PBX crack evolution prediction method fusing discrete element and machine learning
By integrating discrete element and machine learning methods, a crack prediction model for PBX materials is constructed, which solves the problem of accurate characterization of microcrack identification and evolution path of PBX materials in existing technologies, and achieves high-precision and explainable crack prediction, which is suitable for structural optimization and safety assessment of various heterogeneous energetic materials.
Patent Information
- Application Number
- CN202510960214.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-11
- Publication Date
- 2025-10-17
AI Technical Summary
Existing technologies make it difficult to accurately characterize the discrete behavior between particles and the nonlinear damage mechanism of interfaces in PBX materials, identify the starting location and evolution path of microcracks, and machine learning models lack visualization and quantitative interpretation of the contribution of physical characteristics, and have limited generalization capabilities.
By integrating the discrete element method with machine learning, a two-dimensional particle structure model was constructed for mechanical loading simulation, physical descriptors were extracted and feature correlation analysis was performed. The XGBoost algorithm was used to construct a crack classification model, which was combined with the Stacking integrated regression model to predict the crack time. The SHAP method was used for interpretability analysis.
It achieves high-precision prediction and explainability of microcracks in PBX materials, is applicable to a variety of heterogeneous energetic materials, supports structural optimization, safety assessment and virtual testing, and has good versatility and engineering application value.
Smart Images
Figure CN120808956A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the technical field of energetic material structure behavior prediction, and particularly relates to a PBX crack evolution prediction method fusing discrete elements and machine learning. BACKGROUND
[0002] Polymer bonded explosive (PBX) is a typical heterogeneous material composed of energetic crystal particles and binder matrix, which is widely used in high-performance sensitive equipment such as aerospace and weapon systems. The mechanical response of PBX is not only affected by the loading mode, but also significantly dominated by the microstructure characteristics such as particle size, spatial distribution and interfacial bonding strength. Under dynamic or quasi-static loading, the generation and propagation of microcracks usually precede macroscopic failure, which is a key factor affecting the mechanical stability and service safety of PBX materials.
[0003] In practical engineering applications, the crack evolution mechanism of PBX shows obvious multi-scale and multi-mode characteristics. Microcracks often originate from the crystal-matrix interface, interfacial voids and stress concentration zones between particles, and then gradually expand and connect under the driving of local stress redistribution, eventually forming a system-level failure path. Accurate identification of the initiation position, evolution time and spatial path of cracks is of great significance for realizing the reliability prediction, structure safety evaluation and damage control of energetic materials.
[0004] Existing research primarily relies on numerical methods such as finite element (FEM), extended FEM, or continuum damage models to model the deformation and failure processes of PBX materials. However, these methods suffer from the following shortcomings: first, they cannot accurately describe discontinuous behaviors within the grain-level structure; second, they oversimplify or obscure the crack initiation and propagation processes; and third, they have limited accuracy in modeling microscopic behaviors such as stress localization and interfacial debonding, making it difficult to fully characterize the evolutionary process from microcracks to macroscopic failure paths. Furthermore, while experimental methods such as scanning electron microscopy (SEM), X-ray micro-CT, and digital image correlation (DIC) can be used to characterize crack morphology in damaged materials, they are often limited by resolution, test windows, and time segmentation, making it difficult to achieve real-time, high-precision observation of crack formation. In particular, there are still technical limitations in identifying microcracks during the initial loading phase. In recent years, with the advancement of machine learning, data-driven approaches have been introduced into the field of material behavior modeling, demonstrating strong capabilities in predicting properties and fracture behavior. However, most mainstream methods are "black-box" models, lacking the ability to physically interpret prediction results. Furthermore, the model's generalization ability is limited by the quality of the training data, and its application to highly complex, heterogeneous material systems remains insufficient. Furthermore, current crack prediction research often overlooks the coupling between mechanical field characteristics and structural geometric parameters, lacking an interpretable modeling system capable of integrated structure-behavior modeling. This makes it impossible to meet the integrated requirements of "high-precision prediction, mechanism exploration, and engineering deployment" for modern energetic materials.
[0005] Specifically, the limitations of the above methods are mainly reflected in:
[0006] (1) Insufficient particle-scale modeling: It is impossible to accurately characterize the discrete behavior between particles and the nonlinear damage mechanism of the interface in PBX;
[0007] (2) Unclear crack initiation: It is difficult to identify the starting point of microcracks and their evolution path, and empirical criteria are often relied upon;
[0008] (3) Weak interpretability: Most machine learning models are “black boxes” and lack visualization and quantitative explanation of the contribution of physical features;
[0009] (4) Limited generalization capability: Traditional models have poor adaptability to new structures or new load forms and lack universal expansion capabilities.
[0010] Therefore, there is an urgent need for an interpretable method that integrates physical modeling and data-driven technology, which can not only characterize the evolution process of cracks, but also improve the prediction accuracy and scalability to meet the needs of structural optimization and risk assessment. Summary of the Invention
[0011] The technical problem solved by the present application is to provide a PBX crack evolution prediction method combining discrete elements and machine learning, which is suitable for micro-crack behavior modeling and failure path prediction of multi-particle heterogeneous materials under complex loading conditions, can identify the location and time of crack occurrence, reveal the crack evolution mechanism, and provide technical support for structure optimization, safety evaluation and virtual testing of energetic materials.
[0012] To achieve the above-mentioned purpose, the technical scheme adopted by the present application is as follows:
[0013] A PBX crack evolution prediction method combining discrete elements and machine learning, comprising:
[0014] A two-dimensional particle structure model is established based on the actual PBX material image, the model contains the particle size, position, spatial distribution and adjacency relationship of the particles, and is imported into a discrete element simulation platform for mechanical loading simulation;
[0015] Physical descriptors are extracted from the simulation results, feature correlation analysis is performed, and variables with physical independence are selected to construct a feature data set required for crack prediction;
[0016] According to the feature data set, an XGBoost algorithm is used to construct a crack classification model to predict whether each particle will crack within a given time window, and output the probability of cracking;
[0017] A Stacking integrated regression model containing XGBoost, CatBoost and LightGBM as base learners is used to predict the time point of particle cracking, wherein the crack probability output by the classification model is used as the input feature of the regression model;
[0018] The SHAP method is used to perform interpretability analysis on the output results of the classification model and the regression model, quantify the average influence of each input variable on the model output, and identify the key physical features that dominate crack evolution.
[0019] Preferably, the two-dimensional structure model is obtained by image recognition and geometric reconstruction through a scanning electron microscope (SEM) image.
[0020] Preferably, the discrete element simulation uses a parallel key model, and the crack failure modes include tensile failure, tensile-shear composite failure and compression-shear composite failure, and the failure criterion is:
[0021] (1) When the normal stress is greater than the tensile strength, tensile failure occurs;
[0022] (2) The shear stress increases with the normal stress, and when the shear stress exceeds the shear strength, shear failure occurs.
[0023] As preferred, the physical descriptors include: horizontal stress, shear stress, axial stress, maximum principal stress, minimum principal stress, principal stress direction angle, particle radius, particle type, particle X coordinate, particle Y coordinate.
[0024] As preferred, the feature correlation analysis employs Pearson correlation coefficient analysis, with a correlation threshold of 0.8.
[0025] As preferred, the output probability p of the classification model is in the range of [0, 1].
[0026] The present application is not only suitable for PBX materials with a specific particle size distribution, but also has good generality in model structure, feature system and interpretation logic, and can be extended to various heterogeneous energetic material systems with different particle sizes, different interface types or different loading paths. Compared with traditional crack modeling methods, the present application has three advantages of high prediction accuracy, good scalability and physical consistency. Its comprehensive performance is sufficient to support the rapid prediction of material damage evolution, structure sensitivity screening and reliability evaluation in engineering, and has significant practical application value and industrial transformation potential. BRIEF DESCRIPTION OF DRAWINGS
[0027] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or prior art description will be briefly introduced below. Obviously, the drawings in the following description are only embodiments of the present application, and those skilled in the art can obtain other drawings according to the provided drawings without creative labor.
[0028] Figure 1 The overall flowchart of the PBX crack evolution prediction method described in the present application;
[0029] Figure 2 The HMX-PBX two-dimensional particle structure modeling process based on scanning electron microscope images;
[0030] Figure 3 The crack propagation path visualization and crack type labeling diagram generated by the discrete element simulation;
[0031] Figure 4 The Pearson correlation analysis heat map between the constructed physical features;
[0032] Figure 5 The regression performance comparison diagram of the Stacking model for predicting crack nucleation time;
[0033] Figure 6 The SHAP explainability analysis results, including the important feature ranking diagram and the SHAP value distribution diagram. DETAILED DESCRIPTION
[0034] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.
[0035] In order to make the above objectives, characteristics and advantages of the present application more apparent, further specific embodiments of the present application will be described in detail below with reference to the drawings and embodiments.
[0036] Embodiment 1
[0037] As shown in the drawings, the embodiments of the present application provide a PBX material crack prediction method, comprising: Figure 1
[0038] A two-dimensional particle structure model is established based on an actual PBX material image, the model contains particle size, position, spatial distribution and adjacency relationship of the particles, and is imported into a discrete element simulation platform for mechanical loading simulation;
[0039] Physical descriptors are extracted from the simulation results, feature correlation analysis is performed, and variables with physical independence are screened to construct a feature data set required for crack prediction; the input features provided by the embodiments are shown in the following table, part of which are original features, part of which are engineering features derived based on the original features, and finally there are two target features. It should be noted that the first target feature crack_o is also used as an input feature in the second stage of machine learning. The input features in Table 1 are only used for the case of the present application.
[0040] Table 1
[0041]
[0042]
[0043] According to the feature data set, an XGBoost algorithm is used to construct a crack classification model, whether each particle occurs crack in a given time window is predicted, and the occurrence probability is output;
[0044] A Stacking integrated regression model containing XGBoost, CatBoost and LightGBM as base learners is used to predict the time point of crack occurrence of the particles, wherein the crack probability output in the classification model is used as an input feature of the regression model;
[0045] The SHAP method is used for explainability analysis on the output results of the classification model and the regression model, quantifying the average influence of each input variable on the model output, and identifying the key physical features of dominant crack evolution.
[0046] As an embodiment of the present application, the two-dimensional structure model is obtained by image recognition and geometric reconstruction through a scanning electron microscope (SEM) image, and the particle boundary topography and the actual spatial topological relationship are maintained.
[0047] As an embodiment of the present application, the discrete element simulation adopts a parallel key model, the crack failure mode includes tensile failure, tensile-shear composite failure and compression-shear composite failure, and the failure criterion is:
[0048] (1) When the normal stress is greater than the tensile strength, tensile failure occurs.
[0049] (2) The shear stress increases with the normal stress, and when the shear stress exceeds the shear strength, shear failure occurs.
[0050] As an embodiment of the present application, the physical descriptors include horizontal stress, shear stress, axial stress, maximum principal stress, minimum principal stress, principal stress direction angle, particle radius, particle type, particle X coordinate, particle Y coordinate, etc.
[0051] As an embodiment of the present application, the feature correlation analysis adopts Pearson correlation coefficient analysis, the correlation threshold is set to 0.8, and the feature variables with strong physical mechanism independence are retained.
[0052] As an embodiment of the present application, the output probability p of the classification model is in the range of [0, 1], which is one of the input features of the Stacking regression model, and is used to enhance the generalization ability and robustness of the time prediction model.
[0053] As an embodiment of the present application, the meta-learner of the Stacking model is an XGBoost regression model based on gradient boosting trees, the base learners include XGBoost, LightGBM and CatBoost, all of which are built based on the gradient boosting framework, and the Stacking model is fused by multiple learners to improve the accuracy and robustness of crack occurrence time prediction.
[0054] Embodiment 2:
[0055] As shown in Figure 1 , the present application provides a PBX material crack prediction method combining physical simulation and explainable machine learning, which comprises:
[0056] Firstly, a two-dimensional microstructure model of HMX-based PBX material was established. The model was based on the images of PBX material obtained in actual experiments, and the particle boundaries were obtained by image recognition algorithm, and the particle geometry was processed and reconstructed. The spatial coordinates, radius and adjacency relationship of each particle were recorded to form a structure input model. The structure model was imported into the discrete element simulation platform, and the parallel key model was used as the theoretical basis of the interparticle interaction force, and the normal and tangential contact stiffness between each particle were defined. The simulation load was applied in the form of uniaxial quasi-static compression loading, and the boundary deformation was applied at a constant displacement rate. The contact state between particles, stress tensor, particle motion information (displacement, rotation) and key fracture events were recorded in real time during the simulation. The criterion for key fracture was:
[0057] τ s PB = c PB -σ PB tanφ PB
[0058] wherein, c PB represents the cohesion, and φ PB is the friction angle. If the normal stress σ PB exceeds the tensile strength σ t , tensile failure occurs, otherwise the shear strength τ PB increases with the increase of the normal stress diagram. When the shear force τ PB exceeds the shear strength, shear failure occurs. For each crack event, mark its occurrence time step, spatial position, particle ID and fracture type (tensile, shear or mixed mode).
[0059] During the simulation, the multi-dimensional physical state information of each particle at each time step was recorded, including but not limited to: normal stress, shear stress, maximum principal stress, minimum principal stress, principal stress direction angle, particle radius, particle type, position coordinates, number of contact pairs, normal contact force, tangential contact force, etc. 15 descriptors. In order to prevent feature redundancy and collinearity from affecting modeling performance, Pearson correlation analysis was used to calculate the correlation coefficient of all feature variables two by two, and the correlation threshold was set to 0.8. If the correlation of two features is higher than this value, the one with stronger explanatory power is retained. At the same time, it is also necessary to consider whether each feature corresponds to a different physical mechanism, and they also play different roles in the crack nucleation process. This ensures that the feature set not only can make accurate prediction, but also maintains interpretability within the framework of fracture mechanics.
[0060] All features were processed by center standardization to have uniform scale in each dimension. The standardization method is:
[0061]
[0062] where x i,j is the original value of the jth feature in the ith sample, μ j is the feature mean, and σ j is the standard deviation.
[0063] The crack prediction modeling adopts a two-stage machine learning structure. The first stage is a binary classification problem of whether a crack occurs or not. An XGBoost classifier is used to train the classifier, and the output is the prediction result and its probability value of whether a crack occurs in each particle within a specified time window. The second stage is a regression problem to predict the first occurrence time point of the crack. A stacking ensemble regression model is used, which includes three base learners XGBoost, CatBoost, and LightGBM, and the crack probability output by the classification stage is input as an auxiliary feature. The meta-learner uses ridge regression for fusion, and outputs the final time prediction value.
[0064] The model training adopts a 5-fold cross-validation method. The overall data set is divided into three segments, which are the training set (72%), the test set (16%), and the validation set (10%). In each round, the training set and the test set are input into the model, and then the training set is used for training, and the test set is used for the first evaluation of the model. The independent validation set is also used for the second evaluation of the model. For the regression model, the coefficient of determination (R 2 ), mean squared error (MSE), and (RMSE) are the main indicators.
[0065] For the classification model, the following indicators are mainly used as evaluation criteria:
[0066] (1) Accuracy: measures the proportion of correctly predicted samples in the total samples.
[0067]
[0068] where TP represents the number of samples that are actually positive and predicted as positive, TN represents the number of samples that are actually negative and predicted as negative, FP represents the number of samples that are actually negative but predicted as positive, and FN represents the number of samples that are actually positive but predicted as negative.
[0069] (2) Recall: measures the recognition ability of the model for actual positive samples.
[0070]
[0071] (3) F1-score: the harmonic mean of accuracy and recall, used to comprehensively evaluate the classification performance.
[0072]
[0073] For regression models, the main evaluation indicators include:
[0074] (1) Determination coefficient (R 2 ): reflects the degree of linear fitting between the predicted value and the true value of the model, the closer to 1 indicates the better the model fitting effect.
[0075]
[0076] where y i is the true crack occurrence time, is the model predicted value, is the true value mean, and n is the sample size.
[0077] (2) Mean square error (MSE): the average of the square of the error between the predicted value and the true value.
[0078]
[0079] (3) Root mean square error (RMSE): the square root of the mean square error, with the same dimension as the original quantity.
[0080]
[0081] Further explainability analysis of the regression model, introduce SHAP method to evaluate the average contribution of each input feature to the model output. The analysis results show that the normal stress, shear stress, spatial position coordinates, particle radius and contact pair number of the particles are the main factors affecting the crack evolution time, and the contribution ranking is stable, which is consistent with the mechanical law.
[0082] The application fuses the discrete element method and the PBX crack evolution prediction method of machine learning, and is suitable for the evolution modeling and time sequence prediction of micro-cracks in non-homogeneous energetic materials. The method first constructs a two-dimensional particle structure model based on a scanning electron microscope image, imports the structure model into a discrete element simulation platform, applies a loading condition, and extracts multi-dimensional features such as stress state, geometric parameters and contact information at multiple time steps. The XGBoost classifier is used to determine whether a crack occurs, and the occurrence probability is output. The probability is used as an auxiliary feature to input a Stacking integrated regression model (including XGBoost, CatBoost and LightGBM base learners), and the crack occurrence time is further predicted. In order to enhance the transparency and physical interpretation ability of the model, the SHAP method is introduced to quantitatively analyze the influence of each input variable on the model output, and to identify the key physical mechanism of the dominant crack nucleation behavior. Compared with the traditional numerical modeling method, the application has the advantages of multi-source feature driving, high-precision prediction and good interpretability, can accurately identify the evolution behavior of cracks in PBX materials under the driving of multi-scale features, realize the traceable explanation of the evolution path and time sequence process, and is suitable for the structure safety evaluation, damage monitoring and virtual test platform construction of energetic materials, and has good application prospect.
[0083] In order to verify the effectiveness of the crack prediction method, the HMX-based PBX material is taken as the research object, and the complete prediction process, model construction method and result analysis are shown in the complete process as shown in Figure 1 A PBX material crack prediction method combining physical simulation and interpretable machine learning includes the following steps:
[0084] S1: Based on the scanning electron microscope image, the particle distribution information in the PBX is obtained by image boundary detection and geometric reconstruction method. The identified particles are simplified as two-dimensional discs, and the radius, position coordinates and adjacency relationship are recorded to form a two-dimensional particle structure model. The model contains the real particle size distribution, particle density and spatial arrangement, representing the actual charge microstructure. As shown in Figure 2 The structure modeling process is shown in the figure, which retains the real distribution density, size distribution and topological connection information of the particles.
[0085] S2: The structure model is imported into the discrete element simulation platform, and the parallel key model is used to describe the bonding relationship between the particles. In the simulation, a uniaxial compression load is applied, and the loading mode is to fix the upper boundary to apply a constant speed displacement, the lower boundary is fixed, and the lateral boundary is free. The position, radius, stress and key fracture information of the particles are recorded in real time during the simulation. The crack type is determined according to the dominant stress of the damage, which is divided into tensile crack, tensile-shear crack and compression-shear crack. As shown in Figure 3 The crack propagation path and fracture labeling diagram output by the simulation (yellow for shear crack, blue for tensile-shear crack, and green for compression-shear crack) are shown in the figure.
[0086] S3: Extract the multi-dimensional feature variables of each particle at each time step during the simulation, including horizontal stress, shear stress, axial stress, maximum and minimum principal stress, principal stress direction angle, particle radius, particle type, spatial coordinates, etc., a total of 14 descriptors. These features cover the typical state information of particles in geometry, mechanics, and topology. To ensure the stability and interpretability of subsequent modeling, the correlation between features is quantitatively evaluated using the Pearson correlation coefficient. Set the threshold to 0.8, when the absolute value of the correlation between any two features is greater than the threshold, delete one of them, and keep the variable with clear explanation of physical behavior. In addition, in the feature retention decision, auxiliary judgment is made in combination with fracture mechanics knowledge. For example, although the horizontal stress and the principal stress may have correlation, but they represent different physical mechanisms, so retaining both does not constitute redundancy. This "correlation + physical mechanism explanation" double-standard screening strategy ensures the physical independence and modeling effectiveness of the features during model training. As shown in Figure 4 , a correlation heat map between all features is shown, where the remaining features have low correlation or complementary information with each other.
[0087] S4: The 14 input features are centered and standardized to eliminate the scale effect caused by different physical dimensions. The model is divided into two stages: the first stage is a crack classification model, which uses XGBoost to build a classifier and outputs the probability of each particle cracking; the second stage is a time prediction model, which uses a Stacking ensemble regressor to integrate XGBoost, CatBoost, and LightGBM three base learners. The output probability of the classification model is added as an auxiliary feature to the regression stage. The meta-learner of the regression uses tree regression for integration. As shown in Figure 5 , the model architecture diagram is shown, including the structure of the two-stage model and the composition of the input features.
[0088] S5: Use 5-fold cross-validation to train and test the model. The total data set is divided into training set (72%), test set (16%) and validation set (10%). The model performance evaluation is divided into two stages, where the classifier has a classification accuracy of 99.20%, F1 score of 0.9366, and recall rate of 0.9213 on the test set; the regressor has R 2 2 of 0.9238, MSE of 0.2175, and RMSE of 0.0473 on the test set, and R 2 2 of 0.9032, MSE of 0.2355, and RMSE of 0.0555 on the validation set. The predicted crack distribution and its main crack morphology are shown in Figure 5 .
[0089] S6: Introduce SHAP explanation method to classification and regression models, quantify the average contribution of each feature to the model output. Analysis shows that shear stress, particle radius, first principal stress are the most critical factors. The model has good consistency with the physical mechanism of crack, and enhances the prediction reliability. SHAP explanation results and feature ranking chart can be seen in Figure 6 .
[0090] S7: The embodiments of the present application are not only suitable for PBX materials of specific particle size, but also have good adaptability to other non-homogeneous material systems. It can be used for high-throughput material design, crack evolution sensitivity analysis, risk warning system development and virtual test scene construction, and has high engineering practicability and scientific research value.
[0091] The above-described embodiments are only a description of the preferred mode of the present application, and do not limit the scope of the present application. Without departing from the design spirit of the present application, various modifications and improvements to the technical solutions of the present application made by those skilled in the art shall fall within the protection scope determined by the claims of the present application.
Claims
1. A PBX crack evolution prediction method integrating discrete element method and machine learning, characterized in that: include: A two-dimensional particle structure model was established based on the actual PBX material image. The model included the particle size, position, spatial distribution, and adjacency relationship, and was imported into the discrete element simulation platform for mechanical loading simulation. Extract physical descriptors from simulation results, perform feature correlation analysis, and screen variables with physical independence to build the feature dataset required for crack prediction; Based on the feature data set, the XGBoost algorithm is used to build a crack classification model to predict whether each particle will crack within a given time window and output its occurrence probability; A stacking ensemble regression model, which includes XGBoost, CatBoost, and LightGBM as base learners, is used to predict the time point when particles will crack. The crack probability output from the classification model is used as the input feature of the regression model. The SHAP method was used to perform interpretability analysis on the output results of the classification model and regression model, quantify the average influence of each input variable on the model output, and identify the key physical features that dominate crack evolution.
2. The PBX crack evolution prediction method integrating discrete element and machine learning as claimed in claim 1, characterized in that: The two-dimensional structural model is obtained by performing image recognition and geometric reconstruction on a scanning electron microscope (SEM) image.
3. The PBX crack evolution prediction method integrating discrete element and machine learning as claimed in claim 2, characterized in that: The discrete element simulation adopts the parallel bond model. The crack failure modes include tensile failure, tensile-shear combined failure and compression-shear combined failure. The failure criterion is: (1) When the normal stress is greater than the tensile strength, tensile failure occurs; (2) Shear stress increases with normal stress, and when the shear stress exceeds the shear strength, shear failure occurs.
4. The PBX crack evolution prediction method integrating discrete element and machine learning as claimed in claim 3, characterized in that: The physical descriptors include: horizontal stress, shear stress, axial stress, maximum principal stress, minimum principal stress, principal stress direction angle, particle radius, particle type, particle X coordinate, and particle Y coordinate.
5. The PBX crack evolution prediction method integrating discrete element and machine learning as claimed in claim 3, characterized in that: The feature correlation analysis was performed using the Pearson correlation coefficient, with the correlation threshold set at 0.
8.
6. The PBX crack evolution prediction method integrating discrete element and machine learning as claimed in claim 3, characterized in that: The output probability of the classification model is p∈[0,1].