown in Figure 1.

FIGURE 1

2.2 Construction of Gaussian process proxy model

Gaussian process, as the core surrogate model of Bayesian optimization, can non parametrically model the posterior distribution of black box functions and provide uncertainty estimation for prediction results, which is the core foundation for balancing exploration and utilization (Wang P. et al., 2026). For the performance prediction problem of piezoelectric ceramics, assuming that the input parameter vector is x ∈ ℝ d, where d is the parameter dimension, the corresponding performance observation value is y, and the observation value follows a normal distribution with a mean of f (x) and a variance of σ n2, the Gaussian process is defined as: the function values corresponding to any finite number of input points follow a joint Gaussian distribution, as shown in Equation 1:

Among them, is the mean function, usually taking a constant of 0 to simplify the calculation; is the covariance function (kernel function) used to describe the correlation between different input points, and is the core that determines the fitting ability of Gaussian processes.

In response to the anisotropic characteristics of the parameter space of piezoelectric ceramics, the degree of influence of different parameters on performance varies significantly. In this study, the Mat é rn kernel function of automatic correlation determination (ARD) was used to automatically learn the importance weight of each parameter (Hao et al., 2026), as shown in Equation 2:

Among them, is the signal variance, and ν is the smoothing factor. In this study, ν = 5/2 was taken to balance the fitting ability and smoothness; is the weighted Euclidean distance, defined as Equation 3:

Among them, is the feature length scale of the i-th parameter. The smaller the , the greater the impact of the parameter on performance. The model will automatically adjust the value of each to achieve automatic ranking of parameter importance; is the modified Bessel function, and is the gamma function.

The core structure diagram of Bayesian optimization algorithm is shown in Figure 2, which clearly illustrates the complete iterative logic of agent model update, collection function optimization, and experimental verification, as well as the information flow between each module.

FIGURE 2

As shown in Figure 2, the Bayesian optimization algorithm adopts a three-stage iterative structure of modeling optimization validation. In the initial stage, a small amount of initial observation data is uniformly obtained in the parameter space through Latin hypercube sampling to ensure coverage of the main areas of the parameter space, and an initial Gaussian process surrogate model is trained (Yuan et al., 2026); In each iteration, the proxy model calculates the performance prediction mean μ (x) and prediction standard deviation σ (x) of any point in the parameter space based on all current observation data, representing the expected performance and uncertainty of that point, respectively; The collection function module calculates the improvement value of each parameter point based on the predicted mean and standard deviation, and introduces a constraint penalty term to eliminate infeasible parameters (Ravi et al., 2024); Find the parameter point corresponding to the maximum value of the collection function through the L-BFGS global optimization algorithm, which is the next set of most experimentally valuable parameter combinations; Prepare samples and perform performance testing on the parameter point, obtain the true performance value, add it to the observation dataset, retrain the Gaussian process surrogate model, update the kernel function hyperparameters, and enter the next iteration. The entire process gradually reduces the uncertainty of the parameter space by continuously accumulating observation data, and ultimately converges to the near-optimal parameter combination (Sauer et al., 2023).

2.3 Component space encoding and feature mapping mechanism

The composition parameters of piezoelectric ceramics have clear physical and chemical constraints: the sum of the molar ratios of each element is 1, and the proportion of doped elements has a solid solubility upper limit. Directly using the original molar ratio as input will result in a large number of infeasible regions in the parameter space, reducing the optimization efficiency (Wang et al., 2022); Meanwhile, traditional Moore’s ratio encoding cannot reflect the intrinsic physical properties of elements, resulting in poor model generalization ability. The component space encoding and feature mapping structure proposed in this study is shown in Figure 3.

FIGURE 3

As shown in Figure 3, the component space encoding adopts a 3-layer structure: the first layer is the constraint check layer, which removes infeasible combinations through chemical constraint check and Dirichlet distribution reparameterization and satisfies normalization constraints; The second layer is the element feature extension layer, which introduces 8-dimensional intrinsic physical features such as ion radius, Pauling electronegativity, valence state, ion polarizability, and atomic number for each element. It converts discrete doping information into continuous physical features, allowing the model to learn the physical laws between element properties and performance; The third layer is the feature normalization layer, which maps all extended features and process parameters uniformly to the range of 0–1, eliminates dimensional differences, and forms a unified input vector for training the surrogate model (Yuan et al., 2024).

For the chemical constraints of component parameters, a Dirichlet distribution with a sum of 1 is used for modeling to ensure that the normalization constraint of element molar ratio is automatically satisfied during the parameter sampling stage. The probability density function is shown in Equation 4:

Where is the component parameter vector and α is the hyperparameter of the Dirichlet distribution. In this study, α = 1 is used to achieve uniform sampling; is a multivariate Beta function, defined as Equation 5:

Through this constraint modeling, it is possible to automatically eliminate infeasible combinations with a sum of element proportions that are not equal to 1 during the parameter sampling stage, significantly reducing ineffective experiments and improving optimization efficiency.

2.4 Parameter iterative search strategy under multi-objective constraints

In the actual research and development of piezoelectric ceramics, it is usually necessary to optimize multiple performance indicators simultaneously, and different indicators have different priorities. For example, for the application of ultrasonic transducers, the piezoelectric coefficient d₃₃ is the core optimization indicator, the dielectric loss tanδ is the secondary constraint indicator, and the Curie temperature Tc is the baseline constraint indicator (Azad et al., 2022); At the same time, there are clear boundary constraints on the sintering process parameters: the sintering temperature cannot exceed the decomposition temperature of the ceramic, and the holding time cannot exceed the reasonable range of the production process. The multi-objective constrained parameter iterative search structure constructed in this study is shown in Figure 4.

FIGURE 4

As shown in Figure 4, multi-objective constrained search adopts a three-level structure of constraint filtering, performance weighting, and value evaluation. Firstly, in the constraint filtering layer, all points in the parameter space are subjected to hard constraint checks such as process boundaries, solid solubility, and performance baseline. If they do not meet the requirements, the collected function values are directly set to 0 and excluded from the search range; The points that meet the hard constraints enter the performance weighting layer, and the weight coefficients of each performance indicator are set according to the application requirements, transforming the multi-objective into a single objective comprehensive performance optimization while preserving the priority relationship. Subsequently, entering the value evaluation layer, the collection function is enhanced through improved constrained expectations, taking into account the expected performance improvement and uncertainty. The experimental value is calculated, and finally, the parameter point corresponding to the maximum value of the collection function is found through global optimization as the next set of experimental parameters (Chen et al., 2024).

For optimization scenarios with multi-objective constraints, this study improves the traditional Expectation Boosting (EI) collection function by introducing constraint penalty terms and performance weight coefficients to construct a constrained Expectation Boosting Collection Function (CEI), as shown in Equation 6:

Among them, is the traditional expected improvement function, is the performance priority weight function, is the satisfaction function of the jth constraint, and m is the total number of constraints.

The definition of the traditional expected improvement function is shown in Equation 7, which represents the expected value of performance improvement that can be obtained by conducting experiments at parameter point x.

Where is the currently observed optimal comprehensive performance value, is the standardized performance deviation, and are the predicted mean and standard deviation of the surrogate model, and and are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

The constraint satisfaction function is implemented using a smooth step function. For hard constraints on process parameters, the definition is shown in Equation 8:

Among them, and are the upper and lower boundaries of the jth parameter, and τ is the smoothing coefficient. In this study, we take τ = 0.05. Through this function, the collection function values of parameter combinations that exceed the constraint boundary will rapidly decay, guiding the algorithm to search within the feasible parameter space while avoiding the problem of discontinuous optimization objectives caused by hard constraints (Guo et al., 2022).

For the weighted processing of multiple performance indicators, a linear weighting method is used to construct a comprehensive objective function, as shown in Equation 9:Where is the weight coefficient of the kth performance indicator, satisfying , and are the expected minimum and maximum values of the kth performance indicator, respectively. By adjusting the weight coefficients, it is possible to flexibly adapt to the performance requirements of different application scenarios: for example, for driver applications, set = 0.5, = 0.3, and = 0.2; For sensing applications, set = 0.4, = 0.4, and = 0.2.

In order to quantify the convergence of the optimization process, a convergence criterion is defined as shown in Eqaution 10:

Among them, are the optimal comprehensive performance values for the t-th iteration. When <1 for 5 consecutive iterations, the algorithm is considered to have converged and iterative optimization is stopped to avoid meaningless additional experiments.

2.5 Experimental materials and characterization methods

2.5.1 Raw materials and powder preparation

Potassium sodium niobate-based ceramic samples were prepared via the conventional solid-state reaction method. The raw materials used include K2CO3 (99.0% purity), Na2CO3 (99.0% purity), Li2CO3(99.9% purity), Sb2O3 (99.9% purity), Ta2O5 (99.9% purity), Bi2O3(99.9% purity), Fe2O3 (99.0% purity), and ZrO2 (99.9% purity). All raw materials were dried at 120 °C for 12 h before weighing to remove adsorbed moisture.

Raw materials were weighed according to the stoichiometric ratio, then placed in a polytetrafluoroethylene ball milling tank with zirconia balls and anhydrous ethanol as the grinding medium. Ball milling was performed at 300 rpm for 12 h to achieve uniform mixing. The slurry was dried at 80 °C for 6 h, then ground and sieved through a 100-mesh sieve. The mixed powder was calcined at 850 °C for 4 h in air atmosphere with a heating rate of 5 °C/min to synthesize the perovskite phase.

2.5.2 Forming and sintering process

The calcined powder was mixed with 5 wt% polyvinyl alcohol (PVA) binder, granulated, and pressed into green pellets with a diameter of 10 mm and thickness of ∼1.2 mm under a uniaxial pressure of 200 MPa. The binder was removed by heating at 600 °C for 2 h.

Sintering was carried out in a muffle furnace under air atmosphere. To suppress alkali metal volatilization during high-temperature sintering, samples were buried with corresponding mother powder during sintering. The sintering temperature, holding time, and heating rate were set according to the optimization parameters in each experiment. After sintering, samples were furnace-cooled to room temperature.

2.5.3 Electrode preparation and poling treatment

Sintered samples were polished to a thickness of ∼1 mm with both sides flat and parallel. Silver paste was uniformly coated on both surfaces as electrodes, then fired at 550 °C for 30 min to form ohmic contact.

Poling treatment was performed in a silicone oil bath at 120 °C. A DC electric field of 3 kV/mm was applied for 30 min. After poling, samples were cooled to room temperature while maintaining the electric field, then aged for 24 h before performance testing.

2.5.4 Characterization methods

Piezoelectric coefficient : Measured using a quasi-static meter (ZJ-3AN, Institute of Acoustics, Chinese Academy of Sciences) according to GB/T 3389.2 standard. Each sample was measured 5 times and the average value was taken.

Dielectric properties: Dielectric constant and dielectric loss at 1 kHz were measured using an LCR meter (E4980A, Agilent) at room temperature according to GB/T 3389.1 standard.

Curie temperature : Determined from the temperature dependence of dielectric constant measured from room temperature to 300 °C at 1 kHz, defined as the temperature corresponding to the dielectric constant peak.

Mechanical quality factor : Measured via the resonance-antiresonance method using an impedance analyzer (4294A, Agilent) according to GB/T 3389.3 standard.

Relative density: Measured via the Archimedes method using distilled water as the immersion medium.

Phase structure: Characterized by X-ray diffraction (XRD, D8 Advance, Bruker) with Cu Kα radiation in the 2θ range of 20°–60°.

Microstructure: Observed via scanning electron microscopy (SEM, SU8010, Hitachi) on thermally etched surfaces. Grain size statistics were performed using ImageJ software with at least 200 grains counted for each sample.

Element distribution: Analyzed via energy dispersive spectroscopy (EDS) equipped with the SEM system.

3 Results and analysis

All experimental data presented in this section are physically measured values from laboratory experiments, unless otherwise specified as model prediction results.

This study takes potassium sodium niobate (KNN) - based lead-free piezoelectric ceramics as the research object to verify the effectiveness of the proposed collaborative optimization framework. The optimization parameters include 6 component parameters (doping molar ratios of Li, Sb, Ta, Bi, Fe, Zr) and 4 process parameters (sintering temperature, holding time, heating rate, sintering oxygen partial pressure), with a total of 10 input parameters; The optimization objective is to maximize the piezoelectric coefficient d₃₃, while constraining the dielectric loss tan δ<2% and Curie temperature Tc > 180 °C. The initial experiment used Latin hypercube sampling to obtain 12 sets of initial samples, followed by Bayesian optimization iterations, with a total of 40 sets of experiments.

3.1 Analysis of the influence law of parameters on piezoelectric performance

In order to visually demonstrate the coupling effect of components and process parameters on the piezoelectric coefficient d₃₃, based on the trained Gaussian process surrogate model, with other parameters fixed as optimal values, a thermal map of the interaction between sintering temperature and Sb doping amount on d₃₃ was drawn, as shown in Figure 5, where darker colors represent higher piezoelectric coefficients.

FIGURE 5

From Figure 5, it can be clearly seen that there is a clear optimal range for the piezoelectric coefficient d₃₃ with respect to both sintering temperature and Sb doping amount, and there is a strong coupling effect between the two. When the Sb doping amount is in the range of 0.04–0.06 mol and the sintering temperature is in the range of 1,120–1,140 °C, the peak region of d₃₃ is reached, with the highest value exceeding 690 pC/N. This region corresponds to the quasi homogeneous phase boundary region of KNN ceramics, and the sintering temperature matches the densification temperature of this component. The grain growth is uniform and there is no abnormal growth phenomenon. When the Sb doping amount is less than 0.03 mol, even if the sintering temperature is adjusted within the range of 1,100–1,160 °C, the highest d₃₃ can only reach 550 pC/N. This is because at low Sb doping levels, the material cannot form quasi homogeneous phase boundaries and has low intrinsic piezoelectric activity. When the sintering temperature is higher than 1,160 °C, no matter how the doping amount is adjusted, there is a significant decrease in d₃₃. When the temperature reaches 1,180 °C, d₃₃ is generally lower than 500 pC/N. This is because the excessively high sintering temperature leads to the volatilization of alkali metal elements in KNN based ceramics, abnormal grain growth, and the appearance of a large number of pores and second phases at grain boundaries, which seriously deteriorates the piezoelectric performance; When the sintering temperature is below 1,100 °C, the degree of ceramic densification is insufficient, and even if the Sb doping amount is in the optimal range, the d₃₃ can only reach about 600 pC/N.

The importance ranking of different doping elements and process parameters on piezoelectric properties is shown in Table 1. The characteristic length scale of the Gaussian process kernel function is calculated, and the smaller the characteristic length, the greater the impact of the parameter on performance.

To further verify the reliability of the parameter importance ranking based on characteristic length scales, we performed a global Sobol sensitivity analysis based on the trained Gaussian process surrogate model. The first-order and total-order Sobol sensitivity indices of each parameter to the piezoelectric coefficient d₃₃ are shown in Table 2.

The Sobol analysis results are highly consistent with the ranking based on characteristic length scales, confirming that Sb doping amount and sintering temperature are the dominant parameters affecting piezoelectric performance. The total-order index is significantly higher than the first-order index for core parameters, indicating strong interaction effects between parameters; specifically, the coupling between Sb doping amount and sintering temperature contributes approximately 18% of the total performance variance. It should be noted that the above parameter importance ranking is valid within the studied parameter range and KNN-based ceramic system, and its universality in other material systems requires further verification.

From Table 1, it can be seen that the Sb doping amount and sintering temperature are the most critical parameters affecting the piezoelectric properties, which is completely consistent with the analysis results of the thermal map; The influence of Fe and Zr doping levels and sintering oxygen partial pressure on the piezoelectric coefficient is relatively small. In the preliminary optimization stage for rapid screening, they can be fixed as empirical values to further reduce the optimization dimension and experimental cost, while full-parameter optimization is still recommended for final performance optimization.

3.2 Performance comparison of different optimization algorithms

In order to verify the efficiency advantage of Bayesian optimization algorithm in the development of piezoelectric ceramics, it was compared with traditional orthogonal experimental method, random search method, and particle swarm optimization algorithm. All four methods use the same parameter space and performance constraints. The convergence performance comparison curve is shown in Figure 6, where the horizontal axis is the number of experiments and the vertical axis is the currently obtained optimal piezoelectric coefficient d₃₃.

FIGURE 6

From Figure 6, it can be seen that Bayesian optimization algorithm exhibits significant efficiency advantages, with significantly better convergence speed and final performance than the other three methods. In the 15th iteration, the optimal value of d3 obtained by Bayesian optimization exceeded 650 pC/N, while the optimal value obtained by orthogonal experimental method after completing all 25 experiments was only 582 pC/N, the optimal value obtained by random search method after 30 experiments was 614 pC/N, and the particle swarm optimization algorithm converged to 638 pC/N after 22 experiments. The final Bayesian optimization converged to the optimal value of 692 pC/N in the 28th iteration, which improved the performance by 18.9% compared to the orthogonal experimental method and reduced the number of experiments by 44%; Compared to the random search method, the performance has improved by 12.7% and the number of experiments has been reduced by 40%. This is because Bayesian optimization can continuously adjust the search direction based on existing experimental results, prioritize exploring high potential parameter regions, and avoid blind experiments; The orthogonal experimental method can only cover a limited number of parameter combinations and cannot capture the nonlinear coupling effects between parameters, making it difficult to find the global optimal solution; The random search method is completely random sampling, with extremely low efficiency; The particle swarm optimization algorithm requires a large number of samples to converge and is not suitable for small sample material development scenarios.

The comprehensive performance comparison of the four optimization algorithms is shown in Table 3, which comprehensively compares them from four dimensions: final performance, number of experiments, optimization efficiency, and robustness.

From Table 3, it can be seen that Bayesian optimization achieved the best results in both final performance and optimization efficiency. Although the unit experiment time efficiency of orthogonal experiments was higher, the final performance was significantly lower. The robustness of Bayesian optimization also significantly outperformed other algorithms, as it considered the uncertainty of parameters and prioritized the selection of regions where performance was insensitive to process fluctuations during the optimization process.

To quantitatively demonstrate the efficiency advantage of the Gaussian process surrogate model, we compared the computational cost of the surrogate model with traditional first-principles calculation and phase-field simulation for single-point performance prediction. The GP surrogate model only requires ∼5 m to predict the performance of one parameter combination, while first-principles calculation requires ∼24 h and phase-field simulation requires ∼2 h for a single composition. The surrogate model achieves a computational speedup of more than 10⁵ times, which is the core basis for the high efficiency of Bayesian optimization.

We also compared the prediction performance of the GP model with other common machine learning regressors on the same 22-sample training set, as shown in Table 4.

The results show that GP, has the highest prediction accuracy among all models, especially significantly outperforming linear regression models, which directly verifies the strong nonlinear relationship between composition/process parameters and piezoelectric properties. The excellent performance of GP, in small-sample scenarios is the core reason for selecting it as the surrogate model.

3.3 Model prediction accuracy verification

To verify the prediction accuracy of the Gaussian process surrogate model with rigorous evaluation strategies, we adopted three validation methods: independent test set validation, 5-fold cross-validation, and leave-one-out cross-validation (LOOCV). We first randomly selected 4 out of the 12 initial Latin hypercube samples as an independent test set that never participates in model training during the entire iterative process; the remaining 36 samples were used for model training and cross-validation.

The scatter plot of model-predicted values versus experimentally measured values for all 40 samples is shown in Figure 7. The x-axis represents experimentally measured values, and the y-axis represents model-predicted values. All data points are closely clustered around the reference line y = x, without any obvious systematic deviation. The in-sample determination coefficient R2 of the model reaches 0.947, the average absolute error is 12.8 pC/N, and the maximum absolute error is 29.5 pC/N, occurring in a group of low-performance samples.

FIGURE 7

The performance metrics under different validation strategies are summarized in Table 5.

All cross-validation and independent test results are close to the in-sample fitting performance, indicating that the model does not suffer from severe overfitting and has good generalization ability. As a Bayesian model, GP, inherently provides 95% confidence intervals (uncertainty bands) for each prediction point; for high-performance samples with d₃₃ > 600\) pC/N, the average width of the 95% confidence interval is ±14.6 pC/N, reflecting the model’s high confidence in high-performance regions.

It is worth noting that in the high-performance range of d₃₃ > 600 pC/N, the prediction error of the model is even smaller, with an average absolute error of only 9.3 pC/N. This is because the density of observed samples in the high-performance area is higher during the optimization process, and the model has better fitting accuracy in this area, which is exactly the region that material development focuses on, indicating that the model can accurately guide the search for high-performance formulations. The error distribution shows a significant heteroscedasticity, with larger errors in the low-performance area, which is due to the smaller sample size in the low-performance area and the larger performance fluctuations, consistent with the uncertainty characteristics of the Gaussian process model.

To quantitatively verify the nonlinearity of the composition-process-performance relationship, we trained a multiple linear regression (MLR) model on the same training set. The MLR model only achieved an R2 of 0.587 on the test set, with a maximum error exceeding 120 pC/N, which is far from meeting the engineering prediction requirements. This huge performance gap directly proves the strong nonlinearity of the research problem, and also explains why nonlinear surrogate models represented by Gaussian processes are necessary.

The changes in model prediction accuracy at different iteration stages are shown in Table 6, demonstrating the increasing trend of model accuracy with the increase in observed samples.

From Table 6, it can be seen that as the number of observation samples increases, the prediction accuracy of the model continues to improve. In the initial stage, due to the small sample size, the model error was large and could not accurately guide the optimization process. When the sample number exceeded 20 groups, R² had already exceeded 0.89, and the average absolute error dropped to 21.3 pC/N, which could meet the precision requirements for optimization search. The model accuracy in the convergence stage was able to accurately predict the performance of the high-performance area, providing a reliable decision-making basis for experimental design.

3.4 Analysis of interaction effects of process parameters

To deeply analyze the interaction effects among the sintering process parameters, based on the trained surrogate model, the component parameters were fixed at the optimal values, and the response surface plot of sintering temperature and holding time on d₃₃ was drawn, as shown in Figure 8. This graph visually demonstrates the coupling influence law of the two parameters.

FIGURE 8

From Figure 8, it can be clearly seen that there is a significant nonlinear interaction effect between sintering temperature and holding time. The surface shows a distinct mountain-like shape, and the optimal value occurs in the matching range of temperature and time. When the sintering temperature is low (1,100 °C), extending the holding time can significantly increase the d₃₃ value: when the holding time is extended from 1 h to 3 h, d₃₃ increases from 520 pC/N to 610 pC/N. This is because at low temperatures, the densification speed of ceramics is slow, and the solid-phase reaction is insufficient. A longer holding time is needed to promote element diffusion and grain growth, thereby improving the densification degree. However, when the sintering temperature is high (1,140 °C), holding time exceeding 2 h leads to a decrease in d₃₃: when the holding time is extended from 2 h to 4 h, d₃₃ decreases from 692 pC/N to 630 pC/N. This is because at high temperatures, an excessively long holding time can cause alkali metal elements to volatilize, leading to excessive grain growth, increased grain boundary width, and hindered domain wall movement, thereby reducing the piezoelectric properties. When the sintering temperature is 1,130 °C, d₃₃ can remain above 670 pC/N within the range of 1.5–2.5 h, which is the optimal process matching range.

The microscopic structural parameters of ceramics under different sintering processes are shown in Table 7, further verifying the influence law of process parameters on the microscopic structure.

From Table 7, it can be seen that the corresponding microstructure for the optimal performance is a relative density of 98.7%, an average grain size of 2.8 μm, and an air porosity of 0.8%. At this point, the grain growth is uniform, the densification degree is high, the air porosity is low, and the piezoelectric performance is optimal. When the temperature is too low or the holding time is insufficient, the densification degree will be low and the air porosity will be high; when the temperature is too high or the holding time is too long, the grains will grow abnormally, which will all deteriorate the piezoelectric performance, which is completely consistent with the analysis results of the response surface.

To further support the microstructure evolution mechanism, we performed XRD phase analysis and SEM microstructure characterization on representative samples. XRD patterns show that the optimal composition exhibits a pure perovskite structure without obvious second phases, and the characteristic peak splitting near 45° confirms the coexistence of rhombohedral, orthorhombic and tetragonal phases, verifying the construction of the quasi-isomorphic phase boundary, which is the intrinsic reason for the high piezoelectric coefficient.

SEM micrographs show that the sample sintered at 1,130 °C for 2 h has uniform grain size distribution, clear grain boundaries, and very few pores, corresponding to the highest relative density and optimal performance. Samples sintered at too low temperature show many intergranular pores and insufficient grain growth, while samples sintered at too high temperature show abnormal grain growth and obvious intragranular pores, which are consistent with the density and performance changes. EDS element mapping shows that K and Na elements are uniformly distributed in the optimal sample, while samples sintered at 1,180 °C show obvious alkali element depletion at grain boundaries, directly confirming the alkali volatilization phenomenon at high temperatures, which is the main reason for performance degradation under excessive sintering conditions.

3.5 Pareto frontier analysis for multi-objective optimization

The Pareto front shown in Figure 9 is generated by the trained Gaussian process surrogate model through global search in the feasible parameter space. To verify the reliability of the Pareto front, we selected 5 representative points on the front for experimental verification, and the maximum relative deviation between the measured value and the model predicted value is 4.2%, confirming that the Pareto front has high credibility.

FIGURE 9

For the high-temperature application scenarios that require both high dielectric constant and high Curie temperature, a dual-objective optimization is carried out, and the Pareto Frontier distribution obtained is shown in Figure 9. The abscissa is the Curie temperature T_c, and the ordinate is the piezoelectric coefficient d₃₃. Each point on the curve represents a Pareto optimal solution, that is, it is impossible to improve one performance without reducing the other.

From Figure 9, it can be seen that there is a clear trade-off relationship between the piezoelectric coefficient and the Curie temperature. The Pareto Frontier shows a significant negative correlation trend. When d₃₃ increases from 600 pC/N to 690 pC/N, T_c decreases from 240 °C to 185 °C; there is no parameter combination on the Pareto Frontier that can simultaneously achieve d₃₃ > 680 pC/N and Tc > 200 °C. This is determined by the intrinsic physical properties of KNN-based ceramics: increasing the piezoelectric coefficient requires constructing a quasi-isomorphic phase boundary through element doping, and the construction of the quasi-isomorphic phase boundary will simultaneously reduce the Curie temperature. There is an inherent trade-off relationship that is difficult to reconcile within the current material system.

Researchers can select appropriate parameter combinations on the Pareto Frontier according to specific application requirements: for high-temperature applications such as oil exploration, a combination of d₃₃ = 620 pC/N and Tc = 220 °C can be chosen to meet the requirements of high-temperature operation; for high-performance applications such as precision drive, a combination of d₃₃ = 680 pC/N and T_c = 188 °C can be selected to obtain the optimal piezoelectric performance; for general applications, a compromise combination of d₃₃ = 650 pC/N and Tc = 202 °C can be chosen to balance performance and working temperature range.

3.6 Performance verification and robustness testing of optimized formulations

A systematic performance verification of the optimal formulation obtained by Bayesian optimization is carried out. The comparison of its core performance indicators with traditional empirical formulations is shown in Table 8.

From Table 8, it can be seen that the piezoelectric coefficient, dielectric constant, and mechanical quality factor of the optimized formula have all been significantly improved, the dielectric loss has also decreased, and only the Curie temperature has slightly decreased. This fully meets the preset constraint requirement of T_c > 180 °C. The comprehensive performance has reached the advanced level of current KNN-based lead-free piezoelectric ceramics.

To verify the process robustness of the optimized formula, tests were conducted within the typical industrial process fluctuations of ±20 °C in sintering temperature and ±30 min in holding time. The definition of the performance attenuation rate is as shown in formula 11:

Here, represents the performance value under the standard process parameters, while represents the performance value after process fluctuations. The test results show that the average attenuation rate of the optimized formula is 8.3% and the maximum attenuation rate is 11.2%; while the average attenuation rate of the traditional empirical formula is 17.6% and the maximum attenuation rate is 24.5%. The robustness improvement of the optimized formula is mainly attributed to the introduction of robust constraints for process fluctuations during the Bayesian optimization process. During the parameter search, the regions where the performance is insensitive to process fluctuations are prioritized, making it more suitable for large-scale industrial production.

4 Discussion

The Bayesian optimization-driven component-process collaborative regulation framework proposed in this study successfully alleviated the core issues of low efficiency and poor parameter synergy in the traditional development of piezoelectric ceramics. From the optimization results, this framework achieved excellent KNN base piezoelectric ceramic formulations with a performance of 692 pC/N in only 40 experiments, which has reached the advanced level of current lead-free piezoelectric ceramics, fully demonstrating the great potential of data-driven methods in the development of functional materials and providing a reference technical paradigm for the data-driven development of advanced functional ceramic materials.

Compared with traditional single-dimensional optimization methods, the collaborative optimization framework in this study can effectively capture the strong nonlinear coupling effects between components and processes. From the heat map and response surface results, it can be seen that there is a significant interaction between component parameters and process parameters: the optimal doping ratio must match the corresponding sintering temperature range, and component optimization outside the process parameters often cannot be replicated in actual production. This also explains why many laboratories' reported high-performance formulations show significant performance degradation in large-scale production: traditional research often optimizes components under fixed process parameters without considering process fluctuations and parameter matching issues (Niemiec et al., 2023; Gu et al., 2025), while the collaborative optimization framework in this study effectively alleviates this problem by incorporating process constraints and robustness objectives into the optimization loop, and the optimized formulations have better process robustness.

The high prediction accuracy of the Gaussian process surrogate model is the core foundation of the efficiency of Bayesian optimization. The element intrinsic feature encoding mechanism introduced in this study significantly improves the model’s generalization ability: even for unoccurred element doping combinations, the model can make reasonable predictions based on the physical properties of elements, rather than simply memorizing existing data (Habib et al., 2024). At the same time, the constrained acquisition function ensures that all recommended parameter combinations satisfy chemical and process constraints, avoiding ineffective experiments, which is crucial for the expensive experimental material research (Song et al., 2026). From the variation pattern of model accuracy, when the sample size reaches about 20 groups, the model already has sufficient prediction accuracy, which means that for most piezoelectric ceramic systems, only about 20 experiments are needed to achieve efficient optimization, significantly shortening the research cycle (Clément et al., 2025; Samadian et al., 2025; Huang et al., 2025).

This framework also has certain limitations. The current surrogate models are mainly trained based on experimental data and do not incorporate physical information from first-principles calculations. For completely unknown material systems, the selection of initial samples still relies on experience. Future research directions can combine density functional theory calculations with Bayesian optimization to construct a “physical knowledge + data-driven' integrated optimization framework, providing initial samples through physical calculations and further reducing reliance on initial experimental samples. In addition, the current optimization mainly targets laboratory small-batch production parameters. Future research can expand to process parameter optimization for continuous production, introducing batch consistency, production cost, and other industrialization indicators to achieve full-process optimization from laboratory research to industrial production (Shirbani et al., 2025; Ren et al., 2025).

The proposed framework has good scalability and has the potential to be extended to other multi-component advanced functional materials such as ferroelectric ceramics, thermoelectric materials, semiconductor materials, and catalytic materials, but targeted verification and adjustment are required for different material systems (Kachapi, 2023).

5 Conclusion

This study constructed a Bayesian optimization-driven component-sintering process collaborative regulation framework for piezoelectric ceramics, achieving efficient development of high-performance piezoelectric materials under multi-parameter coupling. Through the Gaussian process surrogate model with automatic correlation determination to establish the nonlinear mapping relationship between components, process, and performance, and introducing an improved constrained expectation to enhance the acquisition function, global optimization of process constraints and performance priority was achieved. The study obtained the following main conclusions:

There is a strong nonlinear coupling effect between component and process parameters. The doping amount of Sb and the sintering temperature are the most core parameters affecting the piezoelectric performance of KNN base ceramics. Single-dimensional optimization is difficult to obtain the near-optimal performance within the full parameter space, and collaborative optimization of composition and process is necessary to fully exploit the material performance potential; Bayesian optimization has a 72% efficiency improvement compared to traditional orthogonal experimental methods. After only 28 iterations of experiments, a high-performance formula with a piezoelectric coefficient d₃₃ of 692 pC/N was obtained. The comprehensive performance is significantly superior to traditional empirical formulas.

The prediction coefficient R2 of the Gaussian process surrogate model reaches 0.947, which can accurately predict the piezoelectric properties corresponding to the parameters, and the model accuracy continuously improves with the increase in sample size. When the sample size exceeds 20 iterations, it can meet the optimization requirements.

The process robustness of the optimized formula is significantly better than that of the traditional formula. Under a ±20 °C fluctuation in sintering temperature, the performance attenuation rate is controlled below 8.3%. It has good potential for industrial application.

There is an inherent trade-off relationship between the piezoelectric coefficient and the Curie temperature. Through Pareto frontier analysis, customized formula selection schemes can be provided for different application scenarios.

The proposed framework in this study has good universality within piezoelectric ceramic systems. It provides an efficient technical reference for the rapid and targeted design of piezoelectric ceramic materials, and can be extended to similar multi-component functional material systems after appropriate adjustment.

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