A method for rapidly improving the flexoelectric performance of functional ceramics based on a Gaussian machine learning process model

Abstract

This invention discloses a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, comprising the following steps: S1: Preparation of input dataset; S2: Establishment of a Gaussian process GP regression model; S3: Establishment of a Bayesian optimization machine learning model; S4: Synthesis and characterization of the optimal compound; S5: Flexural coefficient analysis of multi-matrix composite ceramic materials; S6: Analysis of machine learning prediction capabilities. This invention employs the aforementioned method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model. It can not only predict the quasi-linear relationship between the flexural coefficient and the compound, but also accelerate the discovery of the currently predicted optimal material with high flexural electrical properties, increasing the accuracy of experimental design, reducing costs, avoiding time-consuming experimental operations, and providing necessary theoretical and experimental basis for guiding experimental design in predicting the properties of new ceramic materials.

Description

Technical Field

[0001] This invention relates to the field of machine learning technology, and in particular to a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model. Background Technology

[0002] Flexoelectricity describes the coupling effect between strain gradient and polarization, as well as the coupling effect between electric field gradient and elastic stress. Flexoelectricity has two prominent characteristics: (1) it is a universal electromechanical coupling effect present in all dielectric materials; (2) the bending electric effect can be larger at the nanoscale because the strain gradient is inversely proportional to the sample size. Compared with piezoelectricity, flexoelectricity can reduce the dependence of devices on materials, thereby expanding the application range of devices. In recent years, many researchers have measured the flexoelectricity of various materials, but the results show that the flexoelectric coefficient obtained by experimental methods is much larger than the flexoelectric coefficient estimated by researchers. In addition, some scholars have also focused on the experimental measurement of the buckling electric coefficient of various materials. However, it is worth noting that the measured values ​​of the flexoelectric coefficient of many materials are still very small, which may not provide a basis for the fabrication of flexoelectric devices. Therefore, the premise for designing macroscopic flexoelectric devices is that the flexoelectric value of the material should be increased or can be improved. At present, most studies rely on trial and error or empirical methods, which are limited to the degree of freedom of structural interactions, microstructure and chemical substances.

[0003] In recent years, machine learning techniques have been widely applied in materials informatics, achieving significant success in high-throughput computing and accelerating materials discovery. This machine learning algorithm can effectively learn from initial training data and construct a statistical model that connects target properties with material descriptors. The constructed inference model is then used to predict target properties of unexplored materials. The updated inference model can be quantized to generalize new compositions of the target material. By carefully selecting features or descriptors, target properties of unexplored materials can be predicted based on a vast unexplored space. Here, an active machine learning method known as Bayesian optimization (BO) or Gaussian process (GP) has been widely applied to predict material properties such as large electrical strain band gaps, large piezoelectricity, and binding energies of perovskite compositions. BO can adaptively search for new compounds to guide the next experiment and provides potential value for selecting materials for the next experimental measurement by utilizing uncertainty and maximizing active learning functions (such as expectation improvement functions, upper confidence bound functions, etc.). The BO machine learning method balances the trade-off between exploration (uncertainty) and development (using only the best properties predicted by the inference model), providing an optimal standard for designing the next experiment. Summary of the Invention

[0004] The purpose of this invention is to provide a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model. This method is efficient and accurate, and can not only predict the quasi-linear relationship between the flexural electrical coefficient and the compound, but also accelerate the discovery of the best materials with the highest predicted flexural electrical properties. It increases the accuracy of experimental design, reduces costs, avoids time-consuming experimental operations, and provides necessary theoretical and experimental basis for guiding the experimental design of new ceramic material properties.

[0005] To achieve the above objectives, this invention provides a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, comprising the following steps:

[0006] S1: Input dataset preparation;

[0007] S2: Establish a Gaussian process GP regression model;

[0008] S3: Establish a Bayesian optimized machine learning model;

[0009] S4: Synthesize and characterize the optimal compound;

[0010] S5: Perform flexural conductivity analysis on multi-component composite ceramic materials;

[0011] S6: Perform machine learning prediction capability analysis.

[0012] Preferably, in step S1, the input dataset is used to design experiments using different elements at the A and B sites of the multi-based composite ceramic material, their corresponding dielectric constant values, and five characteristic values.

[0013] Preferably, in step S2, a Gaussian process GP regression model is established, expressed using a mean function and a covariance function. Given a function f(x), it follows a Gaussian process with a mean m(x) and a covariance function k(x,x'), expressed as:

[0014] f(x)~GP(m(x),k(x,x')) (1)

[0015] Suppose there exists a dataset D t ={(x i y i )}, i = 1, 2, ..., t, where y i =f(x) i ).

[0016] Preferably, the Gaussian covariance function, as the popular squared exponent kernel, is expressed as:

[0017]

[0018] Where θ is a hyperparameter, determined using maximum likelihood estimation. The unconstrained optimization problem is solved by the following expression:

[0019]

[0020] Preferably, the initial training dataset D is used for predicting compounds through a GP regression model. t ={(x i y i )} and new experimental point (x) t+1 y t+1 The joint distribution of ) is expressed as:

[0021]

[0022] Where, k = [k(x1, x t+1 ),k(x2,x t+1 ),…,k(x t ,x t+1 )] T .

[0023] Preferably, based on the Sherman-Morrison-Woodbury formula, the predicted distribution of the function values ​​is expressed as:

[0024]

[0025] in

[0026] μ t (x t+1 )=k T K -1 y 1:t (6)

[0027]

[0028] Preferably, in step S3, the Bayesian optimization machine learning model is established by using a GP regression model to construct a functional relationship between the initial sample set and the vector of mole fraction x and the flexural coefficient y, i.e., y = f(x). The optimization problem is described as follows:

[0029] x * =argmax x∈X f(x) (8)

[0030] Where X is the field of x;

[0031] The expected improvement to the EI function is adopted, and the improved function is defined as follows:

[0032] I(x) = max{0, f t+1 (x)-f(x * )} (9)

[0033] Where, x * =argmaxf(x i ).

[0034] Preferably, the improved EI function is expressed analytically as follows:

[0035] EI(x)=zσ(x)Φ(z)+σ(x)φ(z) (10)

[0036] in, Φ(z) is the standard normal cumulative distribution function and the standard normal probability density function.

[0037] Preferably, in step S5, the flexural conductivity of the multi-component composite ceramic material is analyzed and converted to average polarization using the following formula:

[0038]

[0039] Where Q is the polarization charge;

[0040] The average strain gradient in the thickness direction is calculated using the following formula:

[0041]

[0042] According to the definition of flexural conductivity, the flexural conductivity is obtained using formulas (11) and (12):

[0043]

[0044] Preferably, in step S6, a machine learning prediction capability analysis is performed, and the mean squared error (MSE) and cross-validation error (CVE) are calculated using the following formulas to evaluate the prediction capability of BO:

[0045]

[0046]

[0047] Where n is the total number of compounds in the training dataset, y i It is the measured value of the flexural coefficient. It is the average of the predicted values ​​from 1000 computational samples generated from a virtual space of all possible compounds. It is the predicted value of the i-th compound.

[0048] Therefore, this invention adopts the above-mentioned method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, and develops an efficient and high-precision strategy. It can not only predict the quasi-linear relationship between the flexural electrical coefficient and the compound, but also accelerate the discovery of the best material with the highest predicted flexural electrical properties. This increases the accuracy of experimental design, reduces costs, avoids time-consuming experimental operations, and provides necessary theoretical and experimental basis for guiding the experimental design to predict the properties of new ceramic materials.

[0049] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0050] Figure 1 This invention presents a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model. The BO method accelerates the discovery of the maximum flexural electrical coefficient through iterative guided experiments.

[0051] Figure 2 This is a schematic diagram of the first iteration of the method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, using the BO method to predict the synthesis and characterization of compounds.

[0052] Figure 3 This is a schematic diagram of the second iteration of the synthesis and characterization of a compound based on the BO method, which is a method for rapidly improving the flexural electrical properties of functional ceramics according to the Gaussian machine learning process model of the present invention.

[0053] Figure 4 This invention provides a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model. The method uses the BO method to predict the synthesis and characterization of compounds in the third iteration.

[0054] Figure 5 This invention relates to a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, which uses three iterations of Gaussian process machine learning.

[0055] Figure 6 This invention relates to an experimental apparatus for measuring the flexural coefficient of a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model.

[0056] Figure 7 This invention relates to a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, which addresses cross-validation errors performed by different regression models. Detailed Implementation

[0057] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0058] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0059] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention, and no reference numerals in the claims should be construed as limiting the scope of the claims.

[0060] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can be appropriately combined to form other embodiments that can be understood by those skilled in the art. These other embodiments are also covered within the scope of protection of this invention.

[0061] It should also be understood that the specific embodiments described above are only used to explain the present invention, and the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

[0062] Techniques, methods, and apparatus known to those skilled in the art may not be discussed in detail, but where appropriate, they should be considered part of the specification.

[0063] All prior art documents cited in this specification are incorporated herein by reference in their entirety and are therefore part of the disclosure of this invention.

[0064] Example 1

[0065] like Figure 1As shown, this invention provides a method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model. The first step is to input a dataset, using two groups (A-site and B-site) of different elements in a multi-matrix composite ceramic material, their corresponding dielectric constant values, and five characteristic values ​​for experimental design. Group A consists of Ba, Ca, Sr, and Cd. Group B consists of Ti, Zr, Sn, and Hf. The dielectric constant is set at an experimental temperature of 25℃ and a frequency of 1kHz. The five characteristic values ​​are ionic radius, bond distance, electronegativity, ionic displacement, and conductivity.

[0066] Second, establish a Gaussian process (GP) regression model. A Gaussian process is a stochastic process in which each finite set of random variables follows a joint Gaussian distribution. In fact, a GP process can be viewed as a multivariate Gaussian distribution over an infinite-dimensional vector. The GP model can be represented by a mean function and a covariance function. Given a function f(x), which follows a Gaussian process with a mean m(x) and a covariance function k(x,x'), it can be expressed as:

[0067] f(x)~GP(m(x),k(x,x')) (1)

[0068] Suppose there exists a dataset D t ={(x i y i )}, i = 1, 2, ..., t, where y i =f(x) i In this context, when the mean m(x) is assumed to be 0, the Gaussian process depends only on the covariance function k(x,x'). It is noteworthy that the kernel k(x,x') plays the most crucial role in the Gaussian process because it reveals prior knowledge related to the properties of the modeling function and also reflects the correlation between any two arbitrary points. Choosing the Gaussian covariance function as the popular squared exponent kernel, it is expressed as:

[0069]

[0070] Here, θ is a hyperparameter that significantly affects the accuracy of the GP regression model; therefore, maximum likelihood estimation is used to determine θ. The unconstrained optimization problem is then solved by the following expression:

[0071]

[0072] It is important to note that the optimal parameter θ should be searched through global optimization. opt And an improved DIRECT algorithm is used. To consider using the GP regression model for compound prediction, the initial training dataset D... t ={(x i y i)} and the joint distribution of the new experimental point (x t+1 , y t+1 ) can be expressed as:

[0073]

[0074] where k = [k(x1, x t+1 ), k(x2, x t+1 ), …, k(x t , x t+1 )] T .

[0075] Based on the Sherman-Morrison-Woodbury formula, the predictive distribution of the function value can be expressed as:

[0076]

[0077] where

[0078] μ t (x t+1 ) = k T K -1 y 1:t (6)

[0079]

[0080] It can be seen from equations (6) and (7) that the predictive variance of the GP regression model depends only on the covariance function and the input data set x 1:t . At the same time, the marginal distribution of any point is a univariate Gaussian model. Because of the properties of the multivariate Gaussian distribution, this predictive model can make the calculation of the acquisition function easier.

[0081] The GP model can be constructed according to the above formula, and then the target properties such as the flexoelectric coefficient can be predicted according to the experimental data set.

[0082] Third, establish a Bayesian optimization machine learning model. To accelerate the discovery of the maximum flexoelectricity in the Ba x Sr (1-x) TiO3 ceramic cluster, an optimization compound was synthesized from the design steps using machine learning methods. The search space is defined by the compounds given by Ba x Sr (1-x) TiO3, where only a vector of one mole fraction x is considered, which can be restricted to 0.6 < x < 0.9. First, a series of combinations are selected to require experimental design. The initial sample set consists of a vector of mole fraction x and the flexoelectric coefficient y. Then, the GP regression model is used to construct the functional relationship between x and y, that is, y = f(x).

[0083] Bayesian machine learning is a widely used method for searching for the extrema of an unknown objective function f(x) to predict the optimal combination. Therefore, the optimization problem can be formally described as:

[0084] x * =argmax x∈X f(x) (8)

[0085] Where X is the domain of x. BO consists of two main components: the first is a meta-model, which can evaluate at any point of uncertainty, using a common GP regression model as described in step two. The second is an active learning function, which determines the components to be evaluated next. This technique is used experimentally to provide the optimal criteria for designing the next experiment. This invention employs the most widely used Expected Improvement (EI) function because of its simple analytical formula. First, the improvement function is defined as:

[0086] I(x) = max{0, f t+1 (x)-f(x * )} (9)

[0087] Where, x * =argmaxf(x i ).

[0088] The expected improvement (EI) function can be expressed analytically as follows:

[0089] EI(x)=zσ(x)Φ(z)+σ(x)φ(z) (10)

[0090] in, Φ(z) represents the standard normal cumulative distribution function and the standard normal probability density function. EI acquisition maintains a trade-off between development and exploration. Compounds can be synthesized and characterized using the mean and variance of each composition from candidate samples. The acquisition function should be maximized to search for the composition of the next evaluation function. EI sampling is inexpensive and will be computed using efficient global optimization methods, such as the optimized DIRECT algorithm.

[0091] The process of selecting the next compound for experimentation using the BO optimization algorithm based on the acquisition function is as follows:

[0092] (1) Define the initial experimental dataset: D t ={(x 1:t y 1:t )};

[0093] (2) Construct the GP regression model: y = f t (x)~GP(m(x),k(x,x'));

[0094] (3) Determine the upper and lower limits X of the search space;

[0095] (4) Expand the training dataset:

[0096] (5) Perform Bayesian optimization using EI functions:

[0097] a. For t = 1: t upper

[0098] b. Use the optimized EI function equation (10) to find the next combination x t+1

[0099] c. Calculate this new combination x t+1 The objective function y t+1 =f t+1 (x)+ε

[0100] d. Increase {x} t+1 y t+1 Add to the training dataset:

[0101] e. Update the GP regression model

[0102] (6) End

[0103] Subsequently, the predicted compound with the largest EI is synthesized and characterized, completing one iteration from BO, and the new combination, along with the measurement properties associated with it, are augmented into the training dataset. Therefore, the next iteration begins by updating the new GP regression model from the augmented training dataset. The total number of iterations depends on the algorithm designer and is limited by the cost of synthesizing new compounds.

[0104] Fourth, based on the above, the BO method guides the next experiment, providing feedback for enhancing the dataset by measuring and characterizing the synthesized new materials. Through preliminary experiments, the two sets of elements and their corresponding dielectric constants are used as the training dataset. Then, the experimental design is extended to a virtual space with approximately 500,000 possible compounds, which cannot be handled through traditional trial and error. Therefore, BO machine learning is used to learn from the existing data and iteratively guide the experiment with the optimal compounds. A GP regression model is used to construct the relationship between composition and dielectric constant based on the initial training dataset.

[0105] Experiments yielded results from three iterations of predictive compound synthesis and characterization using machine learning with a Gaussian process exhibiting uncertainty. The maximum flexural constant was 10⁵ in the first iteration, 118 in the second, and 139 in the third. A total of three iterations were performed. After each iteration, the dielectric constant of the new compound was measured, and the results were augmented into the training dataset for a new round of regression and statistical analysis. A possible stopping criterion is to stop after a fixed number of iterations. Figure 2 – Figure 4 The results of three iterations of predicting the synthesis and characterization of compounds using the BO method are shown.

[0106] from Figure 5 It can be seen that the predicted compound composition is Ba, Sr, and Ti, and the peak dielectric constant at the Curie temperature point is 10200 during the first iteration. After measuring the dielectric constant value of the compound, the experimental results and the predicted results are found to be in close agreement. The above results are used to guide the next experiment in the synthesis and characterization of the compound. In sequence, the GP regression model can be updated using the augmented dataset. In the second iteration, the predicted dielectric constant value of the compound is 3200, and after three iterations, the maximum dielectric constant of the compound at the Curie temperature point is 16000. Through BO machine learning, the quasi-linear relationship between the flexural coefficient and the composition ratio of the compound's dielectric constant is derived sequentially, and the maximum flexural coefficient is obtained by obtaining the maximum dielectric constant. It also shows that only three experiments are needed to accelerate the discovery of multi-component composite ceramic materials (Ba 1-x-y Ca x Sr z (Ti) 1-u-v Zr u Sn v The maximum flexural conductivity in O3.

[0107] Fifth, we conducted flexural conductivity analysis on multi-component composite ceramic materials. We analyzed the flexural conductivity of (Ba... 1-x-y Ca x Sr z (Ti) 1-u- v Zr u Sn v O3 was subjected to a three-point bending experiment. Figure 6 The experimental setup for measuring the flexural conductivity coefficient is shown. A sinusoidal oscillating force at a frequency of 1 Hz on was applied to the center point of the upper surface of the beam. The two supporting edges were separated by L = 10 mm. The vertical displacement δ of the center point on the upper surface of the beam was recorded. The polarization charge Q was detected using a charge amplifier and an oscilloscope, which can be converted to the average polarization using the following formula:

[0108]

[0109] The average strain gradient in the thickness direction can be calculated using the following formula:

[0110]

[0111] According to the definition of flexural conductivity, the flexural conductivity is obtained using formulas (11) and (12):

[0112]

[0113] Sixth, analyze the predictive power of machine learning. Since the oscillations in the training dataset significantly affect the accuracy of the selected inference model, it is necessary to investigate the effectiveness of other learning strategies. An SVM model is constructed based on the experimental design of the initial compounds to predict the mean and uncertainty. Through efficient global optimization, the compound with the largest EI is selected as a candidate compound from the test training dataset. Subsequently, the SVM model can be retrained and new candidates selected, and this process can be repeated cyclically. To evaluate the predictive power of BO, the mean squared error (MSE) and cross-validation error (CVE) can be calculated using the following formulas:

[0114]

[0115]

[0116] Where n is the total number of compounds in the training dataset, y i It is the measured value of the flexural coefficient. It is the average of the predicted values ​​from 1000 computational samples generated from a virtual space of all possible compounds. It is the predicted value of the i-th compound. Figure 7 The results of cross-validation errors for different regression models are shown to determine the predictive power of active machine learning.

[0117] Therefore, this invention adopts the above-mentioned method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, and develops an efficient and high-precision strategy. It can not only predict the quasi-linear relationship between the flexural electrical coefficient and the compound, but also accelerate the discovery of the best material with the highest predicted flexural electrical properties. This increases the accuracy of experimental design, reduces costs, avoids time-consuming experimental operations, and provides necessary theoretical and experimental basis for guiding the experimental design to predict the properties of new ceramic materials.

[0118] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model, characterized in that: Includes the following steps: S1: Input dataset preparation; S2: Establish a Gaussian process GP regression model; In step S2, a Gaussian process GP regression model is established, expressed using the mean function and covariance function, given a function. It follows a mean Sum of covariance functions The Gaussian process is represented as: （1） Suppose there exists a dataset ,in ; The Gaussian covariance function, as a popular squared exponent kernel, is expressed as: （2） in, These are hyperparameters, determined using maximum likelihood estimation. The unconstrained optimization problem is solved by the following expression: （3）； Predictions of compounds using a GP regression model, initial training dataset and new experimental sites The joint distribution is expressed as: （4） in, ; Based on the Sherman-Morrison-Woodbury formula, the predicted distribution of the function values ​​is expressed as: （5） in （6） （7）； S3: Establish a Bayesian optimized machine learning model; S4: Synthesize and characterize the optimal compound; S5: Perform flexural conductivity analysis on multi-component composite ceramic materials; S6: Perform machine learning prediction capability analysis.

2. The method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model according to claim 1, characterized in that: In step S1, the dataset is input, and the experiment is designed using two sets of different elements at the A and B sites in the multi-based composite ceramic material, their corresponding dielectric constant values ​​and five characteristic values.

3. The method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model according to claim 1, characterized in that: In step S3, the Bayesian optimization machine learning model is established by using a GP regression model to construct a functional relationship between the initial sample set and the vector of mole fraction x and the flexural coefficient y. The optimization problem is described as follows: （8） in, yes The domain; The expected improvement to the EI function is adopted, and the improved function is defined as follows: （9） in, .

4. The method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model according to claim 3, characterized in that: The expected improvement to the EI function is expressed analytically as follows: （10） in, It is the standard normal cumulative distribution function and the standard normal probability density function.

5. The method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model according to claim 1, characterized in that: In step S5, the flexural conductivity of the multi-component composite ceramic material is analyzed and converted to average polarization using the following formula: （11） in, Polarized charge; The average strain gradient in the thickness direction is calculated using the following formula: （12） According to the definition of flexural conductivity, the flexural conductivity is obtained using formulas (11) and (12): （13）。 6. The method for rapidly improving the flexural electrical properties of functional ceramics based on a Gaussian machine learning process model according to claim 1, characterized in that: In step S6, machine learning prediction capability analysis is performed. The mean squared error (MSE) and cross-validation error (CVE) are calculated using the following formulas to evaluate the prediction capability of BO: （14） （15） Where n is the total number of compounds in the training dataset, It is the measured value of the flexural coefficient. It is the average of the predicted values ​​from 1000 computational samples generated from a virtual space of all possible compounds. It is the predicted value of the i-th compound.

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