Method and system for performing prediction and formula fitting on secondary electron spectrum by using LASSO model

Abstract

The invention relates to a method and system for performing prediction and formula fitting on a secondary electron spectrum by using an LASSO model, and the method comprises the steps: selecting a plurality of feature dimensions, and constructing a basic prediction formula; constructing a descriptor pool; constructing a data set based on the descriptor pool, building an LASSO model, and determining an optimal hyper-parameter of the LASSO model; and performing secondary electron spectrum prediction by using the LASSO model with the optimal hyper-parameter, and combining a final prediction formula based on a prediction result and the basic prediction formula. Compared with the prior art, the method has the advantages that a symbolized prediction result can be given, the problem that many black box model results are difficult to explain is solved, the prediction result can be used for analyzing the influence and importance of different factors on the secondary electron energy spectrum, the secondary electron emission phenomenon can be further analyzed and researched, and the performance of related devices is improved.

Description

Technical Field

[0001] This invention relates to the field of secondary electron spectroscopy research, and in particular to a method and system for predicting and fitting secondary electron spectroscopy using the LASSO model. Background Technology

[0002] In vacuum electronic devices, secondary electron emission is unavoidable whenever a primary electron beam is used. However, it affects the performance of vacuum electronic devices and instruments in various ways, depending on the specific device design and operating conditions. In some applications, secondary electrons are very useful; in others, they are harmful and must be avoided. For example, in scientific analysis, scanning electron microscopes use specialized secondary electron detectors to obtain important imaging information from secondary electrons scattered from the target material. On the other hand, the low-energy background signal generated by secondary electrons can interfere with spectral analysis in electron spectroscopy. Therefore, the study of secondary electron emission phenomena is essential.

[0003] Chung and Everhart proposed a simple, highly idealized model for calculating the energy distribution of low-energy secondary electrons, but this model only considers the escape process of secondary electrons and does not include the cascade reactions of secondary electrons. Sickafus proved that a piecewise linear relationship exists when a logarithmic function is applied to the secondary electron energy spectrum, and that this functional form is compatible with the solution of the Boltzmann diffusion equation. Ding et al. used Monte Carlo simulations to consider the phenomenon of cascaded secondary electron generation and calculated the full energy distribution of backscattered electrons from the elastic peak to the real secondary electron peak. This simulation model is based on the dielectric function to describe inelastic scattering and secondary electron excitation, and uses the Mott cross section to describe elastic scattering. Although the general shape of the secondary electron energy spectrum can be described by these theoretical methods, they cannot fully describe the material-related secondary electron peak position information, nor do they have formulas for calculating the absolute peak positions. This is partly due to the limited accuracy of secondary electron energy measurements, and partly because these theories do not consider the influence of related electron scattering on secondary electron emission in materials. Therefore, the theory of secondary electron emission involving electronic structure information has not yet been fully and uniformly studied. Only empirical models exist, and there is no accurate, dimensionally consistent model formula.

[0004] In summary, the existing technology has the following main problems: Currently, there is no accurate physical model or descriptive formula for secondary electron emission. Most existing physical formulas are based on fitting empirical formulas or simplifications of complex phenomena. Furthermore, these formulas are relatively complex and suffer from dimensional inconsistencies, failing to accurately describe the secondary electron energy spectrum and the secondary electron emission phenomenon. Summary of the Invention

[0005] The purpose of this invention is to provide a method and system for predicting and fitting secondary electron energy spectra using the LASSO model. In a data-driven manner, a prediction formula for secondary electron energy spectra that is dimensionally consistent and has certain physical meaning is derived without prior knowledge. This will help to understand the influencing factors of secondary electron emission phenomena and enable a better grasp and utilization of these phenomena.

[0006] The objective of this invention can be achieved through the following technical solutions: A method for predicting and fitting formulas for secondary electron energy spectra using the LASSO model includes the following steps: Based on the physical mechanism of secondary electron emission, multiple feature dimensions are selected from the basic material property parameters and experimental condition parameters, and a basic prediction formula is constructed based on the feature dimensions and the principle of dimensional consistency. Based on the aforementioned basic prediction formula, a descriptor pool consisting of multiple dimensionless descriptors is constructed. A dataset is constructed based on the descriptor pool, a LASSO model is built, multiple candidate hyperparameter values ​​are set, cross-validation is performed on each candidate value based on the dataset, and the optimal hyperparameters of the LASSO model are determined based on the validation results. The secondary electron spectrum is predicted using the LASSO model with optimal hyperparameters, and the final prediction formula is formed by combining the prediction results with the basic prediction formula.

[0007] Furthermore, the basic material properties parameters include at least the number of atoms, the number of valence electrons, the work function, and the Fermi energy, and the experimental conditions parameters include at least the incident voltage and the probe current.

[0008] Furthermore, the basic prediction formula is expressed as: In the formula, The intensity of the secondary electron spectrum. B It is a constant. E The energy of the second electron. a , b , m , n , l The power exponent is to be determined. Z Number of atoms For the number of valence electrons, Let work function be used. For Fermi energy, The incident voltage, To detect current.

[0009] Furthermore, the descriptor pool is constructed by combining different undetermined power exponents.

[0010] Furthermore, the cross-validation is ten-fold cross-validation. Determining the optimal hyperparameter based on the validation results includes: calculating the average score of the ten-fold cross-validation for each candidate value, and selecting the candidate value that makes the evaluation index optimal as the optimal hyperparameter.

[0011] Furthermore, the dataset includes secondary electron spectral data for various crystal orientations under different incident energy conditions.

[0012] Furthermore, constructing the dataset includes: Standardize the data so that each feature dimension in the dataset follows a distribution with a mean of 0 and a standard deviation of 1.

[0013] Furthermore, the combination into the final prediction formula is specifically as follows: The importance coefficients are evaluated based on the output of the LASSO model. Items with an absolute value of importance coefficient greater than a set value are selected as important descriptors. The formula types corresponding to the important descriptors are combined and fitted to form the final prediction formula.

[0014] This invention also provides a system for predicting and fitting formulas for secondary electron energy spectra using the LASSO model, comprising: The formula construction module, based on the physical mechanism of secondary electron emission, selects multiple feature dimensions from the basic material property parameters and experimental condition parameters, and constructs a basic prediction formula based on the feature dimensions and the principle of dimensional consistency. The symbol pool generation module constructs a descriptor pool consisting of multiple dimensionless descriptors based on the basic prediction formula. The model configuration module constructs a dataset based on the descriptor pool, builds a LASSO model, sets multiple candidate hyperparameter values, performs cross-validation on each candidate value based on the dataset, and determines the optimal hyperparameters of the LASSO model based on the validation results. The prediction execution module uses the LASSO model with optimal hyperparameters to predict the secondary electron spectrum, and combines the prediction results with the basic prediction formula to form the final prediction formula.

[0015] Furthermore, the model configuration module includes: The parameter setting unit is used to set the number of hyperparameters, convergence accuracy, and maximum number of iterations. A cross-validation unit is used to perform 10-fold cross-validation based on the dataset; An optimization selection unit is used to select the optimal hyperparameters based on the average score of cross-validation.

[0016] Compared with the prior art, the present invention has the following beneficial effects: 1. No prior knowledge dependency: The method automatically selects important features and fits formula parameters using a data-driven approach, without relying on expert experience or pre-set physical assumptions. This reduces the dependence on prior knowledge in the field and expands the applicability of the method in different material systems.

[0017] 2. Enhanced Physical Interpretability: The LASSO model automatically filters key descriptors through a sparse regularization mechanism, ultimately outputting explicit mathematical expressions rather than a black-box model. This gives the prediction results a clear physical meaning, making it easier to analyze the influence mechanism of factors such as the number of atoms, the number of valence electrons, the work function, the Fermi energy, and the incident voltage on the secondary electron emission phenomenon.

[0018] 3. Multi-scale data fusion: The training dataset covers a variety of crystal orientations and a wide range of incident energy conditions, enabling the prediction formula to capture the secondary electron emission patterns under different crystal structures and energy conditions, thereby enhancing the applicability and robustness of the model. Attached Figure Description

[0019] Figure 1 This is a flowchart illustrating the prediction and formula fitting of secondary electron energy spectrum using the LASSO model in this invention. Figure 2 The figures are data distribution diagrams before and after standardization, where (a) is the data distribution diagram before standardization and (b) is the data distribution diagram after standardization. Figure 3 For the prediction result R 2 Distribution map; Figure 4 Importance distribution of LASSO screening results. Detailed Implementation

[0020] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0021] Example 1 This embodiment provides a method for predicting and fitting formulas for secondary electron energy spectra using the LASSO model, such as... Figure 1 As shown, it includes the following steps: S1. Based on the physical mechanism of secondary electron emission, multiple feature dimensions are selected from the basic material property parameters and experimental condition parameters, and a basic prediction formula is constructed based on the feature dimensions and the principle of dimensional consistency.

[0022] In this embodiment, considering factors that may affect the secondary electron energy spectrum, the number of atoms Z and the number of valence electrons are selected. Work function Fermi energy Incident voltage Detecting current As a feature dimension.

[0023] Based on the selected feature dimensions, a dimensionally consistent basic prediction formula is constructed, which is used to measure the intensity of the secondary electron spectrum. Represented as: In the formula, B It is a constant. E The energy of the second electron. a , b , m , n , l The power exponent is to be determined. Z Number of atoms For the number of valence electrons, Let work function be used. For Fermi energy, The incident voltage, To detect current. Except All quantities except those are dimensionless, and the dimensions of both sides of the equation are consistent.

[0024] S2. Construct a descriptor pool. Specifically, based on the dimensionless ratio term in the basic prediction formula, generate multiple power combinations of the dimensionless ratio term, multiply the power combination with the probe current, and form a descriptor pool composed of multiple formula types.

[0025] In practice, Set as the target quantity, the dimensions in the original data include Z, , , , There are six items in total. Calculations are performed using the basic forecasting formula. , , The powers of the terms are obtained by iterating through the powers of each ratio term and combining them. The combined result is then compared with... Multiplying these results in different formula types, forming a new descriptor pool. In this embodiment, the final symbol pool contains 7776 items.

[0026] S3. Data Preprocessing: Construct a dataset based on the descriptor pool, and standardize the data in the dataset so that the mean of each feature dimension is 0 and the standard deviation is 1.

[0027] In practice, the dataset selected included electron spectra of different elements (100, 110, and 111 crystals) at incident energies of 1 keV, 1.2 keV, 1.5 keV, and 2 keV. Electron spectra with incident energies above 2 keV were added as supplementary data to optimize the training results. StandardScaler was used to process the dataset, transforming each feature into a distribution with a mean of 0 and a standard deviation of 1. The data distribution before and after standardization is shown below. Figure 2 As shown, the dataset is divided into two parts: a training set and a validation set.

[0028] S4. Build the LASSO model and perform hyperparameter design.

[0029] Specifically, the hyperparameter design process includes: The model automatically selects multiple hyperparameter α values ​​and uses the processed dataset for 10-fold cross-validation training. The average cross-validation score corresponding to the α value is obtained by averaging the results of the 10-fold cross-validation. Compare the average cross-validation scores for all candidate αs; Choose the α that optimizes the evaluation index as the hyperparameter of the final model; Design other parameters such as training rounds and learning rate.

[0030] In this embodiment, the model automatically selects 150 values ​​of regularization strength parameter α for training, sets the convergence accuracy to 0.01, and sets the maximum number of iterations to 1,000,000. The average cross-validation score corresponding to each α value is obtained by averaging the results of the 10-fold cross-validation. The average cross-validation scores corresponding to all candidate α values ​​are compared, and the α that optimizes the evaluation metric is selected as the final model's hyperparameter. The model's hyperparameter design is shown in Table 1.

[0031] Table 1 Hyperparameter design of the LASSO model S5, Training and Prediction: Standardized data is input into the LASSO model with optimal hyperparameters for training and prediction. Based on the coefficient screening results of the LASSO model, the target formula type is determined from the descriptor pool. Specifically, the importance coefficient is evaluated according to the output results of the LASSO model, and items with an absolute value of importance coefficient greater than a set value are selected as important descriptors. That is, several items with larger absolute values ​​are selected, and the formula types corresponding to the important descriptors are combined and fitted to form the final prediction formula.

[0032] In practice, the standardized data (which can be new data on the target material) is fed into the LASSO model for prediction, and the R-value of the prediction result is obtained. 2 Distribution as Figure 3 As shown. After prediction, items with coefficients whose absolute values ​​are greater than a set value are selected as importance descriptors. The formula types corresponding to the importance descriptors are combined to form the final prediction formula. The importance distribution of the LASSO screening results is shown below. Figure 4 As shown.

[0033] The above method utilizes the LASSO model to predict and fit formulas for secondary electron spectra. It constructs a basic prediction formula from a dimensional analysis perspective and expands the symbol pool. Using a data-driven approach, the LASSO algorithm yields an explicit, dimensionally consistent prediction formula for secondary electron spectra. The core lies in constructing and combining dimensionless variables based on relevant physical quantities during the symbol pool establishment process, thereby achieving dimensionally consistent prediction results. Cross-validation and automatic hyperparameter search are employed during LASSO training to improve prediction accuracy, ensuring the model predicts with optimal parameter settings and avoiding predictions that are difficult to interpret or do not conform to physical laws, as seen in other algorithms.

[0034] If the above methods are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0035] Example 2 This embodiment provides a system for predicting and fitting formulas for secondary electron energy spectra using the LASSO model, including a formula construction module, a symbol pool generation module, a model configuration module, a data processing module, and a training execution module.

[0036] The formula construction module, based on the physical mechanism of secondary electron emission, selects multiple feature dimensions from the basic material property parameters and experimental condition parameters, and constructs a basic prediction formula based on the feature dimensions and the principle of dimensional consistency. The symbol pool generation module constructs a descriptor pool consisting of multiple dimensionless descriptors based on the basic prediction formula. The model configuration module constructs a dataset based on the descriptor pool, builds a LASSO model, sets multiple candidate hyperparameter values, performs cross-validation on each candidate value based on the dataset, and determines the optimal hyperparameters of the LASSO model based on the validation results. The prediction execution module uses the LASSO model with optimal hyperparameters to predict the secondary electron spectrum, and combines the prediction results with the basic prediction formula to form the final prediction formula.

[0037] The model configuration module includes: a parameter setting unit for setting the number of hyperparameters, convergence accuracy, and maximum number of iterations; a cross-validation unit for performing 10-fold cross-validation based on the dataset; and an optimization selection unit for selecting the optimal hyperparameters based on the average score of the cross-validation.

[0038] The rest is the same as in Example 1.

[0039] The above description of the embodiments is provided to enable those skilled in the art to understand and use the invention. It will be apparent to those skilled in the art that various modifications can be made to these embodiments, and the general principles described herein can be applied to other embodiments without inventive effort. Therefore, the present invention is not limited to the above embodiments, and any improvements and modifications made by those skilled in the art based on the disclosure of the present invention without departing from the scope of the invention should be within the protection scope of the present invention.

Claims

1. A method for predicting and fitting formulas for secondary electron energy spectra using the LASSO model, characterized in that, Includes the following steps: Based on the physical mechanism of secondary electron emission, multiple feature dimensions are selected from the basic material property parameters and experimental condition parameters, and a basic prediction formula is constructed based on the feature dimensions and the principle of dimensional consistency. Based on the aforementioned basic prediction formula, a descriptor pool consisting of multiple dimensionless descriptors is constructed. A dataset is constructed based on the descriptor pool, a LASSO model is built, multiple candidate hyperparameter values ​​are set, cross-validation is performed on each candidate value based on the dataset, and the optimal hyperparameters of the LASSO model are determined based on the validation results. The secondary electron spectrum is predicted using the LASSO model with optimal hyperparameters, and the final prediction formula is formed by combining the prediction results with the basic prediction formula.

2. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 1, characterized in that, The basic material properties parameters include at least the number of atoms, the number of valence electrons, the work function, and the Fermi energy, and the experimental conditions parameters include at least the incident voltage and the probe current.

3. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 1, characterized in that, The basic prediction formula is expressed as follows: ， In the formula, The intensity of the secondary electron spectrum. B It is a constant. E The energy of the second electron. a , b , m , n , l The power exponent is to be determined. Z Number of atoms For the number of valence electrons, Let work function be used. For Fermi energy, The incident voltage, To detect current.

4. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 3, characterized in that, The descriptor pool is constructed by combining different undetermined power exponents.

5. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 1, characterized in that, The cross-validation is a 10-fold cross-validation. Determining the optimal hyperparameter based on the validation results includes: calculating the average score of the 10-fold cross-validation for each candidate value, and selecting the candidate value that makes the evaluation index optimal as the optimal hyperparameter.

6. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 1, characterized in that, The dataset includes secondary electron spectral data for various crystal orientations under different incident energy conditions.

7. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 1, characterized in that, The construction of the dataset includes: Standardize the data so that each feature dimension in the dataset follows a distribution with a mean of 0 and a standard deviation of 1.

8. The method for predicting and fitting secondary electron spectra using the LASSO model according to claim 1, characterized in that, The specific combination into the final prediction formula is as follows: The importance coefficients are evaluated based on the output of the LASSO model. Items with an absolute value of importance coefficient greater than a set value are selected as important descriptors. The formula types corresponding to the important descriptors are combined and fitted to form the final prediction formula.

9. A system for predicting and fitting formulas for secondary electron energy spectra using the LASSO model, characterized in that, include: The formula construction module, based on the physical mechanism of secondary electron emission, selects multiple feature dimensions from the basic material property parameters and experimental condition parameters, and constructs a basic prediction formula based on the feature dimensions and the principle of dimensional consistency. The symbol pool generation module constructs a descriptor pool consisting of multiple dimensionless descriptors based on the basic prediction formula. The model configuration module constructs a dataset based on the descriptor pool, builds a LASSO model, sets multiple candidate hyperparameter values, performs cross-validation on each candidate value based on the dataset, and determines the optimal hyperparameters of the LASSO model based on the validation results. The prediction execution module uses the LASSO model with optimal hyperparameters to perform secondary electron spectrum prediction, and combines the prediction results with the basic prediction formula to form the final prediction formula.

10. The system for predicting and fitting secondary electron spectra using the LASSO model according to claim 9, characterized in that, The model configuration module includes: The parameter setting unit is used to set the number of hyperparameters, convergence accuracy, and maximum number of iterations. A cross-validation unit is used to perform 10-fold cross-validation based on the dataset; An optimization selection unit is used to select the optimal hyperparameters based on the average score of cross-validation.

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