Strength prediction method and system for randomly distributed fractured rock

Abstract

The present application provides a strength prediction method and system for randomly distributed fractured rock, and relates to the fields of geotechnical engineering and intelligent prediction technology. The present application constructs a physical prior model based on the conservation of elastic deformation energy, taking confining pressure, number of fractures and diameter as input and output baseline strength. The model parameters are calibrated using experimental data to generate a real residual data set. A convolution residual correction network with a channel attention mechanism is designed to predict residual correction values combined with zero confining pressure indicator characteristics. A two-stage decoupling strategy is used to train the hybrid model, and the contribution rate of each parameter is quantified through grouping replacement importance analysis. After mechanical rationality verification, the final peak strength prediction value is output. This method combines physical mechanism and data-driven to improve prediction accuracy and interpretability.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering and intelligent prediction technology, specifically to a method and system for predicting the strength of randomly distributed fractured rocks. Background Technology

[0002] Rock masses are complex media formed through long-term geological processes, and their interiors are generally characterized by discontinuous structural planes such as fissures, joints, and bedding. In water conservancy and hydropower, mining, and tunnel and underground space engineering, the random distribution of fissures has a significant impact on the strength, deformation, and failure mode of rock masses. In particular, for rock-like materials or fractured rock masses containing randomly distributed fissures, their macroscopic peak strength depends not only on the mechanical properties of intact rock and confining pressure conditions, but also on geometric parameters such as the number of fissures, fissure diameter, or trace length, and their combined effects. Therefore, establishing a method that can accurately predict the peak strength of fractured rocks under different confining pressures and fissure geometric parameters is of great engineering significance for the stability evaluation of engineering rock masses, support design, and disaster prevention.

[0003] Currently, the main methods for predicting the strength of fractured rock masses or rock-like materials include the following: physical model methods based on empirical strength criteria or equivalent continuum theory, such as the Hoek-Brown criterion, the Mohr-Coulomb criterion, and equivalent models based on damage mechanics. These methods have clear physical meaning and are computationally simple, but they typically assume uniform fracture distribution or use statistical averaging, making it difficult to accurately reflect the nonlinear effects of variations in parameters such as the number and diameter of randomly distributed fractures on strength. Furthermore, when fracture geometric parameters vary significantly, model parameter calibration becomes difficult, limiting prediction accuracy.

[0004] Numerical simulation-based methods simulate the initiation, propagation, and penetration of fractures by constructing finite element, discrete element, or numerical manifold models containing random fractures. While these methods can explicitly consider fracture geometry and interactions, they are complex to model, computationally expensive, and highly sensitive to mesh quality, contact constitutive parameters, and random number generation, making them difficult to rapidly apply to engineering field operations or large-scale parameter analysis.

[0005] Data-driven methods based on machine learning, such as artificial neural networks, support vector regression, random forests, and deep neural networks, learn the complex mapping relationship between input parameters and peak intensity through experimental samples, exhibiting strong nonlinear fitting capabilities. However, data-driven methods typically rely on a large number of high-quality training samples, making them prone to overfitting when the sample size is insufficient. Furthermore, the models lack physical interpretability, easily generating extrapolation results that violate the laws of mechanics outside the scope of the training data.

[0006] To balance the interpretability of physical models with the fitting capabilities of data-driven methods, a hybrid physics-data modeling approach has emerged in recent years. This approach utilizes a physical prior model to provide baseline predictions, followed by data-driven modeling to correct the prediction residuals. However, existing hybrid prediction methods still have several shortcomings: physical prior models have limited ability to represent key geometric parameters such as the number and diameter of randomly distributed fractures; the training processes of the physical model and the residual correction model are often coupled or directly end-to-end trained, which can weaken physical constraints and cause the model to deviate from the fundamental laws of rock mechanics; furthermore, existing methods rarely quantify the contributions of key working condition parameters such as confining pressure, number and diameter of fractures, and lack mechanical rationality verification of contribution ranking, making it difficult to ensure that the model has reliable parameter sensitivity and interpretability in practical applications.

[0007] In summary, there is an urgent need for a method and system for predicting the strength of randomly distributed fractured rocks. This system should be able to reduce reliance on a large number of training samples by utilizing a physical prior model, effectively correct the residuals of the physical model, and quantify and verify the contributions of key working condition parameters to improve prediction accuracy, generalization ability, and physical interpretability.

[0008] The information disclosed in the background section is only intended to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0009] The purpose of this invention is to provide a method and system for predicting the strength of randomly distributed fractured rocks, so as to solve the problems mentioned in the background art.

[0010] To achieve the above objectives, the present invention provides the following technical solution: A method and system for predicting the strength of randomly distributed fractured rocks, the specific steps of which include: S1: Construct a physical prior model of fractured rock strength based on the conservation of elastic deformation energy. The physical prior model of fractured rock strength takes the target working condition parameters as input, including confining pressure, number of fractures and fracture diameter, and outputs the predicted baseline strength value of fractured rock. S2: The learnable parameters of the physical prior model of fractured rock strength are calibrated using rock mechanics test samples. Based on the calibrated physical prior model of fractured rock strength, the predicted baseline strength value corresponding to each test sample is calculated. The real residual dataset is generated based on the difference between the predicted baseline strength value and the measured peak strength value of the corresponding sample, which serves as the training label for the subsequent convolutional residual correction network. S3: Construct a convolutional residual correction network with channel attention mechanism. The convolutional residual correction network takes the target working condition parameters and zero confining pressure indication features as input and outputs the predicted residual correction value of the physical prior model of fractured rock strength. S4: A two-stage decoupling strategy is used to train the hybrid prediction model: In the first stage, all parameters of the physical prior model of fractured rock strength are fixed; in the second stage, the real residual dataset obtained in step S2 is used as the training label to train the convolutional residual correction network to obtain the trained hybrid prediction model. S5: For the trained hybrid prediction model, the group permutation importance method is used to quantify the contribution rate of each target working condition parameter to the final peak intensity prediction value, and the contribution rate ranking result is verified to obtain a qualified hybrid prediction model. The target working condition parameter to be predicted is input into the qualified hybrid prediction model, and the corresponding prediction baseline intensity value and prediction residual correction value are superimposed to obtain the final peak intensity prediction value and output it.

[0011] Furthermore, the construction of the physical prior model of fractured rock strength based on the conservation of elastic deformation energy specifically involves: establishing the energy balance equation of rock mass unit based on the conservation of elastic deformation energy, deriving the equivalent elastic modulus of randomly distributed fractured rock mass, and constructing the physical prior model by combining the definition of damage strength and the confining pressure strengthening criterion. Specifically, the energy balance equation for the rock mass unit is established as follows: the rock mass unit containing random fractures is regarded as a homogeneous equivalent continuous medium. Under the action of far-field axial stress, the total equivalent elastic strain energy of the rock mass consists of two parts: the elastic strain energy of the intact rock matrix and the additional strain energy caused by the elastic deformation of the fractures in the rock mass.

[0012] Furthermore, the logic for obtaining the additional strain energy caused by elastic deformation of fractures within the rock mass is as follows: the total released energy of a single fracture is obtained by integrating along the fracture front; the average released strain energy of a single fracture is obtained by statistically averaging the isotropic and uniform distribution of fractures in the normal direction; then, the fracture density and fracture number correction coefficients are introduced to calculate the additional strain energy caused by elastic deformation of fractures within the rock mass; the obtained additional strain energy caused by elastic deformation of fractures within the rock mass is substituted into the energy balance equation of the rock mass unit to derive the equivalent elastic modulus of the fractured rock mass; and, combined with the confining pressure strengthening criterion, a priori physical model of fractured rock strength is established.

[0013] Furthermore, the learnable parameters of the prior physical model for fractured rock strength are calibrated using rock mechanics test samples. These learnable parameters include: a fracture number correction coefficient and a confining pressure strengthening coefficient. Specifically, the intrinsic parameters of intact rock are first obtained through basic mechanical tests, including the uniaxial compressive strength, elastic modulus, and Poisson's ratio of intact rock. An initial confining pressure strengthening coefficient is calculated based on the confining pressure-strength data of intact specimens. Representative specimen strength data covering different confining pressures, fracture diameters, and fracture numbers are selected. The measured peak strength value is used as the true value. The least squares method is used to globally optimize and calibrate the fracture number correction coefficient and the confining pressure strengthening coefficient in the prior physical model for fractured rock strength. The optimization objective is to minimize the mean square error between the predicted baseline strength value and the measured peak strength value. After calibration, all parameters of the prior physical model for fractured rock strength remain fixed.

[0014] Furthermore, the convolutional residual correction network with channel attention mechanism specifically adopts a three-level structure of convolutional feature extraction, channel attention weighting, and fully connected output. This reshapes the input features into a 1D sequence, adapting it to 1D convolutional operations. Two layers of 1D convolutional kernels are used to perform nonlinear transformation on the input features, generating a 16-channel high-dimensional abstract feature embedding. The ReLU activation function is used to retain nonlinear fitting ability. Weights are calculated for each of the 16 channel features, and the weights are normalized to the 0~1 range using the sigmoid function. The channel attention weight vector is multiplied element-wise with the convolutional output features, mapping the weighted high-dimensional features to a 1D output, i.e., the predicted residual correction value.

[0015] Furthermore, the training of the convolutional residual correction network uses the true residuals in the true residual dataset calculated by S2 as training labels, and the target operating condition parameters and zero confinement pressure indication features as network inputs. The zero confinement pressure indication features are Boolean input features; when the confinement pressure in the target operating condition parameters is 0, the zero confinement pressure indication feature takes a value of 1; when the confinement pressure in the target operating condition parameters is not 0, the zero confinement pressure indication feature takes a value of 0. The Adam optimizer is used to train the CNN-Attention network. During the training process, the ReduceLROnPlateau adaptive learning rate decay strategy is adopted, and the learning rate is reduced when the validation set loss does not decrease for 10 consecutive rounds. An early stopping strategy is set, and training is terminated when the validation set loss does not increase for 20 consecutive rounds to prevent overfitting. After training, the convolutional residual correction network outputs the corresponding predicted residual correction value according to the input target operating condition parameters and zero confinement pressure indication features, thus completing the correction.

[0016] Furthermore, the grouping permutation importance method is used to quantify the contribution rate of each target working condition parameter to the final peak strength prediction value. Specifically, different permutation strategies are used for the confining pressure, number of fractures and fracture diameter among the target working condition parameters. Among them, the confining pressure is replaced by a global random permutation; the number of fractures is replaced by a global random permutation; the fracture diameter is replaced by a conditional permutation based on the confining pressure level, that is, the diameter value is randomly shuffled only within the same confining pressure level. Quantitative calculation of feature contribution rate: A fixed number of random permutations are performed on confining pressure, number of fractures and fracture diameter. The importance score of each target working condition parameter is calculated by the difference in the average absolute error of the model prediction before and after the permutation. The importance score of each target working condition parameter is normalized to obtain the percentage contribution rate of each target working condition parameter.

[0017] Furthermore, the contribution rate ranking results are verified, that is, the contribution rate percentages of confining pressure, number of fractures, and fracture diameter are ranked for mechanical rationality verification. The specific rules for mechanical rationality verification are: the contribution rate of confining pressure is greater than the contribution rate of fracture diameter, and the contribution rate of fracture diameter is greater than the contribution rate of the total number of fractures. If the verification fails, the training set is re-divided and the training hyperparameters are adjusted, and the hybrid prediction model is retrained until the contribution rate ranking meets the verification rules, thus obtaining a qualified hybrid prediction model.

[0018] The present invention also provides a strength prediction system for randomly distributed fractured rocks, which is used in the strength prediction method for randomly distributed fractured rocks described in any of the above claims, and includes: Physical Prior Model Construction Module: Used to construct a physical prior model of fractured rock strength based on the conservation of elastic deformation energy. The physical prior model of fractured rock strength takes target working condition parameters as input, including confining pressure, number of fractures and fracture diameter, and outputs the predicted baseline strength value of fractured rock. The parameter calibration and data generation module is used to calibrate the learnable parameters of the physical prior model of fractured rock strength using rock mechanics test samples. Based on the calibrated physical prior model of fractured rock strength, it calculates the predicted baseline strength value corresponding to each test sample and generates a real residual dataset based on the difference between the predicted baseline strength value and the measured peak strength value of the corresponding sample, which serves as the training label for the subsequent convolutional residual correction network. Residual network construction module: used to construct a convolutional residual correction network with channel attention mechanism. The convolutional residual correction network takes the target working condition parameters and zero confining pressure indication features as input and outputs the predicted residual correction value of the physical prior model of fractured rock strength. Model decoupling training module: used to train the hybrid prediction model using a two-stage decoupling strategy: the first stage freezes all parameters of the physical prior model of fractured rock strength, and the second stage uses the real residual dataset obtained in step S2 as training labels to train the convolutional residual correction network to obtain the trained hybrid prediction model. Contribution rate analysis module: This module is used to quantify the contribution rate of each target working condition parameter to the final peak intensity prediction value of the trained hybrid prediction model using the group permutation importance method, and to verify the contribution rate ranking results to obtain a qualified hybrid prediction model.

[0019] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention constructs a physical prior model based on the conservation of elastic deformation energy and the Mohr-Coulomb strength criterion, and freezes all physical parameters throughout the training process through a two-stage decoupled training strategy, ensuring that the strength prediction results always conform to the basic laws of rock mechanics and meet mechanical constraints such as equivalent elastic modulus reduction, compliant damage degree values, and monotonically increasing strength with confining pressure. This fundamentally avoids the problem of physical mismatch that is prone to occur in pure data-driven models, and the prediction results have a clear mechanical interpretation basis. This invention employs a convolutional residual correction network with channel attention mechanism to specifically correct the nonlinear system bias of the physical prior model under boundary conditions such as low confining pressure and small-diameter dense cracks. Validated by indoor triaxial test data covering confining pressures of 5–60 MPa, 1–200 cracks, and crack diameters of 1–10 mm, the mixture model's coefficient of determination is [value missing]. The mean value can reach 0.9786, which is a significant improvement in prediction accuracy compared to a single physical prior model. Compared to residual networks without attention mechanisms, both prediction stability and accuracy are significantly improved.

[0020] This invention employs a grouped permutation importance method to quantitatively output the contribution rates of confining pressure, fracture diameter, and fracture number to strength prediction results. Tests showed that the average contribution rates for confining pressure, fracture diameter, and fracture number were 44%, 40%, and 16%, respectively, breaking the "black box" limitation of purely data-driven models. The distribution pattern of these contribution rates aligns with the theoretical understanding in rock mechanics that "confining pressure dominates the strengthening effect, fracture size affects stress concentration, and fracture number affects statistical damage accumulation," providing quantitative support for the analysis of damage mechanisms and parameter optimization in engineering rock masses.

[0021] All input parameters of this invention can be obtained through standard uniaxial and triaxial rock mechanics tests without the need for special test equipment; the model has high inference efficiency and short single-sample prediction time, and can be widely used in scenarios such as rapid assessment of the stability of surrounding rock in deep-buried tunnels and reverse design of crack parameters of similar material specimens. Compared with traditional numerical simulation methods, the calculation efficiency is significantly improved, and it is suitable for a variety of engineering needs. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of the overall method flow of the present invention; Figure 2 The above are the convergence curves of the training loss and validation loss of the residual network in an embodiment of the present invention. Figure 3This is a scatter plot comparing the predicted and measured values ​​of the three models in this embodiment of the invention. Figure 4 A scatter plot showing the trend of residual changes with confining pressure before and after correction.

[0023] Figure 5 This is a block diagram of the system module structure of the present invention. Detailed Implementation

[0024] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to specific embodiments.

[0025] It should be noted that, unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0026] Example: Please see Figures 1-4 The present invention provides a technical solution: A method and system for predicting the strength of randomly distributed fractured rocks, the specific steps of which include: S1: Construct a physical prior model of fractured rock strength based on the conservation of elastic deformation energy. The physical prior model of fractured rock strength takes the target working condition parameters as input and outputs the predicted baseline strength value of fractured rock.

[0027] The construction process in this step is based on the conservation of elastic deformation energy to establish the energy balance equation of rock mass unit, derive the equivalent elastic modulus of randomly distributed fractured rock mass, and combine the definition of damage strength and confining pressure strengthening criterion to construct a physical prior model.

[0028] Establish the energy balance equation for the rock mass unit. Treat the rock mass unit containing randomly distributed coin-shaped fractures as a homogeneous equivalent continuous medium, with a unit volume of... The mechanical basis for this equivalent treatment is that when the number of fractures is sufficient and their spatial orientation is random, the disturbance of the macroscopic mechanical response of the rock mass by the fractures tends to be isotropic in a statistical sense. Therefore, the macroscopic deformation behavior of fractured rock masses can be described with equivalent parameters within the framework of continuum mechanics. Under the action of uniform axial stress σ in the far field, the total equivalent elastic strain energy of the rock mass consists of two parts: the elastic strain energy of the intact rock matrix and the additional strain energy caused by the elastic deformation of the fractures within the rock mass.

[0029] Based on this, an energy balance equation is established: in, σ This represents the far-field axial stress, measured in MPa. This is the equivalent elastic modulus of the fractured rock mass, expressed in GPa. The elastic modulus of intact rock is expressed in GPa. The total additional strain energy generated by the elastic deformation of all fractures in the rock mass is expressed in J. The volume of a rock mass unit is expressed in mm³. The left side of the equation represents the far-field stress of the fractured rock mass. The total equivalent elastic strain energy stored is given by the equation. The first term on the right represents the baseline elastic strain energy stored in a intact rock matrix under the same stress level, while the second term represents the additional strain energy generated due to the presence of fissures. The core physical meaning of this equation is that the presence of fissures reduces the macroscopic stiffness of the rock mass, resulting in a higher elastic strain energy stored in the rock mass under the same stress level compared to intact rock. The excess is the additional strain energy contributed by the elastic deformation of the fissures.

[0030] The expression on the left side of the equation comes from the strain energy formula for a linear elastic body: a uniformly stressed linear elastic body under stress... The elastic strain energy per unit volume under the following conditions is Multiply by the volume of the rock mass unit The total elastic strain energy is obtained, and the equivalent elastic modulus of the fractured rock mass is used to obtain the total elastic strain energy. Calculations show that the intact rock matrix... The difference between the two is the additional contribution of the fracture. The reason for using the energy balance equation instead of solving the stress field for each fracture is that the contribution of the fracture to the macroscopic response of the rock mass is reflected in the release of strain energy. Based on energy conservation, the analytical relationship between the fracture geometric parameters and the macroscopic stiffness reduction can be directly established, avoiding detailed stress field analysis for each fracture and ensuring the simplicity and feasibility of the model at the macroscopic prediction level.

[0031] Calculate the average released strain energy of a single randomly oriented fracture. Specifically, for the additional strain energy caused by elastic deformation of fractures within the rock mass, the logic for obtaining this energy is as follows: integrate along the fracture front to obtain the total released energy of a single fracture; statistically average the isotropically uniform distribution of the fracture along its normal direction to obtain the average released strain energy of a single fracture.

[0032] Specifically, for a three-dimensional coin-shaped fracture, based on Irwin's fracture mechanics energy release rate formula, integration is performed along the circumferential and radial directions at the fracture leading edge to obtain the total released strain energy of a single fracture under far-field stress. The derivation is carried out as follows: Determine the energy release rate at the fracture front. Based on Irwin's fracture mechanics theory, under plane strain conditions, the energy release rate at the opening fracture front... With open-type stress intensity factor The following conditions must be met: in, The energy release rate represents the elastic strain energy released per unit area as the fracture front expands. It is the open-type stress intensity factor; Poisson's ratio for intact rock; Let be the elastic modulus of intact rock. This equation shows that the release rate of elastic strain energy at the fracture front is directly proportional to the square of the stress intensity factor and inversely proportional to the elastic modulus. This relationship is derived from the crack closure integral in linear elastic fracture mechanics.

[0033] To determine the stress intensity factor of a coin-shaped crack, for a coin-shaped crack subjected to a uniform tensile stress σ in the far field, the stress intensity factor at any point on its leading edge is: in, The polar angle used to describe the circumferential position of the fracture front edge; For follow The varying dimensionless distribution function reaches a maximum value of 1 at the location corresponding to the direction of stress application. This equation shows that the stress intensity factor is related to the far-field stress. Proportional to the crack radius It is proportional to the square root of the energy release rate. Substituting the energy release rate relationship into the equation, we can obtain the energy release rate at each point on the fracture front as a function of the square root of the energy release rate. Proportional.

[0034] Integrating along the fracture front yields the total released strain energy of a single fracture. (Regarding the polar angle...) Energy release rate Perform circumferential integration along the fracture front edge, where the integration arc length is related to the fracture radius. Proportional to the total released strain energy of a single crack, its dimensions are: : The term originates from the fact that the energy release rate is proportional to the square of the stress. The term originates from the energy release rate and Proportional to, integral arc length and Proportional, strain energy conversion and then introduction The cumulative contribution The term originates from the decrease in stored strain energy at the same stress level when the rock stiffness increases.

[0035] The first step involves statistically averaging the isotropic and uniform distribution of the crack normal. Since the crack normal is isotropic and uniformly distributed in space, the total released strain energy obtained in the third step is integrated and averaged with equal probability over all possible orientations to eliminate the influence of the randomness of the crack orientation, thus obtaining the average released strain energy of a single randomly oriented crack. in, Let be the average released strain energy of a single randomly oriented fracture. This formula shows that the released strain energy of a single fracture is proportional to the square of the far-field stress, proportional to the cube of the fracture radius, and inversely proportional to the elastic modulus of the intact rock. Simultaneously, a dimensionless coefficient related to Poisson's ratio reflects the contribution of fracture opening deformation to the released energy. Where the dimensionless coefficient is... The numerical coefficients obtained by oriented statistical averaging depend only on the Poisson's ratio of the intact rock, for example, when When the value is 0.25, the coefficient is approximately 0.88. When the value is 0.30, the coefficient is approximately 0.87. It decreases gradually with increasing Poisson's ratio, indicating that the influence of Poisson's ratio on this coefficient is weaker than the influence of fracture geometry parameters. The technical advantage of this formula lies in its ability to represent the geometric characteristics of the fracture (radius) ) and rock material parameters ( , Explicit decoupling allows the calculation of total additional strain energy to be based solely on the sum of the contributions of each crack according to the number of cracks, providing a basis for the derivation of the closed analytical expression of the equivalent elastic modulus.

[0036] The total additional strain energy of the rock mass is calculated by introducing correction coefficients for fracture density and fracture number based on the average released strain energy of a single fracture, and then calculating the additional strain energy caused by the elastic deformation of fractures in the rock mass.

[0037] Define fracture density The formula is used to characterize the contribution of fractures per unit volume to the damage of rock mass, and is as follows: in, The total number of fractures within a rock mass unit, dimensionless, fracture density. The number of fractures, fracture radius, and rock mass unit volume are integrated into a dimensionless parameter, reflecting the spatial density of fractures. Fracture density is calculated using... The reason for this form is that the dimension of the average strain energy released by a single crack is... The number of cracks with radius cubed Merging allows the total additional strain energy to be substituted into the energy balance equation and then combined with the unit volume. After natural reduction, the final expression for the equivalent elastic modulus depends only on the dimensionless combination. This ensures the simplicity and dimensional consistency of the equations; if the crack density is in another form, the equivalent elastic modulus expression will retain the element volume term, making the model sensitive to the choice of element size, which is not conducive to engineering applications.

[0038] Introducing a crack number correction factor This study comprehensively corrects for the geometric deviations between actual and ideal coin-shaped fractures, including the equivalent effects of factors such as fracture thickness, surface roughness, and edge damage. An ideal coin-shaped fracture assumes a smooth mathematical plane surface and a regularly circular fracture front. However, actual rock fracture surfaces often exhibit roughness, irregular propagation at the fracture front, and edge damage zones. These factors alter the actual magnitude of strain energy released by the fracture. A fracture number correction factor is introduced to address this issue. The released strain energy of the actual crack is equivalent to The released strain energy of the ideal coin-shaped crack is, after correction, the equivalent number of cracks is: Thus, the total additional strain energy of all fractures within the rock mass unit is obtained: Deriving the equivalent elastic modulus of fractured rock mass. The additional strain energy caused by the elastic deformation of fractures within the rock mass is substituted into the energy balance equation of the rock mass element to derive the equivalent elastic modulus of the fractured rock mass.

[0039] Substituting the expression for the total additional strain energy into the energy balance equation, and dividing both sides of the equation by... ,get ;Will Substitute in, and combine with fracture density By rearranging the expression for the average released strain energy of a single crack, we obtain Taking the reciprocal of both sides of the equation, we obtain the ratio of the equivalent elastic modulus of fractured rock mass to that of intact rock: This formula reflects the reduction law of fracture parameters to rock mass stiffness: fracture density The larger the correction factor The larger the value, the larger the denominator, and the greater the ratio of equivalent elastic moduli. The smaller the value, the more significant the reduction in rock mass stiffness and the higher the degree of damage. When =0 or When =0, This degenerates into the case of intact rock. The derivation shows that the degree of equivalent stiffness reduction is determined by the elastic modulus of intact rock. Poisson's ratio ν and dimensionless parameters Together, they determine that all the geometric information of the fracture is determined by the fracture density. The parameters are entered into the model in a dimensionless form, and each parameter has a clear mechanical meaning. In the subsequent calibration stage, only two learnable parameters need to be calibrated, and the model complexity matches the data requirements.

[0040] Establish a physical prior strength model. Based on the definition of damage using the elastic modulus, the rock mass damage degree is defined as follows: This definition is based on the expression of the elastic modulus damage variable under the assumption of equivalent strain in continuous damage mechanics, and is equivalent to... When the rock mass has no cracks or damage , As the fracture density increases, Reduce The value is increased and approaches 1, ensuring the damage degree is always limited to the range of 0 to 1, thus avoiding non-physical results such as negative strength predictions. Combined with the Mohr-Coulomb strength criterion, the peak strength of intact rock under confining pressure satisfies: in, The uniaxial compressive strength of intact rock; ξ is the confining pressure; ξ is the confining pressure strengthening coefficient, dimensionless, derived from the internal friction angle of the rock. Decide, The derivation of the expression is as follows: The Mohr-Coulomb failure criterion in principal stress space can be written as follows: Using trigonometric identities and , can be obtained .

[0041] This formula shows that when =0° =1, the confining pressure strengthening term equals the confining pressure itself; the internal friction angle of intact rock is greater than 0°, at this time The value is between 0 and 1. It is always greater than 1; for every 1 MPa increase in confining pressure, the increase in peak strength is greater than 1 MPa. The larger, The larger, The larger the confining pressure, the more significant the strengthening effect.

[0042] Substituting the equivalent elastic modulus into the damage intensity formula Based on the confining pressure strengthening criterion, a physical prior model for the strength of fractured rock is established, and the strength calculation formula of the physical prior model is finally obtained. The model output is the predicted baseline strength value σt: in, To predict baseline intensity values; The uniaxial compressive strength of intact rock is obtained through a standard uniaxial compression test. The confining pressure strengthening coefficient is dimensionless and is obtained through the confining pressure-strength data of a complete specimen. The confining pressure is one of the target operating condition parameters. The crack number correction factor is dimensionless and is obtained through global optimization and calibration using representative specimen strength data. The fracture density is dimensionless and is determined by the number of fractures. Crack radius and rock mass unit volume according to The value ν is calculated to be the Poisson's ratio of the intact rock, which is dimensionless and obtained through a standard uniaxial compression test.

[0043] The molecular part of this physical prior model The peak strength of intact rock under confining pressure is represented by the denominator, which represents the strength reduction effect of fracture damage. When the fracture density... As the denominator increases, the predicted baseline intensity value increases. The decrease reflects the weakening effect of fractures on the strength of the rock mass; when the confining pressure... As the molecular weight increases, the predicted baseline intensity value increases. The increase reflects the strengthening effect of confining pressure on the rock mass. Based on the conservation of elastic deformation energy, the model, through rigorous mechanical derivation, ensures that the predicted baseline strength value always meets the basic mechanical constraints such as the peak strength increasing monotonically with confining pressure and the damage degree ranging from 0 to 1.

[0044] S2: The learnable parameters of the physical prior model of fractured rock strength are calibrated using rock mechanics test samples. Based on the calibrated physical prior model of fractured rock strength, the predicted baseline strength value corresponding to each test sample is calculated. The real residual dataset is generated according to the difference between the predicted baseline strength value and the measured peak strength value of the corresponding sample, which serves as the training label for the subsequent convolutional residual correction network.

[0045] To obtain complete basic mechanical parameters of the rock, learnable parameters include the fracture number correction factor and the confining pressure strengthening factor. Specifically, the intrinsic parameters of the intact rock are first obtained through basic mechanical tests, including the uniaxial compressive strength, elastic modulus, and Poisson's ratio of the intact rock. Then, the initial confining pressure strengthening factor is calculated based on the confining pressure-strength data of the intact specimen.

[0046] Specifically, the uniaxial compressive strength of intact rock is determined by a standard uniaxial compression test. Elastic modulus Ea and Poisson's ratio The test methods shall be performed in accordance with the recommended methods of the International Society for Rock Mechanics or the standard GB / T 50266 Test Methods for Engineering Rock Mass: The axial stress value corresponding to the peak point of the uniaxial compressive stress-strain curve is taken; Ea is taken as the slope of the approximately straight line segment of the stress-strain curve, that is, the ratio of the axial stress increment to the axial strain increment in this segment; ν is taken as the absolute value of the ratio of the transverse strain increment to the axial strain increment in the approximately straight line segment. Through conventional triaxial compression tests, the peak strength of intact specimens is measured under multiple different confining pressure levels (e.g., 5 MPa, 10 MPa, 20 MPa, 30 MPa), obtaining a confining pressure-strength data sequence. Based on this data sequence, the initial confining pressure strengthening coefficient is calculated. The calculation logic of the initial confining pressure strengthening coefficient ξ is as follows: the confining pressure-strength data sequence is fitted with a least-squares linear model, and the slope of the fitted line is the initial estimate of ξ; simultaneously, the fitted slope is used to calculate the initial value of ξ. Calculate the internal friction angle of the intact rock This is used to check whether the initial estimate is within a reasonable range for the internal friction angle of common rocks. The initial confining pressure strengthening coefficient serves only as the starting point for subsequent global calibration iterations, and its final value is determined by the global calibration.

[0047] The learnable parameters are globally calibrated by selecting representative specimen strength data covering different confining pressures, different fracture diameters, and different numbers of fractures. The measured peak strength value is used as the true value. The least squares method is used to globally optimize and calibrate the fracture number correction coefficient and confining pressure strengthening coefficient in the physical prior model of fractured rock strength. The optimization objective is to minimize the mean square error between the predicted baseline strength value and the measured peak strength value.

[0048] The optimization objective formula is: in, The number of test samples for calibration; For the first The measured peak strength values ​​of each specimen; For the first The predicted baseline strength value of each specimen; the reason for using the squared error form of the objective function is that the squared loss imposes a higher penalty on samples with larger deviations, and the calibration process prioritizes compressing samples with large deviations, so that the calibration results maintain a low deviation level across the entire working condition range. In addition, the squared loss is a smooth convex function, which is convenient for numerical algorithms to solve stably.

[0049] The Levenberg-Marquardt algorithm was used to solve the problem. The initial value is 1. The initial value is taken from the aforementioned initial confining pressure strengthening coefficient. In each iteration, two parameters are updated along the direction of MSE decrease until the relative change in MSE between two adjacent steps is less than a preset threshold (e.g., 1×10). -6 (or reaches the preset iteration limit; constraints are applied throughout the solution process.) >0、 1. Constraints The basis for >1 is that the confining pressure strengthening coefficient corresponds to the Mohr-Coulomb criterion. The value of this ratio is always greater than 1 when the internal friction angle is positive; constraint The reason for the value >0 is that the correction factor represents the equivalent multiple of the actual fracture relative to the ideal coin-shaped fracture, and physically, there are no negative values. After calibration, the coefficient of determination of the physical prior model is shown in 127 sets of test data covering confining pressures from 5 MPa to 60 MPa, 1 to 200 fractures, and fracture diameters from 1 mm to 10 mm. The result reached 0.8231, indicating that the physical prior model has captured the intensity-dominant variation law, and the remaining bias is corrected by the convolutional residual correction network.

[0050] When selecting representative specimens, it is necessary to ensure that the specimens cover a sufficiently wide range of confining pressure, crack diameter, and crack number to ensure that the calibrated parameters have good applicability under different working conditions. For example, the confining pressure can cover 5MPa to 60MPa, the crack diameter can cover 1mm to 10mm, and the number of cracks can cover 1 to 200.

[0051] After calibration, all parameters of the physical prior model of fractured rock strength are kept fixed.

[0052] The reason for keeping all parameters of the physical prior model fixed after calibration is to prevent unconstrained drift of physical parameters due to data-driven optimization, thus ensuring mechanical consistency. If physical parameters were allowed to continue to be updated in subsequent training phases, data-driven optimization might adjust the physical parameters to values ​​that violate mechanical laws in order to reduce residuals. For example, adjusting the confining pressure strengthening coefficient ξ to a negative value would cause the predicted baseline strength value to decrease with increasing confining pressure, thus compromising the mechanical consistency of the model. Therefore, by freezing all parameters of the physical prior model after calibration and subsequently training only the residual correction network, the mechanical meaning of the physical prior model is guaranteed from being destroyed by the data training process.

[0053] Generate a true residual dataset. Substitute the target load parameters of all training samples—confining pressure, number of fractures, and fracture diameter—into the calibrated physical prior model to calculate the predicted baseline intensity value for each sample, and then calculate the true residual for each sample accordingly. in, For the first The true residuals of each sample represent the prediction bias of the physical model. The true residuals of all samples form the true residual dataset, which serves as the training labels for the subsequent convolutional residual correction network. This reflects the systematic bias of the physical prior model under specific working conditions. This bias stems from the difference between the physical model assumptions and the actual rock mass, such as the non-isotropic spatial distribution of fractures, the interaction effects between fractures, and frictional slip on fracture surfaces—factors not explicitly considered in the physical prior model. By using the actual residuals as training labels, the convolutional residual correction network can specifically learn the bias patterns of the physical prior model, rather than relearning the complete intensity mapping relationship.

[0054] S3: Construct a convolutional residual correction network with channel attention mechanism. The convolutional residual correction network takes the target working condition parameters and zero confining pressure indication features as inputs and outputs the predicted residual correction value of the physical prior model of fractured rock strength.

[0055] Determine the network input features. The network input contains two types of features, totaling four dimensions. The first type is the target operating condition parameters, including confining pressure. Number of cracks Crack diameter ,in That is, the fracture diameter is twice the fracture radius. The second type is the zero confining pressure indication feature. The zero confining pressure indication feature is a Boolean input feature. When the confining pressure in the target working condition parameter is 0, the zero confining pressure indication feature takes the value of 1; when the confining pressure in the target working condition parameter is not 0, the zero confining pressure indication feature takes the value of 0.

[0056] The zero confining pressure indicator feature is designed to help the network identify the unique mechanical modes of boundary conditions without confining pressure, thereby improving the accuracy of boundary condition corrections. Under uniaxial compression conditions with zero confining pressure, the frictional slip mechanism of the fracture surface differs fundamentally from that under conditions with confining pressure, and the deviation pattern of the physical prior model under this boundary condition also differs from other conditions. By introducing the zero confining pressure indicator feature, the network can directly distinguish between uniaxial and triaxial compression conditions at the input level, thus learning different residual correction modes for the two conditions. Without this feature, the network would only distinguish conditions based on the confining pressure value itself, potentially leading to discontinuous correction values ​​or decreased correction accuracy in the transition region where the confining pressure approaches zero.

[0057] A three-level network structure is constructed. The convolutional residual correction network with channel attention mechanism specifically adopts a three-level structure of convolutional feature extraction, channel attention weighting, and fully connected output.

[0058] The input layer reshapes the 4-dimensional input features into a 1-dimensional sequence, adapting it for 1-dimensional convolution operations. Specifically, it organizes the confining pressure, number of fractures, fracture diameter, and zero confining pressure indicator features into a 1-dimensional sequence of length 4 and number of channels 1 in a fixed order. Because the input features have a low dimensionality, directly using a fully connected network can easily lead to insufficient parameters and inadequate feature interaction. However, by reshaping the input features into a 1-dimensional sequence, sliding convolutions can be performed along the sequence dimension using 1-dimensional convolutional kernels to extract local correlation patterns between features in a weight-sharing manner.

[0059] The convolutional feature extraction layer uses two 1D convolutional kernels to perform nonlinear transformation on the input features, generating a 16-channel high-dimensional abstract feature embedding. The activation function is ReLU, which preserves the nonlinear fitting ability.

[0060] The first convolutional layer employs a 1D convolution with a kernel size of 3, a stride of 1, and a zero-padding width of 1. It has 1 input channel and 4 output channels, transforming a 4-length, 1-channel 1D sequence into a feature map of length 4 and 4 channels, followed by a ReLU activation function. The function of this layer is to slide a 3-size convolutional kernel across the original feature sequence, capturing the local interactions between adjacent input features. For example, confinement and zero-confinement indicate that features are adjacent in the sequence, and their combined interaction pattern can be directly captured by the convolutional kernel. The stride of 1 combined with the zero-padding width of 1 ensures that the sequence length remains unchanged before and after convolution, and the corresponding feature positions are not lost. The 4 output channels expand the single input sequence into 4 different feature maps, each emphasizing different combinations of adjacent features.

[0061] The second convolutional layer employs a 1D convolution with a kernel size of 3, a stride of 1, and a zero-padding width of 1. It has 4 input channels and 16 output channels, transforming the output of the first convolutional layer into a feature map of length 4 and 16 channels, followed by a ReLU activation function. The purpose of this layer is to extract higher-order feature combination patterns based on the features from the first convolutional layer and expand the number of channels to 16, providing sufficient candidate feature channels for the channel attention mechanism. The kernel size of 3 is chosen because the input feature sequence length is 4, and a kernel of size 3 can cover the interaction relationships of three adjacent positions. Larger kernels offer limited benefits and increase the number of parameters with a limited sequence length. The 16-channel designation is based on the fact that the input features are only 4-dimensional; a 16-channel high-dimensional embedding can provide sufficient feature representation capabilities without excessively increasing the number of parameters, enabling the network to extract residual-related information from the input features from different perspectives.

[0062] The reason for choosing ReLU as the activation function is that ReLU's gradient is always 1 in the positive interval, avoiding the gradient vanishing problem in deep networks; ReLU sets the negative response to zero, which is equivalent to sparsifying the convolution output features and retaining only the responses that contribute to the residual correction; at the same time, the nonlinear characteristics of ReLU enable the network to fit the nonlinear mapping relationship between the residual and the input features.

[0063] The channel attention layer calculates weights for each of the 16 channel features. Specifically, it performs global average pooling along the sequence length dimension on the feature map output by the second convolutional layer (4 length, 16 channels), averaging the values ​​at four positions for each channel to compress it into a 16-dimensional channel statistical vector, representing the overall activation intensity of each channel. This channel statistical vector is then passed sequentially through a first fully connected layer and a second fully connected layer. The first fully connected layer compresses the 16-dimensional vector into a 4-dimensional vector and activates it using ReLU. The second fully connected layer restores the 4-dimensional vector to 16 dimensions, which is then normalized to the 0-1 range using the sigmoid function, resulting in a 16-dimensional channel weight vector. Finally, the channel weight vector is multiplied channel-by-channel by the 4-length, 16-channel feature map to obtain the attention-weighted feature map. The role of global average pooling is to characterize the overall activation intensity of a channel by the average response of each channel across all sequence positions, filtering out the interference of random fluctuations in the response at a single position on the judgment of channel importance; two fully connected layers form a compression-reduction structure, compressing the channel statistical vector to 4 dimensions and then restoring it to 16 dimensions. The compression process forces the network to learn a compact representation of the correlation between channels, suppressing overfitting when there are many channels; the sigmoid function normalizes the weights to the 0~1 range, giving the weights a clear meaning of the channel importance.

[0064] The channel attention mechanism adaptively focuses on feature channels that contribute significantly to residual correction while suppressing redundant channels. Among the 16 channels, different channels may extract different types of feature patterns; for example, some channels are sensitive to changes in confining pressure, some to changes in the number of fractures, and some show a significant response to zero confining pressure conditions. Through the channel attention mechanism, the network can automatically learn the importance weight of each channel for the residual correction task, assigning weights close to 1 to channels with high contributions and amplifying their responses, while assigning weights close to 0 to channels with low contributions and suppressing their responses, thereby improving feature utilization efficiency. The attention weight distribution also provides observable intermediate variables for residual source analysis.

[0065] The flattening layer flattens the attention-weighted feature map of length 4 and number of channels 16 into a 64-dimensional feature vector, eliminating the constraints of the convolutional structure on the organization of position and channels. The fully connected output layer maps the 64-dimensional feature vector into a 1-dimensional output, namely the predicted residual correction value, in MPa. The output layer uses a linear activation function and does not restrict the range of output values, allowing the predicted residual correction value to take values ​​in both negative and positive ranges, corresponding to the characteristic that the true residual can be positive or negative. That is, the physical prior model overestimates the intensity in some operating conditions and underestimates the intensity in others.

[0066] After the above layers, the total number of network parameters is on the order of several hundred, with low training and inference overhead, making it suitable for small sample rock mechanics test data scenarios. If the number of network parameters is significantly larger than this order of magnitude, obvious overfitting will occur on the training set of more than a hundred samples. Therefore, this embodiment limits the network depth and the number of channels.

[0067] S4: A two-stage decoupling strategy is used to train the hybrid prediction model: In the first stage, all parameters of the physical prior model of fractured rock strength are fixed. In the second stage, the real residual dataset obtained is used as the training label to train the convolutional residual correction network, thus obtaining the trained hybrid prediction model.

[0068] The first phase involves freezing the parameters of the physical model, including all parameters of the prior physical model. All are frozen and will no longer participate in subsequent gradient updates.

[0069] The reason for freezing is to ensure that the mechanical meaning of physical parameters is not destroyed by data training.

[0070] If a joint training strategy is adopted, that is, updating both the physical prior model parameters and the residual network parameters simultaneously, data-driven optimization may adjust the physical parameters to values ​​that violate the laws of mechanics in order to reduce the overall loss. The essential difference between the two-stage decoupling strategy and the joint training strategy is that joint training treats the physical parameters as ordinary parameters that can be freely adjusted, while the two-stage decoupling strategy treats the physical parameters as fixed constants with clear mechanical meanings. The data-driven part only makes local corrections to the residuals of the physical model, rather than replacing the physical model itself.

[0071] Dataset partitioning and hyperparameter optimization. A stratified random sampling method was used to partition the training, validation, and test sets. Specifically, all experimental samples were stratified according to confining pressure level and fracture number level. For example, confining pressure was divided into three levels: less than 20 MPa, 20 MPa to 40 MPa, and greater than 40 MPa; fracture number was divided into three levels: less than 50, 50 to 150, and greater than 150. All samples fell into a two-dimensional stratified grid composed of confining pressure level and fracture number level. Within each grid, samples were randomly drawn at a fixed ratio and divided into training, validation, and test sets at a ratio of 60%, 20%, and 20%, respectively, ensuring that all three sets in each stratified grid contained samples. The reason for stratified sampling is to ensure that the test set covers all combinations of confining pressure and fracture quantity levels, avoiding random sampling bias that could cause the test set to concentrate in a certain range of conditions, resulting in overestimation or underestimation of generalization performance. The reason for dividing the dataset into 60%, 20%, and 20% sets is that the training set needs sufficient sample size to support network training, the validation set is used for early stopping monitoring during training, and the test set is only used for final evaluation after training; the three sets do not overlap. After the dataset is divided, a Bayesian optimization algorithm is used to search for the hyperparameters of the residual network. The search space includes the learning rate, L2 regularization coefficient, and number of convolutional kernels. The optimization objective is to minimize the root mean square error of 10-fold cross-validation to obtain the optimal combination of hyperparameters. The execution method is as follows: the training set and validation set are merged and then randomly divided into 10 folds. Nine folds are used alternately as training data, and the remaining fold is used as validation data, for a total of 10 training and validation iterations. The mean square root error (RMSE) of the 10 validation iterations is taken as the evaluation metric for this hyperparameter combination. A Gaussian process surrogate model is used to fit the mapping relationship between the hyperparameter values ​​and the corresponding cross-validation RMSE. The expected improvement acquisition function is used to select the next hyperparameter combination to be evaluated in the search space. This evaluation and fitting process is repeated a preset number of times (e.g., 25 times), and the hyperparameter combination with the smallest cross-validation RMSE is taken as the optimal hyperparameter combination. The search space is set to a learning rate of 1×10⁻⁶ on a logarithmic scale. -4 Up to 1×10 - ² values ​​within the interval, the L2 regularization coefficient on a logarithmic scale of 1×10 -6 Up to 1×10 -The hyperparameter values ​​are taken within the range of ², and the number of convolutional kernels is taken from a preset set {8, 16, 32, 64}. Bayesian optimization is used instead of grid search because residual networks consume computational resources with each training iteration. Bayesian optimization uses the Gaussian posterior distribution of the evaluated hyperparameter points to guide the next set of search positions, determining a better hyperparameter combination with far fewer evaluations than grid search. The root mean square error (RMSE) of cross-validation is used as the evaluation metric because cross-validation can comprehensively evaluate the network's stable performance under different data partitions. The RMSE has the same dimensions as the residual training task (both are MPa), facilitating direct comparison of the merits of different hyperparameter combinations. The 10-fold rule is based on the fact that increasing the number of folds improves evaluation stability but exponentially increases training overhead; a 10-fold rule strikes a balance between evaluation stability and computational cost. The hyperparameter search range covers multiple orders of magnitude of the learning rate and regularization strength, ensuring that the search range includes optimal configurations suitable for small sample datasets.

[0072] The second stage involves specialized training of the residual network. The generated real residual dataset serves as the training label, while the target operating condition parameters and zero confinement pressure indication features are used as network inputs to train the convolutional residual correction network.

[0073] Before training, the input features are standardized. The three continuous features—confining pressure, number of fractures, and fracture diameter—are standardized using the mean and standard deviation of the training set samples, respectively. This means each feature is subtracted from the training set mean and divided by the training set standard deviation, ensuring the three continuous features have similar numerical scales. The zero-confining-pressure indicator feature is a Boolean input feature, retaining its 0 / 1 value and not standardized. The reason for standardization is that the numerical scales of confining pressure (maximum 60 MPa), number of fractures (maximum 200), and fracture diameter (maximum 10 mm) differ significantly. Without standardization, features with larger numerical scales dominate gradient updates, while features with smaller numerical scales are difficult to learn effectively. Standardization brings the features to similar orders of magnitude, which is beneficial for stable convergence of the gradient descent process and for the comparability of channel attention weights across different feature channels. During training, the mean squared error loss function is used, consistent with the objective function form in the calibration stage, ensuring that the training objective of the residual network corresponds to the dimensions of the true residuals. The weights of each layer of the network are initialized using He, and the initial weight distribution is adaptively determined based on the number of neurons in the previous layer, so that the output of each layer maintains a stable variance in the early stage of training. The ReLU activation function is used to avoid gradient vanishing in the early stage of training.

[0074] The Adam optimizer was used to train the CNN-Attention network. The Adam optimizer was chosen because it combines the advantages of momentum and adaptive learning rates. It maintains first-order and second-order moment estimates for each parameter, and can adaptively adjust the learning rate based on historical gradient information, exhibiting good convergence performance in scenarios with sparse gradients and non-stationary objective functions. For residual correction networks, which have a small number of parameters but non-linear input-output relationships, the Adam optimizer can achieve stable convergence within a relatively small number of training epochs. The initial learning rate of the Adam optimizer during training was set to 1×10⁻⁶. - ³, First-order moment estimation of attenuation coefficient Using 0.9, the second moment is used to estimate the attenuation coefficient. Take 0.999, numerical stability constant Take 1×10 -8 The batch size is 32 samples. The initial learning rate is 1×10⁻⁶. - The basis for setting ³ is that Adam adaptively scales each parameter according to the second moment of its historical gradient. When the initial learning rate is set to this order of magnitude, the convergence speed in the early stage of training is moderate and it is not easy to overshoot the optimal solution. , The value of ensures that the estimation of the first and second moments has appropriate memory length and smoothness; To prevent division by zero errors when the second-order moment estimate is zero, the batch size of 32 is set based on the following: when the total sample size is small, a large batch size leads to smooth gradient estimation and the network is prone to getting trapped in local optima; a small batch size results in excessive gradient estimation noise and oscillations during training. 32 ensures a balance between stability and randomness in gradient estimation in small-sample scenarios. The number of convolutional kernels and the L2 regularization coefficient are determined by Bayesian optimization. L2 regularization applies a sum of squares penalty to the network weights to suppress excessively large individual weights and prevent the network from overfitting to noise in the training samples. The maximum number of training epochs is set to 200 to prevent infinite iteration when the training process fails to converge.

[0075] The training process employs the ReduceLROnPlateau adaptive learning rate decay strategy, which reduces the learning rate when the validation set loss does not decrease for 10 consecutive rounds.

[0076] The purpose of this strategy is to avoid oscillations caused by an excessively large learning rate or slow convergence caused by an excessively small learning rate. In the early stages of training, a larger learning rate helps to quickly approach the vicinity of the optimal solution. When the validation set loss does not decrease for several consecutive rounds, it indicates that the parameter update step size under the current learning rate is too large, causing oscillations around the optimal solution and preventing further convergence. In this case, reducing the learning rate can decrease the parameter update step size, allowing the parameters to more precisely approximate the optimal solution. The specific method of learning rate decay is to multiply the current learning rate by a decay factor of 0.5 and set the lower limit of the learning rate to 1×10⁻⁶. -6A decay factor of 0.5 represents a compromise. An excessively large decay factor leads to a rapid decrease in the learning rate and premature training stagnation, while an excessively small decay factor results in a slow decrease in the learning rate and prolonged oscillations. The lower bound on the learning rate prevents the network from losing its effective update capability after the learning rate decays to an excessively small order of magnitude. The setting of 10 consecutive epochs is based on the fact that too few epochs may cause the learning rate to decrease prematurely, leading to local optima; too many epochs may result in excessively long training times, continuing ineffective updates when oscillations have already occurred.

[0077] An early stopping strategy is implemented, terminating training when the validation set loss does not increase for 20 consecutive rounds to prevent overfitting.

[0078] The purpose of early stopping is to prevent overfitting. As the number of training epochs increases, the network loss on the training set continuously decreases, but the loss on the validation set may first decrease and then increase; the point of increase marks the beginning of overfitting. By monitoring the validation set loss and terminating training when the validation set loss does not increase for 20 consecutive epochs, training can be stopped before overfitting occurs, preserving the model parameters with optimal generalization performance. The 20 consecutive epochs are based on the fact that the validation set loss may exhibit local fluctuations. Too few epochs may lead to premature termination of training at temporary fluctuations in the validation set loss, missing further convergence; too many epochs may lead to continued training after overfitting has occurred, reducing the model's generalization performance. When early stopping is triggered, the training program restores the network weights corresponding to the epoch with the lowest validation set loss, avoiding retaining the parameter state after the validation set loss has begun to increase.

[0079] After training, the convolutional residual correction network outputs the corresponding predicted residual correction value based on the input target operating condition parameters and zero confining pressure indication features, thus completing the correction.

[0080] Please see Figure 2 The curves showing the changes in training and validation losses of the convolutional residual network during training epochs illustrate the variation of training and validation losses of the CNN-Attention residual network with training epochs in this embodiment. The horizontal axis represents the number of training epochs (0-150 epochs), and the vertical axis represents the mean squared error loss (unit: squared MPa). Figure 3It can be seen that in the early stage of training (0 to 30 epochs), the training loss and validation loss decrease rapidly and synchronously, with similar rates of decrease, indicating that the network can effectively learn the residual patterns. At this stage, the network is in an underfitting state and has not yet overfitted. In the middle stage of training (30 to 80 epochs), the rates of decrease for both slow down, but the rate of decrease for the training loss is slightly greater than that for the validation loss. The gap between the training loss and the validation loss gradually widens, indicating that the network has basically captured the main patterns of the residuals. Subsequent parameter updates mainly target the detailed features and noise in the training data, leading to an overfitting trend. In the later stage of training (after 80 epochs), the validation loss reaches its lowest point and then begins to rise slowly, while the training loss continues to decline. At this point, the early stopping strategy comes into play. Training is terminated when the validation loss does not decrease for 20 consecutive epochs. The optimal stopping point is approximately between epochs 85 and 90, and the network weights corresponding to that epoch are restored as the final model weights. The patience value of the early stopping strategy is set to 20 rounds, which strikes a balance between avoiding premature termination due to temporary fluctuations in validation loss and preventing overfitting. During training, a ReduceLROnPlateau adaptive learning rate decay strategy is also employed. When the validation set loss does not decrease for 10 consecutive rounds, the learning rate is multiplied by a decay factor of 0.5, with a lower limit of 1× This strategy allows the system to quickly approach the optimal solution with a large learning rate in the early stages of training, and then to finely approach the optimal solution with a smaller learning rate in the later stages of training, thus avoiding oscillations around the optimal solution.

[0081] During the middle of training (30-80 epochs), the rate of decrease of both training loss and validation loss slowed down, but the rate of decrease of training loss was slightly greater than that of validation loss. This indicates that the network has basically captured the main patterns of the residuals. Subsequent parameter updates are mainly aimed at the detailed features and noise in the training data. The gap between training loss and validation loss gradually widens, indicating the emergence of an overfitting trend.

[0082] In the later stages of training (after 80 rounds), the validation loss reaches its lowest point and then begins to rise slowly, while the training loss continues to decline. This is a typical sign of overfitting. The network's performance on the training data continues to improve, but its performance on the validation data begins to deteriorate. At this point, the early stopping strategy comes into play. When the validation loss does not increase for 20 consecutive rounds, training is terminated, and the network weights corresponding to the round with the lowest validation loss are restored as the final model weights.

[0083] The optimal stopping point, marked by the gray dashed line in the figure, is around rounds 85-90. This position is exactly near the minimum point of the validation loss. The early stopping strategy is based on the fact that the validation loss may fluctuate locally. Too few patience rounds (such as 5 rounds) may cause training to terminate prematurely at temporary fluctuations, missing further convergence. Too many patience rounds (such as 50 rounds) may cause training to continue after overfitting occurs, reducing the model's generalization performance. The patience setting of 20 rounds achieves a good balance between premature and late termination.

[0084] The training process also employs the ReduceLROnPlateau adaptive learning rate decay strategy: when the validation set loss does not decrease for 10 consecutive rounds, the current learning rate is multiplied by a decay factor of 0.5, with a lower limit of 1e-6. The purpose of this strategy is that a larger learning rate in the early stages of training helps to quickly approach the optimal solution; when the validation set loss does not decrease for several consecutive rounds, reducing the learning rate can reduce the parameter update step size, allowing the parameters to more precisely approximate the optimal solution and avoid oscillations around the optimal solution.

[0085] S5: For the trained hybrid prediction model, the group permutation importance method is used to quantify the contribution rate of each target working condition parameter to the final peak intensity prediction value, and the contribution rate ranking result is verified to obtain a qualified hybrid prediction model. The target working condition parameter to be predicted is input into the qualified hybrid prediction model, and the corresponding prediction baseline intensity value and prediction residual correction value are superimposed to obtain the final peak intensity prediction value and output it.

[0086] The importance contribution rate of grouped permutations was quantified. Different permutation strategies were adopted for the confining pressure, number of fractures, and fracture diameter in the target working condition parameters. Specifically, global random permutation was used for confining pressure; global random permutation was used for the number of fractures; and conditional permutation based on confining pressure level grouping was used, that is, the diameter value was randomly shuffled only within the same confining pressure level.

[0087] The differentiated replacement strategy is designed to avoid generating physically unreasonable extreme samples and ensure the accuracy of contribution rate calculations. For confining pressure and the number of fractures, there are no obvious physical constraints on their value ranges; any combination of confining pressure and any number of fractures may correspond to real experimental conditions. Therefore, global random replacement can be used. For fracture diameter, there is a physical correlation with confining pressure. Fracture expansion and closure behaviors differ under different confining pressure levels. Exchanging the fracture diameter under a low confining pressure condition with that under a high confining pressure condition may generate physically unreasonable extreme samples, such as extremely small fracture diameters under high confining pressure conditions or extremely large fracture diameters under low confining pressure conditions. These extreme samples will distort the contribution rate calculation. By grouping conditional replacements by confining pressure level, the diameter values ​​are randomly shuffled only within the same confining pressure level, ensuring that the replaced samples remain within the physically reasonable range of experimental conditions.

[0088] The specific method for quantifying the contribution rate of features is to perform a fixed number of random permutations on the confining pressure, number of fractures, and fracture diameter, and the importance score of each target working condition parameter is obtained by calculating the difference in the average absolute error of the model prediction before and after the permutation.

[0089] The fixed number of permutations is set at 50. This setting is based on the fact that the calculation of the importance score relies on the statistical stability of random permutations. Too few permutations can lead to large fluctuations in the importance score due to the randomness of a single permutation, resulting in unreliable results; too many permutations increase computational costs but offer limited improvement in the statistical accuracy of the importance score. 50 permutations strike a balance between computational cost and statistical stability. Each permutation only changes the correspondence between the evaluated target condition parameters and the zero confining pressure indicator features among the samples, while the values ​​of the other two target condition parameters remain unchanged, thus isolating the permuted parameters from the prediction results. The permutation order is generated using a pre-fixed random seed, ensuring the reproducibility of the importance assessment results.

[0090] The formula for calculating importance score is: in, For the first Importance score of each target operating condition parameter; The mean absolute error of the final peak intensity prediction value output by the hybrid prediction model without replacement; To replace the first After considering the target operating parameters, the hybrid prediction model outputs the mean absolute error of the final peak intensity prediction value.

[0091] The logic behind this formula is that when a feature is randomly replaced, the true correlation between that feature and the output is disrupted. If the feature makes a significant contribution to the prediction result, the prediction error of the model after replacement will increase significantly; if the feature makes a small contribution to the prediction result, the prediction error of the model after replacement will not change much. Therefore, the difference between the mean absolute error before and after replacement can quantify the degree of contribution of that feature to the prediction result.

[0092] The importance scores of each target operating condition parameter are normalized to obtain the percentage contribution rate of each target operating condition parameter.

[0093] The normalization process involves dividing the importance scores of the three parameters by the sum of their importance scores to obtain the percentage contribution of each parameter.

[0094] Mechanical rationality verification involves ranking the contribution percentages of confining pressure, number of cracks, and crack diameter to verify mechanical rationality. The specific rules for mechanical rationality verification are: the contribution rate of confining pressure is greater than the contribution rate of crack diameter, and the contribution rate of crack diameter is greater than the contribution rate of the total number of cracks.

[0095] The mechanical basis of this rule lies in the fact that confining pressure dominates the strengthening effect, fracture size affects the degree of stress concentration, and the number of fractures affects statistical damage accumulation. Specifically, the influence of confining pressure on the peak strength of fractured rock is directly reflected by the confining pressure strengthening term in the Mohr-Coulomb criterion. Increasing the confining pressure can significantly increase the normal stress and frictional resistance of the fracture surface, thereby greatly increasing the peak strength. Therefore, the contribution rate of confining pressure should be the highest. The fracture diameter determines the degree of stress concentration and the magnitude of released strain energy of a single fracture. The larger the fracture diameter, the more significant the weakening of rock mass strength by a single fracture. Therefore, the contribution rate of fracture diameter should be the second highest. The number of fractures affects rock mass strength through statistical damage accumulation. Under the same fracture density, the effect of changes in the number of fractures on strength is relatively indirect. Therefore, the contribution rate of the number of fractures should be the lowest.

[0096] If the validation fails, adjust the way the training set, validation set, and test set are divided, such as changing the stratified sampling ratio or the sampling random seed, and adjust the training hyperparameters, such as the learning rate, L2 regularization coefficient, and number of convolutional kernels. Then, re-execute the cross-validation hyperparameter search and residual network training until the contribution rate ranking meets the validation rules, and obtain a valid hybrid prediction model.

[0097] A failed validation indicates that the input-output mapping learned by the model is inconsistent with the understanding of rock mechanics theory. Possible reasons include an unreasonable training set partitioning leading to the model learning spurious correlation patterns, or improper training hyperparameter settings causing overfitting. By repartitioning the training set and adjusting the training hyperparameters, the learned mapping can be corrected, bringing the contribution rate ranking back to a mechanically reasonable range.

[0098] Target load condition intensity prediction. The target load condition parameters are input into a validated hybrid prediction model. First, the predicted baseline intensity value is calculated using a physical prior model. Then, the predicted residual correction value is calculated through a convolutional residual correction network. The two values ​​are combined to obtain the final predicted peak intensity value: in, This is the final predicted peak compressive strength of fractured rock. The output includes the final predicted peak strength, the predicted baseline strength, the predicted residual correction, and the contribution rate of each target working condition parameter.

[0099] To more intuitively demonstrate the specific prediction effect of the hybrid prediction model in this embodiment under typical working conditions, Table 1 lists the first 20 samples from 127 sets of indoor triaxial test data, covering the working condition range of confining pressure 5.00~58.35 MPa, number of cracks 3~200, and crack diameter 1.84~9.76 mm, as shown in the table below: Table 1. Statistical Table of Indoor Triaxial Test Data and Model Prediction Results Table 1 shows that each sample includes confining pressure, number of fractures, fracture diameter, measured peak intensity, predicted value from the physical prior model, predicted value from the hybrid model in this embodiment, residual correction amount, and percentage of error after correction. The residual correction amount is the difference between the hybrid model prediction and the physical prior model prediction, and the percentage of error after correction is the absolute value of the relative error between the hybrid model prediction and the measured peak intensity. Table 1 shows that there are varying degrees of deviation between the physical prior model prediction and the measured values. The sign and magnitude of the residual correction amount vary with the working conditions. After correction by the convolutional residual correction network, the percentage of error after correction is significantly reduced, with most samples showing a correction error within 5%. This indicates that the hybrid model in this embodiment can effectively correct the systematic bias of the physical prior model and significantly improve prediction accuracy.

[0100] To more intuitively compare the prediction performance of different models on the full set of experimental data, a comparative example was set up: a non-attentional hybrid model with the channel attention layer removed but the rest of the network structure and training parameters identical to this embodiment. The predicted values ​​of the physical prior model, the non-attentional hybrid model, and the hybrid model with channel attention mechanism in this embodiment were compared with 127 sets of measured values ​​using a scatter plot. Figure 3 .

[0101] Figure 3 The comparison between the predicted values ​​of the three models and the measured values ​​of 127 sets of indoor triaxial tests is presented in the form of a scatter plot. The horizontal axis of the figure represents the measured peak strength (unit: MPa), and the vertical axis represents the predicted peak strength (unit: MPa). The 45-degree dashed line represents the ideal situation where the predicted value is completely consistent with the measured value. The scatter plots of the three models are also shown in the figure. It can be seen from the figure that the scatter plot distribution of the physical prior model is relatively discrete, with most scatter plots deviating from the 45-degree reference line. The deviation is more obvious in the high-strength region (greater than 80 MPa) and the low-strength region (less than 30 MPa). This indicates that although the physical model based on the conservation of elastic deformation energy can capture the main trend of the strength of fractured rocks, due to the simplification of the model assumptions, there is a large systematic deviation under extreme conditions.

[0102] The scatter distribution of the attentionless hybrid model is significantly more concentrated near the 45-degree reference line, and the dispersion of the high-intensity and low-intensity regions is significantly reduced. This indicates that the convolutional residual correction network can effectively learn the systematic bias patterns of the physical model and correct them accordingly. The residual correction network uses the prediction residuals of the physical model as the training target, rather than relearning the complete intensity mapping relationship. This "correction rather than replacement" strategy enables the network to achieve effective bias elimination with limited training samples.

[0103] In this embodiment, the scatter distribution of the hybrid model with channel attention mechanism is most densely clustered on both sides of the 45-degree reference line, maintaining good consistency across the entire intensity range. Compared with the hybrid model without attention, the scatter clustering of the model in this embodiment is further improved in the medium intensity region (40-70 MPa), indicating that the channel attention mechanism can adaptively focus on the feature channel that contributes the most to the residual correction and suppress the interference of redundant channels, thereby further improving the correction accuracy in the region where the physical model has already been well fitted.

[0104] Figure 3 This intuitively reflects that the physical prior model provides a mechanically reasonable baseline prediction, ensuring that the prediction results meet the basic mechanical constraints; the residual correction network further refines the prediction based on this, eliminating the systematic bias of the physical model; and the channel attention mechanism further optimizes the correction effect through adaptive weighting of key feature channels. The three work together to achieve a dual guarantee of prediction accuracy and mechanical consistency.

[0105] To further analyze the correction mechanism of the residual correction network, the trends of residual variation with confining pressure before and after correction are plotted on [the graph]. Figure 4 .

[0106] Please see Figure 4 , Figure 4 The trend of residuals before and after correction with confining pressure is shown in scatter plot form. The horizontal axis represents the confining pressure value (unit: MPa), and the vertical axis represents the predicted residual value (unit: MPa). From the figure, it can be seen that the residuals before correction have a significant non-uniform distribution pattern under different confining pressure levels. Specifically, in the low confining pressure region (less than 15 MPa), the residuals are mainly negative (the physical model overestimates the strength). This is because the physical model linearizes the confining pressure effect based on the Mohr-Coulomb criterion. However, under no confining pressure or low confining pressure conditions, the frictional slip mechanism of the fracture surface is fundamentally different from that under confining pressure conditions. The actual rock strength under low confining pressure is lower than the linear extrapolation value. In the medium confining pressure region (15-45 MPa), the residual distribution is relatively uniform. In the high confining pressure region (greater than 45 MPa), the absolute value of the residuals increases slightly, which is related to the nonlinear mechanical behavior caused by the fracture closure effect under high confining pressure.

[0107] The corrected residuals are uniformly distributed near the zero line at all confining pressure levels, without any obvious non-uniform distribution pattern. This indicates that the residual correction network has successfully learned and eliminated the systematic bias of the physical model at different confining pressure levels, achieving a uniform correction effect across the entire confining pressure range (5-60MPa). In particular, in the low confining pressure region, the introduction of the zero confining pressure indicator feature enables the network to directly distinguish between uniaxial compression and triaxial compression at the input level, thereby learning different residual correction patterns for the two conditions and avoiding the problem of discontinuous correction values ​​when the confining pressure is close to 0.

[0108] The dispersion of the corrected residuals remained relatively stable under all confining pressure levels, and there was no significant increase in the dispersion of residuals in a certain confining pressure interval. This indicates that the correction capability of the residual correction network has good uniformity across the entire operating range. There is no uneven correction problem where the correction effect is good under some operating conditions but poor under others. This uniform correction effect benefits from the adaptive weighting of different feature channels by the channel attention mechanism. Under different confining pressure conditions, the attention weights are automatically adjusted to focus on the most relevant feature channels, ensuring the operating condition independence of the correction effect.

[0109] Please see Figure 5 The present invention further provides a strength prediction system for randomly distributed fractured rocks, which is used in the strength prediction method for randomly distributed fractured rocks described in any of the above claims, and includes: Physical Prior Model Construction Module: Used to construct a physical prior model of fractured rock strength based on the conservation of elastic deformation energy. The physical prior model of fractured rock strength takes target working condition parameters as input, including confining pressure, number of fractures and fracture diameter, and outputs the predicted baseline strength value of fractured rock. The parameter calibration and data generation module is used to calibrate the learnable parameters of the physical prior model of fractured rock strength using rock mechanics test samples. Based on the calibrated physical prior model of fractured rock strength, it calculates the predicted baseline strength value corresponding to each test sample and generates a real residual dataset based on the difference between the predicted baseline strength value and the measured peak strength value of the corresponding sample, which serves as the training label for the subsequent convolutional residual correction network. Residual network construction module: used to construct a convolutional residual correction network with channel attention mechanism. The convolutional residual correction network takes the target working condition parameters and zero confining pressure indication features as input and outputs the predicted residual correction value of the physical prior model of fractured rock strength. Model decoupling training module: used to train the hybrid prediction model using a two-stage decoupling strategy: the first stage freezes all parameters of the physical prior model of fractured rock strength, and the second stage uses the real residual dataset obtained in step S2 as training labels to train the convolutional residual correction network to obtain the trained hybrid prediction model. Contribution rate analysis module: This module is used to quantify the contribution rate of each target working condition parameter to the final peak intensity prediction value of the trained hybrid prediction model using the group permutation importance method, and to verify the contribution rate ranking results to obtain a qualified hybrid prediction model.

[0110] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented in software, the above embodiments can be implemented, in whole or in part, as a computer program product. Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution.

[0111] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment, depending on actual needs.

[0112] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A method for predicting the strength of randomly distributed fractured rocks, characterized in that, The specific steps include: S1: Construct a physical prior model of fractured rock strength based on the conservation of elastic deformation energy. The physical prior model of fractured rock strength takes the target working condition parameters as input, including confining pressure, number of fractures and fracture diameter, and outputs the predicted baseline strength value of fractured rock. S2: The learnable parameters of the physical prior model of fractured rock strength are calibrated using rock mechanics test samples. Based on the calibrated physical prior model of fractured rock strength, the predicted baseline strength value corresponding to each test sample is calculated. The real residual dataset is generated based on the difference between the predicted baseline strength value and the measured peak strength value of the corresponding sample, which serves as the training label for the subsequent convolutional residual correction network. S3: Construct a convolutional residual correction network with channel attention mechanism. The convolutional residual correction network takes the target working condition parameters and zero confining pressure indication features as input and outputs the predicted residual correction value of the physical prior model of fractured rock strength. S4: A two-stage decoupling strategy is used to train the hybrid prediction model: In the first stage, all parameters of the physical prior model of fractured rock strength are fixed; in the second stage, the real residual dataset obtained in step S2 is used as the training label to train the convolutional residual correction network to obtain the trained hybrid prediction model. S5: For the trained hybrid prediction model, the group permutation importance method is used to quantify the contribution rate of each target working condition parameter to the final peak intensity prediction value, and the contribution rate ranking result is verified to obtain a qualified hybrid prediction model. The target working condition parameter to be predicted is input into the qualified hybrid prediction model, and the corresponding prediction baseline intensity value and prediction residual correction value are superimposed to obtain the final peak intensity prediction value and output it.

2. The method for predicting the strength of randomly distributed fractured rocks according to claim 1, characterized in that: The construction of the physical prior model of fractured rock strength based on the conservation of elastic deformation energy is specifically as follows: establishing the energy balance equation of rock mass unit based on the conservation of elastic deformation energy, deriving the equivalent elastic modulus of randomly distributed fractured rock mass, and constructing the physical prior model by combining the definition of damage strength and the confining pressure strengthening criterion. Specifically, the energy balance equation for the rock mass unit is established as follows: the rock mass unit containing random fractures is regarded as a homogeneous equivalent continuous medium. Under the action of far-field axial stress, the total equivalent elastic strain energy of the rock mass consists of two parts: the elastic strain energy of the intact rock matrix and the additional strain energy caused by the elastic deformation of the fractures in the rock mass.

3. The method for predicting the strength of randomly distributed fractured rocks according to claim 2, characterized in that: The specific logic for obtaining the additional strain energy caused by elastic deformation of fractures in rock mass is as follows: Integrate along the fracture front to obtain the total released energy of a single fracture; statistically average the isotropic and uniform distribution of fractures in the normal direction to obtain the average released strain energy of a single fracture; then introduce correction coefficients for fracture density and fracture number to calculate the additional strain energy caused by elastic deformation of fractures in rock mass; substitute the obtained additional strain energy caused by elastic deformation of fractures in rock mass into the energy balance equation of rock mass unit to derive the equivalent elastic modulus of fractured rock mass; and combine this with the confining pressure strengthening criterion to establish a priori physical model of fractured rock strength.

4. The method for predicting the strength of randomly distributed fractured rocks according to claim 3, characterized in that: The method involves calibrating the learnable parameters of the prior physical model for fractured rock strength using rock mechanics test samples. These learnable parameters include a fracture number correction coefficient and a confining pressure strengthening coefficient. Specifically, the intrinsic parameters of intact rock, including uniaxial compressive strength, elastic modulus, and Poisson's ratio, are first obtained through basic mechanical tests. An initial confining pressure strengthening coefficient is then calculated based on the confining pressure-strength data of the intact specimens. Representative specimen strength data covering different confining pressures, fracture diameters, and fracture numbers are selected. Using the experimentally measured peak strength value as the true value, the least squares method is employed to globally optimize and calibrate the fracture number correction coefficient and the confining pressure strengthening coefficient in the prior physical model for fractured rock strength. The optimization objective is to minimize the mean square error between the predicted baseline strength value and the measured peak strength value. After calibration, all parameters of the prior physical model for fractured rock strength remain fixed.

5. The method for predicting the strength of randomly distributed fractured rocks according to claim 1, characterized in that: The convolutional residual correction network with channel attention mechanism specifically adopts a three-level structure of convolutional feature extraction, channel attention weighting, and fully connected output. It reshapes the input features into a 1D sequence to adapt to 1D convolution operations. Two layers of 1D convolutional kernels are used to perform nonlinear transformation on the input features to generate a 16-channel high-dimensional abstract feature embedding. The ReLU activation function is used to retain the nonlinear fitting ability. Weights are calculated for each of the 16 channel features, and the weights are normalized to the 0~1 range by the sigmoid function. The channel attention weight vector is multiplied element-wise with the convolution output features to map the weighted high-dimensional features into a 1D output, i.e., the predicted residual correction value.

6. The method for predicting the strength of randomly distributed fractured rocks according to claim 1, characterized in that: The training convolutional residual correction network uses the real residuals in the real residual dataset calculated by S2 as training labels, and the target working condition parameters and zero confinement pressure indication features as network inputs. The zero confinement pressure indication features are Boolean input features. When the confinement pressure in the target working condition parameters is 0, the zero confinement pressure indication feature takes the value of 1. When the confining pressure in the target working condition parameters is not 0, the zero confining pressure indicator feature takes a value of 0; the Adam optimizer is used to train the CNN-Attention network; The training process employs the ReduceLROnPlateau adaptive learning rate decay strategy, which reduces the learning rate when the validation set loss does not decrease for 10 consecutive rounds; an early stopping strategy is also set, which terminates training when the validation set loss does not increase for 20 consecutive rounds to prevent overfitting. After training, the convolutional residual correction network outputs the corresponding predicted residual correction value based on the input target operating condition parameters and zero confining pressure indication features, thus completing the correction.

7. The method for predicting the strength of randomly distributed fractured rocks according to claim 1, characterized in that: The group permutation importance method is used to quantify the contribution rate of each target working condition parameter to the final peak strength prediction value. Specifically, different permutation strategies are used for the confining pressure, number of fractures and fracture diameter in the target working condition parameters. Among them, the confining pressure is replaced by a global random permutation; the number of fractures is replaced by a global random permutation; the fracture diameter is replaced by a conditional permutation based on the confining pressure level, that is, the diameter value is randomly shuffled only within the same confining pressure level. Quantitative calculation of feature contribution rate: A fixed number of random permutations are performed on confining pressure, number of fractures and fracture diameter. The importance score of each target working condition parameter is calculated by the difference in the average absolute error of the model prediction before and after the permutation. The importance score of each target working condition parameter is normalized to obtain the percentage contribution rate of each target working condition parameter.

8. The method for predicting the strength of randomly distributed fractured rocks according to claim 7, characterized in that: The verification of the contribution rate ranking results involves ranking the contribution rate percentages of confining pressure, number of fractures, and fracture diameter for mechanical rationality verification. The specific rules for mechanical rationality verification are: the contribution rate of confining pressure is greater than the contribution rate of fracture diameter, and the contribution rate of fracture diameter is greater than the contribution rate of the total number of fractures. If the verification fails, the training set is re-divided and the training hyperparameters are adjusted. The hybrid prediction model is then retrained until the contribution rate ranking meets the verification rules, resulting in a qualified hybrid prediction model.

9. A strength prediction system for randomly distributed fractured rocks, characterized in that: The strength prediction system for randomly distributed fractured rocks is used to execute the strength prediction method for randomly distributed fractured rocks according to any one of claims 1-8, comprising: Physical Prior Model Construction Module: Used to construct a physical prior model of fractured rock strength based on the conservation of elastic deformation energy. The physical prior model of fractured rock strength takes target working condition parameters as input, including confining pressure, number of fractures and fracture diameter, and outputs the predicted baseline strength value of fractured rock. The parameter calibration and data generation module is used to calibrate the learnable parameters of the physical prior model of fractured rock strength using rock mechanics test samples. Based on the calibrated physical prior model of fractured rock strength, it calculates the predicted baseline strength value corresponding to each test sample and generates a real residual dataset based on the difference between the predicted baseline strength value and the measured peak strength value of the corresponding sample, which serves as the training label for the subsequent convolutional residual correction network. Residual network construction module: used to construct a convolutional residual correction network with channel attention mechanism. The convolutional residual correction network takes the target working condition parameters and zero confining pressure indication features as input and outputs the predicted residual correction value of the physical prior model of fractured rock strength. Model decoupling training module: used to train the hybrid prediction model using a two-stage decoupling strategy: the first stage freezes all parameters of the physical prior model of fractured rock strength, and the second stage uses the real residual dataset obtained in step S2 as training labels to train the convolutional residual correction network to obtain the trained hybrid prediction model. Contribution rate analysis module: This module is used to quantify the contribution rate of each target working condition parameter to the final peak intensity prediction value of the trained hybrid prediction model using the group permutation importance method, and to verify the contribution rate ranking results to obtain a qualified hybrid prediction model.

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